Negative Binomial Distributions for Fixed and Random Parameters

UNIVERSITY OF NAIROBI School of Mathematics College of Biological and Physical sciences

NEGATIVE BINOMIAL DISTRIBUTIONS FOR FIXED AND RANDOM PARAMETERS

NELSON ODHIAMBO OKECH

I56/70043/2011

A PROJECT PRESENTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MATHEMATICAL STATISTICS.

DECEMBER 2014

DECLARATION I declare that this is my original work and has not been presented for an award of a degree to any other university

Signature………………………………………… Date………………………………………….

Mr. Nelson Odhiambo Okech

This thesis is submitted with my approval as the University supervisor

Signature………………………………………………. Date…………………………………………….

Prof. J.A.M. Ottieno

DEDICATIONS This project is dedicated to my mother Mrs. Lorna Okech, my late father Mr. Joseph Okech and my only sister Hada Achieng Okech for having initiated this project, inspired me and sponsored the fulfillment of this dream.

ACKNOWLEDGMENT

I would like to acknowledge with gratitude the effort and guidance from my supervisor Prof. JAM Ottieno. The fulfillment of the project was as a result of his selfless devotion to help.

I’m deeply indebted to the entire School of Mathematics (University of Nairobi) for the support accorded to me at whatever level. The entire academic journey in the mentioned institution was made possible by the relentless support from the dedicated lectures and support staff. Your contribution was immense and may God shower you with his blessings

It cannot be gainsaid that my friends, relatives and well – wishers played a pivotal role in the accomplishment of this project. They provided moral support, encouraged me whenever the journey seemed rocky and gave me the necessary peace of mind to enable me channel all my energy to this remarkable venture: Duncan Okello, Beryl Anyango, Mary Okech, Evelyn Okech, Shadrack Okech, Danson Ngiela and George Oluoch

Last but not least, it is indeed my great pleasure to appreciate the support from my teammates and fellow students during the entire period.

Above all, I thank the Almighty God for the good health and continued protection for the whole period.

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ABSTRACT The first objective of this project is to construct Negative Binomial Distributions when the two parameters p and r are fixed using various methods based on: Binomial expansion; Poisson – Gamma mixture; Convolution of iid Geometric random variables; Compound Poisson distribution with the iid random variables being Logarithmic series distributions; Katz recursive relation in probability; Experiments where the random variable is the number of failures before the rth success and the total number of trials required to achieve the rth success.

Properties considered are the mean, variance, factorial moments, Kurtosis, Skewness and Probability Generating Function.

The second objective is to consider p as a random variable within the range 0 and 1. The distributions used are:

i. The classical Beta (Beta I) distribution and its special cases (Uniform, Power, Arcsine and Truncated beta distribution). ii. Beyond Beta distributions: Kumaraswamy, Gamma, Minus Log, Ogive and two – sided Power distributions. iii. Confluent and Gauss Hypergeometric distributions.

The third objective is to consider r as a discrete random variable. The Logarithmic series and Binomial distributions have been considered. As a continuous random variable, an Exponential distribution is considered for r.

The Negative Binomial mixtures obtained have been expressed in at least one of the following forms.

a. Explicit form b. Recursive form c. Method of moments form.

Comparing explicit forms and the method of moments, some identities have been derived.

For further work, other discrete and continuous mixing distributions should be considered. Compound power series distributions with the iid random variables being Geometric or shifted Geometric distributions are Negative Binomial mixtures which need to be studied.

Properties, estimations and applications of Negative Binomial mixtures are areas for further research.

Table of Contents DECLARATION ...... i DEDICATIONS ...... ii ACKNOWLEDGMENT ...... iii ABSTRACT ...... iv CHAPTER 1 ...... 1 GENERAL INTRODUCTION ...... 1 1.1 Background information ...... 1 Negative Binomial Distribution ...... 1 Distribution Mixtures ...... 2 Uncountable mixtures ...... 3 Mixtures of parametric families ...... 3 Negative Binomial mixtures ...... 4 1.2 Problem statement ...... 4 1.3 Objectives...... 5 1.4 Methodology ...... 5 1.5 Literature review ...... 6 1.6 Significance of the study and its applications ...... 7 Chapter 2 ...... 8 CONSTRUCTIONS AND PROPERTIES OF NEGATIVE BINOMIAL DISTRIBUTION ...... 8 2.1. Introduction ...... 8 2.2. NBD based on Binomial expansion ...... 8 2.3. NBD based on mixtures ...... 9 2.4. Construction from a fixed number of Geometric random variables ...... 11 2.5. Construction from logarithmic series ...... 13 2.6. Representation as compound Poisson distribution ...... 15 2.7. Construction using Katz recursive relation in probability ...... 17 2.7.1. Iteration technique ...... 18 2.7.2. Using the Probability Generating Function (PGF) technique ...... 20 2.7. From experiment ...... 22 Properties ...... 25 Factorial moments of Negative Binomial distribution ...... 27

CHAPTER 3 ...... 31 BETA - NEGATIVE BINOMIAL MIXTURES ...... 31 3.1. Introduction ...... 31 3.2. A brief discussion of the various forms of expressing the mixed distribution...... 32 3.2.1. Explicit form ...... 32 3.2.2. Method of moments ...... 32 3.2.3. Recursion...... 33 3.3. Classical beta – Negative Binomial distribution ...... 33 3.3.1. Construction of Classical Beta Distribution ...... 33 3.3.2. Properties of the Classical Beta distribution ...... 35 3.3.3 The mixture ...... 36 In explicit form ...... 36 Using Method of Moments ...... 36 Mixing using Recursive relation ...... 37 3.3.4 Properties of Beta – Negative Binomial Distribution ...... 41 Mean ...... 41 3.3.5. Moment Generating Function ...... 44 3.4. Special cases of beta – negative Binomial distribution ...... 46 3.4.1. Uniform – Negative Binomial distribution ...... 46 3.4.1.1. Uniform distribution ...... 46 3.4.1.2. The mixtures ...... 47 3.4.2. Power function - negative Binomial distribution ...... 50 3.4.2.1. Power function distribution ...... 50 3.4.2.2 The mixing ...... 51 3.4.3. Arcsine – negative Binomial distribution ...... 54 3.4.4. Negative Binomial – Truncated Beta distribution ...... 59 3.4.4.1 Truncated Beta distribution ...... 59 Construction ...... 59 3.5. Negative Binomial – Confluent Hypergeometric distribution ...... 61 3.5.1. Confluent Hypergeometric ...... 61 3.5.2 Confluent Hypergeometric – Negative Binomial mixing...... 63 3.6. Gauss Hypergeometric – Negative Binomial Distribution ...... 65

3.6.1. Gauss Hypergeometric distribution ...... 65 3.6.2. Gauss Hypergeometric – Negative Binomial mixture ...... 66 CHAPTER 4 ...... 68 NEGATIVE BINOMIAL MIXTURES BASED ON DISTRIBUTIONS BEYOND BETA ...... 68 4.2 Negative Binomial – Kumaraswamy (I) Distribution ...... 69 4.2.1 Kumaraswamy (I) Distribution ...... 69 4.2.2 Negative Binomial – Kumaraswamy (I) distribution mixing ...... 70 4.3 Negative Binomial – Truncated Exponential Distribution...... 72 4.3.1 Truncated Exponential Distribution (TEX(흀, 풃)) ...... 72 4.3.2 Negative Binomial – Truncated Exponential distribution mixing ...... 72 4.4 Negative Binomial – Truncated Gamma Distribution ...... 74 4.4.1 Truncated Gamma Distribution ...... 74 4.4.2 Negative Binomial – Truncated Gamma Distribution mixing ...... 75 4.5. Negative Binomial – Minus Log Distribution...... 76 4.5.1. Minus Log Distribution ...... 76 4.5.2Negative Binomial – Minus Log Distribution mixing ...... 78 4.6 Negative Binomial – Standard Two – Sided Power Distribution ...... 80 4.6.1 Standard Two – Sided Power Distribution ...... 80 4.6.2 Negative Binomial – Standard Two – Sided Power Distribution ...... 81 4.7. Negative Binomial – Ogive Distribution ...... 81 4.7.1. Ogive Distribution ...... 81 4.7.2. Negative Binomial – Ogive Distribution mixing ...... 81 4.8. Negative Binomial – Standard Two – Sided Ogive Distribution ...... 82 4.8.1. Standard Two – Sided Ogive Distribution ...... 82 4.8.2. Negative Binomial – Standard Two – Sided Ogive Distribution mixing ...... 83 4.9. Negative Binomial – Triangular Distribution ...... 84 4.9.1. Triangular Distribution ...... 84 4.9.2. Negative Binomial – Triangular distribution mixing ...... 87 CHAPTER 5 ...... 91 GEOMETRIC DISTRIBUTION MIXTURES WITH BETA GENERATED DISTRIBUTIONS IN THE [0, 1] DOMAIN . 91 5.1. Introduction ...... 91 5.2. Geometric – Classical Beta Distribution ...... 91

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5.2.1. Classical Beta – Geometric distribution from explicit mixing ...... 91 5.2.2. Classical Beta – Geometric distribution from method of moments mixing ...... 92 5.2.3. Classical Beta – Geometric distribution from recursive relation ...... 92 5.2.4. Properties of beta – Geometric distribution ...... 95 5.3. Geometric – Uniform distribution ...... 96 5.3.1. Geometric – uniform distribution from explicit mixing ...... 97 5.3.2. Geometric – uniform distribution from recursive relation ...... 97 5.3.3. Geometric – uniform distribution from Method of moments ...... 98 5.4. Negative Binomial – power function distribution ...... 98 5.4.1. Power function distribution ...... 98 5.4.2. Geometric – Power distribution from explicit mixture ...... 99 5.4.3. Geometric – Power distribution from moments method ...... 99 5.4.4. Geometric – Power distribution in a recursive format ...... 99 5.5. Geometric – Arcsine distribution ...... 101 5.5.2 Geometric – Arcsine from explicit mixing ...... 101 5.5.3. Geometric – Arcsine distribution in a recursive format ...... 102 5.5.4. Geometric – Arcsine distribution from method of moments ...... 102 5.6 Geometric – Truncated beta distribution ...... 102 5.6.1. Truncated beta ...... 102 5.6.2 Geometric – Truncated beta from explicit mixing ...... 103 5.6.3. Geometric – Truncated beta distribution from method of moments ...... 103 5.7 Geometric – Confluent Hypergeometric...... 103 5.7.1. Confluent Hypergeometric ...... 103 5.7.2 Geometric – Confluent Hypergeometric distribution from explicit mixing ...... 105 5.7.2 Geometric – Hypergeometric distribution from Method of Moments ...... 105 5.8. Geometric – Gauss Hypergeometric distribution ...... 106 5.8.1. Gauss Hypergeometric distribution ...... 106 5.8.3. Gauss Hypergeometric – Geometric distribution from Explicit mixing ...... 106 5.8.4. Gauss Hypergeometric – Geometric distribution from method of moments mixing ...... 106 CHAPTER 6 ...... 107 GEOMETRIC MIXTURES BASED ON DISTRIBUTIONS BEYOND BETA FROM NEGATIVE BINOMIAL MIXTURES ...... 107

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6.1. Introduction ...... 107 5.2. Geometric – Kumaraswamy (I) Distribution ...... 107 5.2.1. Kumaraswamy (I) Distribution ...... 107 5.2.2. Geometric – Kumaraswamy (I) distribution from explicit mixing ...... 108 5.2.2. Geometric – Kumaraswamy (I) distribution from method of moments mixing ...... 108 6.3. Geometric – Truncated Exponential Distribution (TEX(흀, 풃)) ...... 108 6.3.1. Truncated Exponential Distribution (TEX(흀, 풃)) ...... 108 6.3.1. Geometric – Truncated Exponential distribution from explicit maxing ...... 108 6.4. Geometric – Truncated Gamma Distribution ...... 109 6.4.1. Truncated Gamma Distribution ...... 109 6.4.2. Geometric – Truncated Gamma distribution from explicit mixing ...... 109 6.4.1. Geometric – Truncated Gamma distribution from moments method mixing ...... 109 6.5. Geometric – Minus Log Distribution ...... 110 6.5.1. Minus Log Distribution ...... 110 6.6. Geometric – Standard Two – Sided Power Distribution ...... 110 6.6.1. Standard Two – Sided Power Distribution ...... 110 6.6.1. Geometric – Standard Two – Sided Power distribution from explicit mixing...... 111 6.7. Geometric – Ogive Distribution ...... 111 6.7.1. Ogive Distribution ...... 111 6.7.2. Geometric – Ogive distribution from explicit mixing ...... 111 6.8. Geometric – Standard Two – Sided Ogive Distribution ...... 112 6.8.1. Standard Two – Sided Ogive Distribution ...... 112 6.8.1. Geometric – Standard Two – Sided Ogive Distribution from explicit mixing ...... 112 6.9. Geometric – Triangular Distribution ...... 113 6.9.1. Triangular Distribution ...... 113 6.9.2. Geometric – Triangular distribution from explicit mixing ...... 114 6.9.2. Geometric – Triangular distribution from method of moments mixing ...... 114 CHAPTER 7 ...... 116 NEGATIVE BINOMIAL MIXTURES WITH P AS A CONSTANT IN THE MIXING DISTRIBUTION ...... 116 7.1. Introduction ...... 116 7.2. Logarithmic distribution/ logarithmic series distribution/ log series distribution ...... 116 7.3. Negative Binomial – exponential distribution ...... 118

CHAPTER 8 ...... 123 SUMMARY AND CONCLUSION ...... 123 Table 1: Negative Binomial mixtures with [0,1] domain distribution priors based on Classical Beta ...... 124 Table 2: Negative Binomial mixtures with [0,1] domain Beyond Beta distribution priors ...... 127 A framework for the mixtures ...... 129 Recommendations ...... 130 REFERENCE ...... 132

CHAPTER 1

GENERAL INTRODUCTION

1.1 Background information

Negative Binomial Distribution Pierre de Montmort first mentioned Negative Binomial distribution in 1713. He considered a series of Binomial trials and came up with a finding of the probability of the number of failures ‘’x’’, before the rth success in the series.

In 1838 Poisson Simeon (1781 - 1840) developed the Poisson regression. He first introduced the new distribution as the limiting case of the Binomial. He later discovered that we can derive Poisson from the Binomial distribution and he demonstrated how the two distributions actually relate. Poisson regression was developed to handle count data, and later became the standard method used to model counts.

It is important to note that Poisson assume equality in its mean and variance. This is a very rare characteristic in real data. Data that has greater variance than the mean is termed as “Poisson over dispersed”, yet more commonly known as “overdispersed”. It is recommended that we apply Negative Binomial distribution instead of Poisson distribution when dealing with count data that is “overdispersed”.

Some of the most prominent and well known discrete distributions are the Binomial, the Poisson and the Negative Binomial distribution. The theoretical connection between these distributions is too close that it is hardly convenient to discuss any one of them without referring to the other.

For instance, the Negative Binomial distribution is based on the other two distributions (Poisson and Binomial) in relation to its construction as you will see later in this project.

Negative Binomial distribution can be expressed in two different ways depending on the definition of the parameter 푟 as follows.

a. 1st Form of Negative Binomial Distribution

Consider a sequence of independent Bernoulli (p) trials. In each trial the probability of success is p. Let the random variable 푋 denote the trial, at which the rth success occurs, where r is a fixed integer, then,

푥 − 1 푃푟 푋 = 푥/푟, 푝 = 푝푟 1 − 푝 푥−푟 푥 = 푟, 푟 + 1, 푟 + 2, … (1.01) 푥 − 푟

And we say that X has a Negative Binomial distribution with parameters r and p expressed as 푋~푁퐵(푟, 푝)

b. 2nd Form of Negative Binomial Distribution

Negative Binomial distribution can as well be expressed as follows.

푥 = 푡푕푒 푁푢푚푏푒푟 표푓 푓푎푖푙푢푟푒푠 푏푒푓표푟푒 푡푕푒 푟푡푕 푠푢푐푐푒푠푠 푖푛 푎푛 푖푛푓푖푛푖푡푒 푠푒푟푖푒푠 표푓

푖푛푑푖푝푒푛푑푒푛푡 푡푟푖푎푙푠 푤푖푡푕 푎 푐표푛푠푡푎푛푡 푝푟표푏푎푏푖푙푖푡푦 표푓 푠푢푐푐푒푠푠 푝.

푥 + 푟 − 1 = 푡푕푒 푛푢푚푏푒푟 표푓 푡푟푖푎푙푠 푒푥푐푙푢푑푖푛푔 푡푕푒 푟푡푕 푠푢푐푐푒푠푠

푥 + 푟 − 1 = 푡푕푒 푛푢푚푏푒푟 표푓 푤푎푦푠 표푓 표푏푡푎푖푛푖푛푔 푥 푓푎푖푙푢푟푒푠 푎푛푑 푟 − 1 푠푢푐푐푒푠푠 푥

Thus the alternative form of the Negative Binomial distribution is expressed as follows

푟 + 푥 − 1 푃푟표푏(푋 = 푥/푟, 푝) = 푝푟 1 − 푝 푥 1.02 푥

푓표푟 푥 = 0,1,2, … 푎푛푑 푟 > 0

Distribution Mixtures

A mixture distribution is the probability distribution of random variable whose values can be interpreted as being derived from an underlying set of other random variables: specifically, the final outcome value is selected at random from among the underlying values, with a certain probability of selection being associated with each. Here the underlying random variables may be random vectors each having the same dimension, in which case the mixture distribution is a multivariate distribution.

In cases where each of the underlying random variable is continuous, the outcome variable will also be continuous and its probability density function is sometimes referred to as a mixture density.

A mixture distribution constitutes a number of components which are often restricted to being FINITE, although in some cases the components may be COUNTABLE. More general cases (i.e. an UNCOUNTABLE set of component distributions), as well as the countable case, are referred to as COMPOUND DISTRIBUTIONS

Finite or countable mixtures 푛

퐹 푥 = 푤푖푃푖 (푥) 푗 =1

푛

푓 푥 = 푤푖푝푖(푥) 푗 =1

The sum is finite and the mixture is called a finite mixture, and in applications, an unqualified reference to a "mixture density" usually means a finite mixture. The case of a countable set of components is covered formally by allowing푛 = ∞.

Uncountable mixtures

Consider a probability density function p(x;a) for a variable x, parameterized by a. That is, for each value of a in some set A, p(x;a) is a probability density function with respect to x. Given a probability density function w (meaning that w is nonNegative and integrates to 1), the function

푓 푥 = 푤(푎)푝(푥; 푎)da 퐴 is again a probability density function for x. A similar integral can be written for the cumulative distribution function.

Mixtures of parametric families

Parameters in a mixture distribution can be grouped together into a parametric family and they don’t follow any arbitrary probability distributions. In such cases, assuming that it exists, the density can be written in the form of a sum as:

푛

푓 푥; 푎1, … … … , 푎푛 = 푤푖푝푖(푥; 푎푖) 푗 =1

3 for one parameter, or

푛

푓 푥; 푎1, … … … , 푎푛 , 푏1, … … … , 푏푛 = 푤푖푝푖(푥; 푎푖푏푖) 푗 =1 for two parameters, and so forth.

Negative Binomial mixtures The Negative Binomial distribution is constituted of two parameters 푟 푎푛푑 푝, and either of the parameters can be randomized to achieve the Negative Binomial mixture.

This project entails the scenarios where the parameter p has a continuous mixing distribution with the probability g(p) such that

1 푟 + 푥 − 1 푓 푥 = 푝푟 1 − 푝 푥 푔(푝)푑푝 0 푥 where f(x) is the Negative Binomial mixture

1.2 Problem statement The project considers the Negative Binomial of the following format

푟 + 푘 − 1 푃푟 푋 = 푘/푝 = 푝푟 1 − 푝 푘 푘

The problem statement is to find the Negative Binomial mixtures in the following distribution

1 푟 + 푘 − 1 i. 푓 푥 = 푝푟 1 − 푝 푘 푔 푝 푑푝 0 푘 푝 is within the [0,1] domain

∞ 푟 + 푘 − 1 ii. 푓 푥 = 푝푟 1 − 푝 푘 푔 푟 푑푟 0 푘

We need to develop these new distributions to help in solving the problem associated with over dispersed data.

Evaluating data that has several factors that affect the final outcome of the analysis need to be fitted using a heterogeneous distribution that will capture a majority of the aspects. This will reduce the risk that is associated with data loss or generality

1.3 Objectives Main objective

To construct Negative Binomial mixtures when the mixing distributions come from the probability of success and the number of success as random variable.

Specific objectives

a. To express the Negative Binomial mixed distributions in explicit forms, recursive forms and expectation forms when the mixing distributions are: i. Classical beta distribution and its special cases. ii. The beyond beta distributions b. To construct the Negative binomial mixtures when the number of successes takes i. Logarithmic distribution ii. Binomial distribution iii. Negative Binomial distribution c. To construct generalized Negative Binomial mixture when the number of successes (r) is Exponential distribution d. To construct Geometric Mixtures as special cases of Negative Binomial distribution

1.4 Methodology The methods applied to construct the Negative Binomial mixtures include:

i. Explicit or direct method ii. Moment Generating Function method iii. Recursive relation method

1.5 Literature review Here, we will analyze and access some of the works that has been done in relation to Negative Binomial distribution and its mixtures

Wang, Z. (2010) has done research on a three parameter distribution which is called the Beta Negative binomial (BNB) distribution. He derived the closed form and the factorial moment of the BNB distribution. A recursion on the pdf of the BNB stopped-sum distribution and a stochastic comparison between BNB and NB distributions are derived as well. He observed that BNB provides a better fit with a heavier tail compared to the Poisson and the NB for count data especially in the insurance company claim data

Li Xiaohu et al (2011) have studied the Kumaraswamy Binomial Distribution. They considered two models of the Kumaraswamy Distribution, derived their pdfs and other basic properties. The stochastic orders and dependence properties are also worked on by the group. Applications based on the incident of international terrorism and drinking days in two weeks were highlighted as well.

Nadarajah, S. et al (2012) proposed a new three – parameter distribution for modeling lifetime date. This is the Exponential – Negative Binomial distribution. It is advocated as most reasonable among the many exponential mixture type distributions proposed in the recent years.

Kotz S. et al (2004) came up with other Continuous families of Distributions that are Beyond Beta with Bounded Support and applications. Properties studied included moments, CDF, Quartiles, maximum likelihood method of estimating parameters amongst others. The distributions highlighted include Triangular distribution, Standard Two sided Power series

Sarabia J. et al (2008) have done some work on construction of multivariate distributions. The paper reviews the following set of methods: (a) Construction of multivariate distributions based on order statistics, (b) Methods based on mixtures, (c) Conditionally specified distributions, (d) Multivariate skew distributions, (e) Distributions based on the method of the variables in common and (f) Other methods, which include multivariate weighted distributions, vines and multivariate Zipf distributions.

Furman E.(2007) has done some work on the generation of the Negative Binomial Distribution from the sum of random variables. He also talks about the reasons why the negative Binomial distribution has been frequently proposed as a reasonable model for the number of insurance claims.

Ghitany et al (2001) has also shown that Hypergeometric generalized negative binomial distribution has moments of all positive orders, is overdispersed, skewed to the right and leptokurtic.

1.6 Significance of the study and its applications

This is an important project both in substance and timing. The objectives of this project as well as the analysis, as scheduled for discussion are important in identifying new distributions, their properties, identities as well as relevant applications.

Statistical distributions are at the core of statistical science and are a leading requisite tool for its applications. Negative Binomial and especially its mixtures are used widely in the insurance industry in the measure of total claims. This can be done by calculating the total claims distribution (according to the different methods known) by spending a reasonable computing time and without incurring in underflow and overflow (this problem could be defined as a “compatibility” problem of the parameters).

A finite mixture of Negative Binomial (NB) regression models has been proposed to address the unobserved heterogeneity problem in vehicle crash data. This approach can provide useful information about features of the population under study. For a standard finite mixture of regression models, previous studies have used a fixed weight parameter that is applied to the entire dataset. However, various studies suggest modeling the weight parameter as a function of the explanatory variables in the data.

Chapter 2

CONSTRUCTIONS AND PROPERTIES OF NEGATIVE BINOMIAL DISTRIBUTION

2.1. Introduction Negative Binomial distribution can be constructed from a variety of methods. Some of the techniques are based on:

1. Binomial expansion 2. Mixtures 3. Compound Poisson distribution 4. Katz Recursive relation in probability 5. Logarithmic series 6. Sums of a fixed number of Geometric random variables 7. From experiment

Below are the brief discussions of these methods.

2.2. NBD based on Binomial expansion Expanding (푎 + 푏)−푟푤푕푒푟푒 푟 > 0, 푤푒 푔푒푡 −푟 −푟 −푟 푎 + 푏 −푟 = 푎−푟 + 푎−푟−1푏1 + 푎−푟−2푏2 + ⋯ 0 1 2

∞ −푟 −푟−푥 푥 = 푥=0 푥 푎 푏 (2.01)

퐿푒푡 푎 = 1 푎푛푑 푏 = −휃

Then

−푟 ∞ −푟 푥 1 − 휃 = 푥=0 푥 (−휃) (2.02)

Dividing both sides by 1 − 휃 −푟

∞ −푟 −휃 푥 1 = 푥 1 − 휃 −푟 푥=0

∞ −푟 = −1 푥 휃 푥 1 − 휃 푟 2.03 푥 푥=0

−푟 푥 푥 푟 ∴ 푝푥 = 푥 −1 휃 1 − 휃 푓표푟 푟 > 0; 푥 = 0,1,2, .. (2.04)

Is a pmf

But

−푟 −푟 −푟 − 1 −푟 − 2 −푟 − 3 … … . . −푟 − 푥 − 2 −푟 − 푥 − 1 = 푥 1 ∙ 2 ∙ 3 … … . 푥 − 1 푥

−푟 −1 x푟 푟 + 1 푟 + 2 푟 + 3 … … . . 푟 + 푥 − 2 푟 + 푥 − 1 = 푥 x! −푟 푟 + 푥 − 1 = (−1)x 푥 푥

Thus

−푟 푟 + 푥 − 1 (−1)x = 푥 푥

Replacing this in equation above −푟 푝 = −1 푥 휃 푥 1 − 휃 푟 푓표푟 푥 = 0,1,2, … 푥 푥 r + 푥 − 1 푝 = 휃푥 1 − 휃 푟 푓표푟 푥 = 0,1,2, … . ; 0 < 휃 < 1 (2.05) 푥 푥

This is the Negative Binomial distribution with parameters 푟 푎푛푑 휃 = 1 − 푝

2.3. NBD based on mixtures Negative Binomial distribution can be developed from mixing Poisson distribution with Gamma distribution. Gamma distribution is considered as the prior distribution while the Poisson distribution a posterior distribution

Poisson distribution

This is expressed as the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independent of time since the last event.

A discrete random variable X is said to have a Poisson distribution with parameter 휆 > 0 , if for x=0, 1, 2, …, the probability mass function of X is expressed as:

푒−휆휆푥 푃푟 푋 = 푥/휆 = ; 푥 = 0,1,2, … ; 휆 > 0 (2.06) 푥! Gamma distribution

휆 − 휆훼−1푒 훽 푔 휆; 훼, 훽 = ; 푓표푟 휆 ≥ 0 푎푛푑훼, 훽 > 0 (2.07) Γ훼훽훼

The mixture

∞ 푓 푥 = 푃푟 푋 = 푥/휆 푔 휆; 훼, 훽 푑휆 0

Inserting the equations

∞ 1 −휆 푥 훼−1 −휆/훽 푓 푥 = 훼 푒 휆 휆 푒 푑휆 푥! Γ훼훽 0

∞ 1 −휆−휆/훽 푥+훼−1 푓(푥) = 훼 푒 휆 푑휆 푥! Γ훼훽 0

∞ 1+훽 1 −휆 훽 푥+훼−1 푓(푥) = 훼 푒 휆 푑휆 푥! Γ훼훽 0 but

∞ 1+훽 훼+푥 −휆 훽 푒 훽 휆푥+훼−1 푑휆 = Γ 훼 + 푥 0 1 + 훽

1 훽 훼+푥 푓 푥 = Γ 훼 + 푥 푥! Γ훼훽훼 1 + 훽

훼 + 푥 − 1 ! 훽 훼+푥 푓(푥) = 푥! 훼 − 1 ! 훽훼 1 + 훽

훼 + 푥 − 1 1 ∝ 1 푥 푓 푥 = 1 − (2.08) 푥 1 + 훽 1 + 훽

푓표푟 푥 = 0,1,2, … . ; 훽, ∝> 0

1 The marginal distribution of X is a Negative Binomial distribution with 푟 = 훼 푎푛푑 푝 = 1+훽

2.4. Construction from a fixed number of Geometric random variables

퐿푒푡 푠푟 = 푋1 + 푋2 + 푋3 + ⋯ + 푋푟 푑푒푛표푡푒 푡푕푒 푠푢푚 표푓 푖푖푑 푟푎푛푑표푚 푣푎푟푖푎푏푙푒푠 푋푖푎푛푑 푟 푖푠 푓푖푥푒푑

The PGF of 푠푁is given by

퐻 푠 = 퐸 푠푠푟 (2.09)

= 퐸 푠푋1+푋2+⋯+푋푟

= 퐸(푠푋1 푠푋2 푠푋3 … … . 푠푋푟 )

= 퐸(푠푋1 )퐸 푠푋2 퐸(푠푋3 ) … … . 퐸(푠푋푟 ) (2.10)

(since 푋푖 are independent and identical)

푋 푟 ′ 퐻(푆) = 퐸 푠 (푏푒푐푎푢푠푒 푋푖 푠 푎푟푒 푖푑푒푛푡푖푐푎푙)

= 퐺 푠 푟 where 퐺 푠 푖푠 푡푕푒 푝푔푓 표푓푋푖

퐿푒푡 푋푖~퐺푒표푚푒푡푟푖푐(푝)

Case 1

The pmf of 푋 is

푥 푝푥 = 푝푞 푓표푟 푥 = 0,1,2, … ; 푞 = 1 − 푝; 0 < 푝 < 1 and the pgf of 푋is 푝 퐺 푠 = 퐸 푠푋 = ; 푠 < 1 − 푝 −1 1 − 푞푠

Therefore

퐻 푠 = 퐺 푠 푟

푝 푟 퐻(푠) = (2.11) 1−푞푠

푝 푟 is the pgf of a Negative Binomial distribution with 0 < 푝 < 1, 푞 + 푝 = 1푎푛푑 푟 > 0 1−푞푠

Proof

푝 푟 퐻(푠) = 1 − 푞푠

= 푝푟 1 − 푞푠 −푟

∞ −푟 = 푝푟 −1 푥 (푞푠)푥 푥 푥=0

∞ 푟 + 푥 − 1 = 푝푟 (푞푠)푥 푥 푥=0

∞ 푟 + 푥 − 1 = 푝푟 푞푥 푠푥 푥 푥=0

∞ 푥 = 푝푥 푠 푥=0

= 퐸 푠푥 where 푝푥 is the pdf of the negative Binomial distribution and hence the above statement is validated

Case 2

The pmf of x is

푥−1 푝푥 = 푝푞 푓표푟 푥 = 1,2,3, … ; 푞 = 1 − 푝; 0 < 푝 < 1

The pgf of X in this case is given by

푝푠 1 퐺 푠 = 푖푓 푠 < 1 − 푞푠 푞

Therefore

퐻 푠 = 퐺 푠 푟 푝푠 푟 퐻(푠) = 1 − 푞푠

푝푠 푟 is the pgf of a Negative Binomial distribution with 0 ≤ 푝 ≤ 1, 푞 + 푝 = 1푎푛푑 푟 > 0 1−푞푠

Proof

푝푠 푟 퐻(푠) = 1 − 푞푠

= 푝푠 푟 1 − 푞푠 −푟

∞ −푟 = 푝푠 푟 −1 푥 (푞푠)푥 푥 푥=0

∞ 푟 + 푥 − 1 = 푝푠 푟 (푞푠)푥 푥 푥=0 푟 − 1 푟 푟 + 1 푟 + 푥 − 1 = 푝푠 푟 (푞푠)0 + 푞푠 1 (푞푠)2 + ⋯ + (푞푠)푥 + ⋯ 0 1 2 푥

∞ 푥 − 1 = 푝푠 푟 푞푠 푥−푟 푥 − 푟 푥=푟

∞ 푥 − 1 = 푝푟 푞푥−푟 푠푥 푥 − 푟 푥=푟

∞ 푥 = 푝푥 푠 푥=푟

= 퐸 푠푥 where 푝푥 is the pdf of the negative Binomial distribution and hence the above statement is validated

푥 − 1 푝 = 푝푟 푞푥−푟 푓표푟 0 ≤ 푝 ≤ 1, 푝 + 푞 = 1, 푥 = 푟, 푟 + 1, 푟 + 2, … 푥 푥 − 푟

2.5. Construction from logarithmic series Construction of Logarithmic distribution.

1 = 1 + 푝 + 푝2 + 푝3 + 푝4 + 푝5 + ⋯ 1 − 푝

Integrating both sides with respect to pwe get

푑푝 = 1 + 푝 + 푝2 + 푝3 + 푝4 + 푝5 + ⋯ dp 1 − 푝

푝2 푝3 푝4 푝5 푝푥 − log 1 − 푝 = 푝 + + + + + ⋯ + + ⋯ (2.12) 2 3 4 5 푥

∞ 푝푥 = 푥 푥=1

∞ 푝푥 1 = −푥 log 1 − 푝 푥=1

And hence the logarithmic distribution takes the form below

푝푥 푝 = 푓표푟 푥 = 1,2,3, … . ; 0 < 푝 < 1 (2.14) 푥 −푥 log 1 − 푝

If we consider the associated power equation 2.12 and find its derivative, we get

1 1 + 푝 + 푝2 + 푝3 + ⋯ + 푝푥 + ⋯ = (2.15) 1 − 푝 multiplying both sides of (2.15) by 1 − 푝

(1 − 푝) + (1 − 푝)푝 + (1 − 푝)푝2 + (1 − 푝)푝3 + ⋯ + (1 − 푝)푝푥 + ⋯ = 1

Note that the generating term of this series (1 − 푝)푝푥 is the pmf of the Geometric distribution where 푥 = 0,1,2, …represents the number of successes before the first failure in a sequence of independent Bernoulli trials with parameter p

Derivative of both sides of equation (2.15)

1 1 + 2푝 + 3푝2 + 4푝3 + ⋯ + 푥푝푥−1 + 푥 + 1 푝푥 + ⋯ = (2.16) (1 − 푝)2

Multiply across by 1 − 푝 2 푎푛푑 consider the associated generating distribution

푥 + 1 푝푥 1 − 푝 2 = 푥 + 1 1 − 푝 2푝푥 ; 푥 = 0,1,2, …

푥 + 1 = 1 − 푝 2푝푥 1 2 + 푥 − 1 = 1 − 푝 2푝푥 2 − 1

14 which is the pmf of the negative Binomial distribution with parameters 2 and p for 푥 = 0,1,2, …

Again take the derivative of (2.16)

2 2 + 3x2푝 + 4x3푝2 + 5x4푝3 + ⋯ + 푥 + 1 푥푝푥−1 + 푥 + 2 푥 + 1 푝푥 + ⋯ = (1 − 푝)3

2 Multiplying across by the reciprocal of and taking the associated pmf (1−푝)3

푥 + 2 푥 + 1 푥 + 2 1 − 푝 3푝푥 = 1 − 푝 3푝푥 2 2 3 + 푥 − 1 = 1 − 푝 3푝푥 3 − 1

푓표푟 푥 = 0,1,2, … which is the pmf of the negative Binomial distribution with parameter 3 and p

Taking the rth derivative of each side the power series, we find a series from which the NBD with parameters r and p can be obtained

푟! 푟 + 1 푘 + 2 ! 푟 + 푥 − 1 ! 푟 − 1 ! 푟 − 1 ! + 푝 + 푝2 + 푝3 + ⋯ + 푝푥 + ⋯ = (2.17) 1! 2! 3! 푥! 1 − 푝 푟

which is associated with NBD with parameters r and p, and hence

푟 + 푥 − 1 ! 푟 + 푥 − 1 1 − 푝 푟 푝푥 = 1 − 푝 푟 푝푥 (2.18) 푟 − 1 ! 푥! 푥

푓표푟 푥 = 0,1,2, … ; 푟 > 0; 0 ≤ 푝 ≤ 1

This is a negative Binomial distribution with parameters r and p

2.6. Representation as compound Poisson distribution

Let 푌푛 푛 = 1,2,3, … denote a sequence of identical and independent distributed random variables each having a logarithmic distribution log 푝 푤푖푡푕 푎 푝푚푓

푝푦 푝 = 푓표푟 푦 = 1,2,3, … . ; 0 < 푝 < 1 (2.19) 푦 −푦 log 1 − 푝

Let N be a random variable independent of the sequence and suppose

푁~푃표푖푠푠표푛 휆 = 푟푙푛 1 − 푝

푙푒푡 푠푁 = 푋1 + 푋2 + 푋3 + ⋯ … … + 푋푁 푏푒 푠푢푚 표푓 푡푕푒 푖푛푑푒푝푒푛푑푒푛푡 푟푎푛푑표푚 푣푎푟푖푎푏푙푒푠

To calculate the pgf 퐻(푠) 표푓 푋 푤푕푖푐푕 푖푠 푡푕푒 푐표푚푝표푠푖푡푖표푛 표푓 푡푕푒Probability Generating functions 퐺푁(푠) 푎푛푑 퐺푦 (푠)

휆 푠−1 퐺푁 푠 = 푒 (2.20) and

ln 1 − 푝푠 1 퐺 푠 = 푠 < (2.21) 푦 ln 1 − 푝 푝

We get

퐻 푠 = 퐺푁 퐺푦 푠

= 푒휆 퐺푦 푠 −1

ln 1 − 푝푠 = exp 휆 − 1 ln 1 − 푝

= exp −푟 ln 1 − 푝푠 − ln 1 − 푝

1 − 푝 푟 퐻(푠) = (2.22) 1 − 푝푠 which is the probability generating function for the negative Binomial distribution

Proof

푞 푟 퐻(푠) = 1 − 푝푠

= 푞푟 1 − 푝푠 −푟

∞ −푟 = 푞푟 −1 푥 (푝푠)푥 푥 푥=0

∞ 푟 + 푥 − 1 = 푞푟 (푝푠)푥 푥 푥=0

∞ 푟 + 푥 − 1 = 푞푟 푝푥 푠푥 푥 푥=0

∞ 푥 = 푝푥 푠 푥=0

= 퐸 푠푥 where 푝푥 is the pdf of the negative Binomial distribution and hence the above statement is validated

2.7. Construction using Katz recursive relation in probability Consider the following recursive relation in probabilities

푓(푥 + 1) 푃(푥) = (2.23) 푓(푥) 푄(푥)

Where 푃 푥 푎푛푑 푄 푥 푎푟푒 푝표푙푦푛표푚푖푎푙푠 표푓 푥

푓(. ) is the probability mass function in particular

Let

푃(푥) 훼+훽푥 = (2 .24) 푄(푥) 1+푥 which is the Katz relationship

This implies

푓 푥 + 1 훼 + 훽푥 = ; 푥 = 0,1,2 … 푓 푥 1 + 푥

Let 훼 ≠ 0 푎푛푑 훽 ≠ 0

Then

훼 + 훽푥 푓 푥 + 1 = 푓 푥 ; 푥 = 0,1,2 … (2.25) 1 + 푥

We shall use two approaches to obtain the negative Binomial distribution.

2.7.1. Iteration technique When 푥 = 0 푓 1 = 훼푓 0

훼+훽 훼+훽 When 푥 = 1 푓 2 = 푓 1 = 훼푓 0 2 2

훼+2훽 훼+2훽 훼+훽 When 푥 = 2 푓 3 = 푓 2 = 훼푓 0 3 3 2

훼+3훽 훼+3훽 훼+2훽 훼+훽 When 푥 = 3 푓 4 = 푓 3 = 훼푓 0 4 4 3 2

훼+ 푘−1 훽 When 푥 = 푘 − 1 푓 푘 = 푓 푘 − 1 푘

훼 + 푘 − 1 훽 훼 + 푘 − 2 훽 훼 + 3훽 훼 + 2훽 훼 + 훽 푓 푘 = … … 훼푓 0 (2.26) 푘 푘 − 1 4 3 2

훼 훼 훼 훼 훼 훽 + 푘 − 1 훽 + 푘 − 2 … … . . 훽 + 2 훽 + 1 훽 푓 0 훽 훽 훽 훽 훽 푓(푘) = 푘!

훼 + 푘 − 1 ! 푘 훽 푓(푘) = 훽 훼 푓 0 − 1 ! 푘! 훽

Therefore 훼 + 푘 − 1 푓 푘 = 훽푘 훽 푓 0 ; 푘 = 0,1,2,3 … (2.27) 푘

Since equation 2.27 is a pmf, then;

∞ 푓 푘 = 1 푘=0

∞ 푓 0 + 푓 푘 = 1 푘=1 ∞ 훼 + 푘 − 1 푓 0 + 훽푘 훽 푓 0 = 1 푘=1 푘

But

∞ 훼 + 푘 − 1 푓 0 1 + 훽푘 훽 = 1 푘=1 푘 1 푓 0 = 훼 + 푘 − 1 ∞ 푘 훽 1 + 푘=1 훽 푘 But 훼 + 푘 − 1 푓 푘 = 훽푘 훽 푓 0 ; 푘 = 0,1,2,3 … 푘

This implies that 푓(푘) will be

훼 + 푘 − 1 훽푘 훽 푘 푓 푘 = 훼 (2.28) + 푘 − 1 ∞ 푘 훽 푘=0 훽 푘 푘 = 0,1,2, …

훼 If 푟 = is a positive integer 훽 훼 + 푘 − 1 푟 + 푘 − 1 −푟 훽 = = −1 푘 푘 푘 푘

∞ 훼 ∞ + 푘 − 1 −푟 훽푘 = −훽 푘 훽 푘 푘=0 푘 푘=0

= 1 − 훽 −푟

0 < 훽 < 1

Conclusion

훼 + 훽푥 푓 푥 + 1 = 푓 푥 ; 푥 = 0,1,2 … (2.29) 1 + 푥 19

For

훼 1. 푟 = is a positive integer 훽 2. 0 < 훽 < 1 3. 훼 > 0

훼 + 푘 − 1 훽푘 훽 훼 + 푘 − 1 푓 푘 = 푘 = 훽 훽푘 1 − 훽 푟 1 − 훽 −푟 푘 푟 + 푘 − 1 푓 푘 = 훽푘 1 − 훽 푟 (2.30) 푘 0 < 훽 < 1; 푘 = 0,1,2,3 ….

This is a Negative Binomial distribution with parameters 푘 푎푛푑 푝 = 1 − 훽

2.7.2. Using the Probability Generating Function (PGF) technique Remember

훼 + 훽푥 푓 푥 + 1 = 푓 푥 ; 푥 = 0,1,2, … ; 훼 > 0; 0 < 훽 < 1 1 + 푥

Let 훼 ≠ 0 푎푛푑 훽 ≠ 0

Then

∞ ∞ [1 + 푥] 푓 푥 + 1 푠푥 = 훼 + 훽푥 푓 푥 푠푥 (2.31) 푥=0 푥=0

Define

∞ 퐺 푠 = 푓(푥) 푠푥 (2.32) 푥=0

∞ 퐺′ 푠 = 푥푓 푥 푠푥−1 푥=0

∞ 퐺′ 푠 = (푥 + 1)푓 푥 + 1 푠푥 푥=0

∞ ∞ 퐺′ 푠 = 훼 푓(푥) 푠푥 + 훽 푥푓(푥) 푠푥 푥=0 푥=0

∞ 퐺′ 푠 = 훼퐺 푠 + 훽푠 푥푓 푥 푠푥−1 (2.33) 푥=0

퐺′ 푠 = 훼퐺 푠 + 훽푠퐺′ 푠

1 − 훽푠 퐺′ 푠 = 훼퐺 푥

휕퐺 푠 1 − 훽푠 = 훼퐺 푠 휕푠

휕퐺 푠 훼 = 휕푠 (2.34) 퐺 푠 1 − 훽푠 훼 ln 퐺(푠) = ln 1 − 훽푠 + ln 퐶 −훽

훼 ln 퐺(푠) = ln 1 − 훽푠 −훽 + ln 퐶

훼 ln 퐺(푠) = ln 퐶 1 − 훽푠 −훽

훼 퐺 푠 = 퐶 1 − 훽푠 −훽 (2.35)

Let 푠 = 1

훼 퐺 1 = 퐶(1 − 훽)−훽

−훼 1 = 퐶(1 − 훽) 훽

1 훼 훽 퐶 = −훼 = (1 − 훽) (2.36) (1 − 훽) 훽

−훼 1 − 훽푠 훽 ∴ 퐺 푠 = 1 − 훽

훼 Let = 푟 be a positive integer 훽

1 − 훽푠 −푟 퐺 푠 = 1 − 훽

1 − 훽 푟 퐺 푠 = 2.37 1 − 훽푠

Suppose 푝 = 1 − 훽 푎푛푑 푝 + 푞 = 1, 푡푕푒푛

푝 푟 퐺 푠 = 2.38 1 − 푞푠

This is a PGF of a NBD with parameters 푟 > 0 푎푛푑 푝

Note: proof is carried out in section 2.4 above under case 1

2.7. From experiment Let X be the number of failures preceding the rth success in an infinite series of independent trials with a constant probability of success p

푟 + 푥 − 1 = 푡푕푒 푡표푎푡푎푙 푛푢푚푏푒푟 표푓 푡푟푖푎푙푠 푒푥푐푙푢푑푖푛푔 푡푕푒 푟푡푕 푠푢푐푐푒푠푠

푟 + 푥 − 1 = 푇푕푒 푛푢푚푏푒푟 표푓 푤푎푦푠 표푓 표푏푡푎푖푛푖푛푔 푥 푓푎푖푙푢푟푒푠 푎푛푑 푟 − 1 푠푢푐푐푒푠푠 푥 푟 + 푥 − 1 ∴ 푃푟표푏 푋 = 푥 = 푝푟−1 1 − 푝 푥 . 푝 푥 푟 + 푥 − 1 = 푝푟 1 − 푝 푥 (2.39) 푥

푓표푟 푟 > 0; 0 ≤ 푝 ≤ 1; 푥 = 0,1,2,3 ….

Alternatively, denote the probability of X=x by 푝푥 푟 , 푘 = 0,1,2, …

We can formulate a difference equation for 푝푥 푟 as follows.

푝푥 푟 = 푇푕푒 푝푟표푏푎푏푖푙푖푡푦 표푓 푡푕푒 푓푖푟푠푡 푡푟푖푎푙 푏푒푖푛푔 푎 푠푢푐푐푒푠푠 푓표푙푙표푤푒푑 푏푦 푎 푝푟표푏 표푓 푎 푓푎푖푙푢푟푒 푤푖푡푕 푟 − 1 푠푢푐푐푒푠푠푒푠

푝푥 푟 = 푇푕푒 푝푟표푏푎푏푖푙푖푡푦 표푓 푡푕푒 푓푖푟푠푡 푡푟푖푎푙 푏푒푖푛푔 푎 푓푎푖푙푢푟푒 푓표푙푙표푤푒푑 푏푦 푥 − 1푓푎푖푙푢푟푒푠 푤푖푡푕 푟 푠푢푐푐푒푠푠푒푠

푝푥 푟 = 푝푝푥 푟 − 1 + 푞푝푥−1 푟 ; 푥 = 1,2,3, … 푎푛푑 푟 ≥ 1 2.40

In terms of pgf,

∞ ∞ ∞ 푥 푥 푥 푝푥 푟 푠 = 푝 푝푥 푟 − 1 푠 + 푞 푝푥−1 푟 푠 2.41 푥=1 푥=1 푥=1

∞ 푥 But 퐺 푠, 푟 = 푥=0 푝푥 푟 푠

∴ 퐺 푠, 푟 − 푝0 푟 = 푝 퐺 푠, 푟 − 1 − 푝0 푟 − 1 + 푞푠퐺 푠, 푟 2.42

푟 푟−1 But 푝0 푟 = 푝 푎푛푑 푝0 푟 − 1 = 푝

∴ 퐺 푠, 푟 − 푝푟 = 푝퐺 푠, 푟 − 1 − 푝푟 + 푞푠퐺 푠, 푟

1 − 푞푠 퐺 푠, 푟 = 푝퐺 푠, 푟 − 1 푝 퐺 푠, 푟 = 퐺 푠, 푟 − 1 1 − 푞푠

푝 Put 푟 = 1 => 퐺 푠, 1 = 퐺 푠, 0 1−푞푠

But

∞ ∞ 푥 푥 퐺 푠, 0 = 푝푥 0 푠 = 푝0 0 + 푝푥 0 푠 푥=0 푥=1

푝푥 0 = 푇푕푒 푝푟표푏 표푓 푥 푓푎푖푙푢푟푒푠 푏푒푓표푟푒 푧푒푟표 푠푢푐푐푒푠푠

= 0

푝푥 0 = 0 푎푛푑 푝0 0 = 1 푓표푟 푥 ≠ 0 푝 푝 퐺 푠, 1 = 퐺 푠, 0 = 1 − 푞푠 1 − 푞푠

푝 푝 2 Put 푟 = 2 퐺 푠, 2 = 퐺 푠, 1 = 1−푞푠 1−푞푠

In general

푝 푟 퐺 푠, 푟 = (2.43푎) 1 − 푞푠

This is the pgf of a NBD where the random variable X is the number of failures before the rth success.

The other case is to consider X to be the total number of trials required to achieve r successes.

Let us denote X=x with Probability 푝푥 푟 . The corresponding difference equation is:

푝푥 푟 = 푝푝푥−1 푟 − 1 + 푞푝푥−1 푟 ; 푥 = 1,2,3, … 푎푛푑 푟 ≥ 1 2.44

In terms of pgf,

∞ ∞ ∞ 푥 푥 푥 푝푥 푟 푠 = 푝 푝푥−1 푟 − 1 푠 + 푞 푝푥−1 푟 푠 2.45 푥=1 푥=1 푥=1

∞ 푥 But 퐺 푠, 푟 = 푥=0 푝푥 푟 푠

∴ 퐺 푠, 푟 − 푝0 푟 = 푝 푠퐺 푠, 푟 − 1 + 푞푠퐺 푠, 푟

푟 푟−1 But 푝0 푟 = 푝 푎푛푑 푝0 푟 − 1 = 푝

∴ 퐺 푠, 푟 − 푝푟 = 푝푠퐺 푠, 푟 − 1 + 푞푠퐺 푠, 푟

1 − 푞푠 퐺 푠, 푟 − 0 = 푝푠퐺 푠, 푟 − 1 푝푠 퐺 푠, 푟 = 퐺 푠, 푟 − 1 1 − 푞푠

푝푠 Put 푟 = 1 => 퐺 푠, 1 = 퐺 푠, 0 1−푞푠

But

∞ 푥 퐺 푠, 0 = 푝푥 0 푠 푥=0

∞ 푥 = 푝0 0 + 푝푥 0 푠 푥=1

푝푥 0 = 푇푕푒 푝푟표푏 표푓 푥 푓푎푖푙푢푟푒푠 푏푒푓표푟푒 푧푒푟표 푠푢푐푐푒푠푠

= 0

푝푥 0 = 0 푎푛푑 푝0 0 = 1 푓표푟 푥 ≠ 0

∴ 퐺 푠, 0 = 1 푝푠 푝푠 ∴ 퐺 푠, 1 = . 1 = 2.46 1 − 푞푠 1 − 푞푠

푝푠 푝푠 2 Put 푟 = 2 퐺 푠, 2 = 퐺 푠, 1 = 1−푞푠 1−푞푠

In general

푝푠 2 퐺 푠, 푟 = (2.43푏) 1 − 푞푠

This is the pgf of a NBD where the random variable X is the number of trials required to achieve r>1 successes.

Properties From the construction of Negative Binomial distribution, we have established that Negative Binomial distribution can be expressed in the following formats

푟 + 푥 − 1 1. 푝 = 푝푟 (1 − 푝)푥 푓표푟 푟 > 0; 0 < 푝 < 1; 푥 = 0,1,2,3 …. 푥 푥

This is a Negative Binomial distribution with parameters 푟 푎푛푑 푝, x represents the total number of failures before the rth success

푥 − 1 2. 푝 = 푝푟 1 − 푝 푥−푟 푥 = 푟, 푟 + 1, 푟 + 2, … 푥 푟 − 1 If p is a sequence of independent Bernoulli trials and random variable 푥 is taken to denote the trial, at which the rth success occurs, where r is a fixed integer

Probability generating function

The probability generating function of a Negative Binomial distribution is given by the following equation ∞ 푥 푥 퐺 푠 = 퐸 푠 = 푝푥 푠 푥=0

∞ 푟 + 푥 − 1 = 푝푟 푞푥 푠푥 2.47 푥 푥=0

∞ 푟 + 푥 − 1 = 푝푟 (푞푠)푥 푥 푥=0

∞ −푟 = 푝푟 −1 푥 (푞푠)푥 = 푝푟 1 − 푞푠 −푟 푥 푥=0

푝 푟 퐺 푠 = 2.48 1 − 푞푠

This is the probability generating function for the negative Binomial distribution with

푞푠 < 1, 0 ≤ 푝 ≤ 1, 푞 + 푝 = 1 푎푛푑 푟 > 0

Consider

푟 + 푥 − 1 푝 = 푝푟 (1 − 푝)푥 푓표푟 푟 > 0; 0 < 푝 < 1; 푥 = 0,1,2,3 …. 푥 푥

The first derivative of the pgf of this distribution is

푟푞푝푟 퐺′ 푠 = 2.49 1 − 푞푠 푟+1

푟푞푝푟 푙푒푡 푠 = 1 퐺′ 1 = 1 − 푞 푟+1 푟푞 퐺′ 1 = 2.50 푝

The second derivative of the pgf of the negative Binomial distribution is

푟 푟 + 1 푞2푝푟 퐺′′ 푠 = 2.51 1 − 푞푠 푟+2

푟 푟 + 1 푞2 푙푒푡 푠 = 1 퐺′′ 1 = 2.52 푝2

Mean

퐸 푋 = 퐺′ 1

푟 1 − 푝 퐸 푋 = 2.53 푝

Variance

2 푣푎푟 푋 = 퐺′′ 1 + 퐺′ 1 − 퐺′ 1 푟푞 푣푎푟 푋 = 2.54 푝2

Factorial moments of Negative Binomial distribution For a natural number r, the rth factorial moment of a probability distribution on the real or complex numbers, or in other words, a random variable X with that probability distribution is

퐸 푋 푘 = 휇푘 푋 = 퐸 푋 푋 − 1 푋 − 2 … 푋 − 푘 + 1 2.55

Where E refers to the expectation and 푥 푘 = 푥 푥 − 1 푥 − 2 … 푥 − 푘 + 1 is the falling factorial.

Thus for the negative binomial distribution

Γ 푟 + 푘 1 − 푝 푘 휇 푋 = 푓표푟 푘 = 1,2,3 … 2.56 푘 Γ푟 푝푘

푟 푟 − 1 ! 푞 푟푞 휇 푋 = 퐸 푋 = = 2.57 1 푟 − 1 ! 푝 푝

Γ 푟 + 2 1 − 푝 2 휇 푋 = 퐸(푋2 − 푋) = 2.58 2 Γ푟 푝2

푞2 μ X = 푟 푟 + 1 2.59 2 푝2

푞2 퐸(푋2 − 푋) = 푟 푟 + 1 2.60 푝2

푞2 푟푞 퐸(푋2) = 푟 푟 + 1 + 푝2 푝 푟푞 퐸(푋2) = 푟 + 1 푞 + 푝 2.61 푝2

3 2 휇3 푋 = 퐸(푋 − 3푋 + 2푋) 2.62

푞3 = 푟 + 2 푟 + 1 푟 2.63 푝3

푞3 퐸(푋3 − 3푋2 + 2푋) = 푟 + 2 푟 + 1 푟 푝3

푞3 퐸(푋3) = 푟 + 2 푟 + 1 푟 + 3퐸(푋2) + 2퐸(푋) 푝3

푞3 푟푞 푞 푟푞 퐸(푋3) = 푟 + 2 푟 + 1 푟 + 3 푟 + 1 + 1 − 2 2.64 푝3 푝 푝 푝 푟푞 푞 = 푟 + 2 푟 + 1 푞2 + 3푝2 푟 + 1 + 1 − 2푝2 푝3 푝 푟푞 푞 = 푟 + 2 푟 + 1 푞2 + 푝2 3 푟 + 1 + 1 푝3 푝 푟푞 퐸(푋3) = 푟 + 2 푟 + 1 푞2 + 푝 3 푟 + 1 푞 + 푝 2.65 푝3

휇4 푋 = 퐸 푋 푋 − 1 푋 − 2 푋 − 3 2.66

= 퐸 (푋4 − 6푋3 + 11푋2 − 6푋)

= 퐸(푋4) − 6퐸 푋3 + 11퐸 푋2 − 6퐸 푋

푞4 휇 푋 = 푟 + 3 푟 + 2 푟 + 1 푟 2.67 4 푝4

Hence

푞4 퐸(푋4) = 푟 + 3 푟 + 2 푟 + 1 푟 + 6퐸 푋3 − 11퐸 푋2 + 6퐸 푋 2.68 푝4

푞4 푟푞 = 푟 + 3 푟 + 2 푟 + 1 푟 + 6 푟 + 2 푟 + 1 푞2 + 푝 3 푟 + 1 푞 + 푝 푝4 푝3 푟푞 푟푞 − 11 푟 + 1 푞 + 푝 + 6 푝2 푝

푟푞 퐸(푋4) = 푟 + 3 푟 + 2 푟 + 1 푞3 + 6푝 푟 + 2 푟 + 1 푞2 + 푝 3 푟 + 1 푞 + 푝 − 11푝2 푟 + 1 푞 + 푝 + 6푝3 2.69 푝4

Skewness and kurtosis

Skewness is a measure of symmetry or the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point.

Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. That is, data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak. A uniform distribution would be the extreme case.

According to Pearson’s moment coefficient of skewness, the skewness of a random variable X is the third Standard moment denoted by 1

Suppose 푋푖, 푖 = 1,2,3, … . 푁 are univariate data that follows a Negative Binomial distribution then

푥 − 휇 3  = 퐸 2.70 1 푠

1 3 = 퐸 푥3 − 3푥2휇 + 3푥휇2 + 휇3 푠

1 3 = 퐸(푥3) − 3휇퐸 푥2 + 3휇2퐸 푥 + 휇3 푠

1 3 = 퐸(푥3) − 3휇 퐸 푥2 + 3휇2 + 휇3 푠

1 3 = 퐸 푥3 − 3휇푠2 + 휇3 푠

∴

3 푟푞 푟푞 푟푞 푟푞 3 푝2 2  = 푟 + 2 푟 + 1 푞2 + 푝 3 푟 + 1 푞 + 푝 − 3 + 2.71 1 푝3 푝 푝2 푝 푟푞

Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right

푘푢푟푡표푠푖푠

푁 4 푋푖 − 휇 퐾푢푟푡표푠푖푠 = 푖=1 푁 − 1 푠4

푥 − 휇 3 = 퐸 2.72 푠

1 4 = 퐸 푋 4−4휇퐸(푋3) + 6휇2퐸(푋2) − 3휇4 푠

푟푞 = 푟 + 3 푟 + 2 푟 + 1 푞3 + 6푝 푟 + 2 푟 + 1 푞2 + 푝 3 푟 + 1 푞 + 푝 푝4 푟푞 − 11푝2 푟 + 1 푞 + 푝 + 6푝3 − 4휇 푟 + 2 푟 + 1 푞2 + 푝 3 푟 + 1 푞 + 푝 푝3 2 푟푞 푟푞 4 푝2 + 6휇2 푟 + 1 푞 + 푝 − 3 푝2 푝 푟푞

CHAPTER 3

BETA - NEGATIVE BINOMIAL MIXTURES

3.1. Introduction From chapter 2, we have identified the following forms of Negative Binomial distribution.

r+푘−1 푟 푘 1. 푝푘 = 푘 푝 1 − 푝 푓표푟 푘 = 0,1,2, … . ; 0 ≤ 푝 ≤ 1 (3.01) with parameters 푟 푎푛푑 푝. k represents the total number of failures before the rth success and;

푘 − 1 2. 푝 = 푝푟 1 − 푝 푘−푟 푘 = 푟, 푟 + 1, 푟 + 2, … ; 0 ≤ 푝 ≤ 1 (3.02) 푘 푟 − 1 If p is a sequence of independent Bernoulli trials and random variable 푘 is taken to denote the number of trials required to produce r successes, where r is a fixed integer

In this chapter we are going to consider the Negative Binomial distribution as given in (3.01) when r is fixed and p is varying between 0 and 1.

The distribution of p is the classical beta distribution is given by

푥훼−1 1 − 푥 훽−1 푓 푥 = 0 < 푥 < 1; 훼, 훽 > 0 3.03 퐵 훼, 훽

This will act as acts as the mixing distribution.

We shall also use the special cases of the classical beta distribution. These are

1. Uniform distribution 2. Power function distribution 3. Truncated beta distribution 4. Arc – sine distribution

Apart from the beta distribution and its special cases, we shall also consider

5. Confluent Hypergeometric distribution 6. Gauss Hypergeometric distribution

The mixed negative Binomial distributions obtained by various mixing (prior) distributions will be expressed

1. Explicitly (where integration is possible) 2. Recursively 3. Using method of Moments

In section 3.2 we shall have a brief discussion of the forms.

For the other sections we shall briefly introduce the mixing distributions before mixing them with the negative Binomial distribution.

3.2. A brief discussion of the various forms of expressing the mixed distribution

3.2.1. Explicit form The mixed distribution is expressed as

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔 푝 푑푝 3.04 푥 0

If the integration is possible then we say that f(x) is expressed explicitly.

However, in most cases this is not possible so we resort to alternative forms.

3.2.2. Method of moments 푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔 푝 푑푝 (3.05) 푥 0

푥 푟 + 푥 − 1 푥 1 푓 푥 = −1 푘 푝 푟+푘 푔 푝 푑푝 (3.06) 푥 푘 푘=0 0

푥 푟 + 푥 − 1 푥 푓 푥 = −1 푘 퐸 푝 푟+푘 푥 푘 푘=0

푥 Γ 푟 + 푥 푥 푓 푥 = −1 푘퐸 푝 푟+푘 3.07 Γ 푟 푥! 푘 푘=0

푓표푟 0 ≤ 푝 ≤ 1; 푟 > 0; 푥 = 0,1,2, …

퐸 푝푟+푘 푖푠 푡푕푒 푚표푚푒푛푡 표푓 표푟푑푒푟 푟 + 푘 푎푏표푢푡 푡푕푒 표푟푖푔푖푛 표푓 푡푕푒 푚푖푥푖푛푔 푑푖푠푡푟푖푏푢푡푖표푛

3.2.3. Recursion One way of obtaining recursions is by considering the ratios of two consecutive probabilities i.e. f(x)/ f(x- 1)

3.3. Classical beta – Negative Binomial distribution

3.3.1. Construction of Classical Beta Distribution Classical beta distribution can be constructed in various ways.

Method 1

We can consider a beta function which is expressed in the following format.

1 퐵 훼, 훽 = 푥훼−1 1 − 푥 훽−1 푑푥 (3.08) 0

If we divide both sides by퐵 훼, 훽 , we get

1 푥훼−1 1 − 푥 훽−1 1 = 푑푥 (3. .09) 0 퐵 훼, 훽

The right hand side of equation 3.06 is a pdf since the integral is equal to 1 and hence the pdf is expressed as follows

푥훼−1 1 − 푥 훽−1 푓 푋 = 푥; 훼, 훽 = ; 0 < 푥 < 1; 훼, 훽 > 0 (3.10) 퐵 훼, 훽

This is the Beta distribution.

Method 2

An alternative way of constructing a beta distribution is shown below.

Let 푥1푎푛푑 푥2 be two stochastically independent random variables that have Gamma distributions and joint pdf

1 1 푓 푥 , 푥 = 푥 훼−1푒−푥1 푥 훽−1푒−푥2 (3.11) 1 2 Γ훼 1 Γ훽 2

1 푓 푥 , 푥 = 푥 훼−1푥 훽−1푒−푥1 푒−푥2 0 < 푥 < ∞, 0 < 푥 < ∞ 1 2 Γ훼Γ훽 1 2 1 2

푥1 Let 푌 = 푥1+푥2 푎푛푑 푝 = 푥1+푥2

Therefore

푥1 = 푦푝, 푥2 = 푦 − 푦푝 = 푦(1 − 푝)

Then

1 푔 푝, 푦 = 푝푦 훼−1 푦 1 − 푝 훽−1푒−푦푝 푒−푦 1−푝 퐽 (3.12) 1 Γ훼Γ훽

Where

푑푥1 푑푥1 푑푦 푑푝 푝 푦 퐽 = = 푑푥2 푑푥2 1 − 푝 −푦 푑푦 푑푝

퐽 = −푦푝 − 푦 1 − 푝 = −푦 = 푦

Therefore equation 3.12 becomes

1 푔 푝, 푦 = 푝훼−1 1 − 푝 훽−1푦훼−1+훽−1+1푒−푦푝 −푦(1−푝) 1 Γ훼Γ훽

1 푔 푝, 푦 = 푝훼−1 1 − 푝 훽−1푦훼+훽+1푒−푦 (3.13) 1 Γ훼Γ훽

푤푖푡푕 0 < 푦 < ∞ 푎푛푑 0 < 푝 < 1

Integrating 푔1 푝, 푦 with respect to y, the results to the marginal pdf is given by

∞ 1 훼−1 훽−1 훼+훽−1 −푦 푔2 푝, 푦 = 푝 1 − 푝 푦 푒 푑푦 (3.14) 0 Γ훼Γ훽

Introducing

Γ(훼 + 훽)

푝훼−1 1 − 푝 훽−1Γ 훼 + 훽 ∞ 푦훼+훽−1푒−푦 푔2 푝, 푦 = 푑푦 Γ훼Γ훽 0 Γ 훼 + 훽

푝훼−1 1 − 푝 훽−1Γ(훼 + 훽) 푝훼−1 1 − 푝 훽−1 푔 푝, 푦 = .1 = 2 Γ훼Γ훽 퐵(훼, 훽)

푝훼−1 1 − 푝 훽−1 푔 푝, 푦 = ; 0 ≤ 푝 ≤ 1, 훼, 훽 > 0 (3.15) 2 퐵 훼, 훽

Equation 3.16 is also called the Classical Beta distribution with parameters 휶 풂풏풅 휷

3.3.2. Properties of the Classical Beta distribution The jth moment of this pdf (classical beta) about the origin is given by

1 푝훼+푖−1 1 − 푝 훽−1 퐸 푃푖 = 푑푝 0 퐵 훼, 훽 퐵 훼 + 푗, 훽 퐸 푃푖 = (3.16푖) 퐵 훼, 훽

훼 + 푗 − 1 ! 훼 + 훽 − 1 ! 퐸 푃푖 = (3.16푖푖) 훼 + 훽 + 푗 − 1 ! 훼 − 1 !

The Mean of the classical beta is therefore 훼 퐸 푃 = (3.17) 훼 + 훽

The 2nd moment about the origin is

훼 + 1 훼 퐸 푃2 = (3.18) 훼 + 훽 훼 + 훽 + 1

And finally the variance of the mixing distribution becomes

푉푎푟 푃 = 퐸 푃2 − (퐸(푃))2

훼훽 푉푎푟 푃 = (3.19) 훼 + 훽 2 훼 + 훽 + 1

3.3.3 The mixture

In explicit form 푟 + 푥 − 1 1 푝훼−1(1 − 푝)훽−1 푓 푥 = 푝푟 1 − 푝 푥 푑푝 푥 0 퐵(훼, 훽)

푟 + 푥 − 1 1 1 = 푝푟+훼−1 1 − 푝 푥+훽−1푑푝 (3.20) 푥 퐵 훼, 훽 0

푟 + 푥 − 1 퐵 푟 + 훼, 푥 + 훽 = 푤푕푒푟푒 푥 = 0,1,2, … (3.21푎) 푥 퐵 훼, 훽

푟 + 푥 − 1 ! 퐵 푟 + 훼, 푥 + 훽 = 푤푕푒푟푒 푥 = 0,1,2, , … … (3.21푏) 푟 − 1 ! 푥! 퐵 훼, 훽

Γ 푟+푥 퐵 푟+훼,푥+훽 = ; 푥 = 0,1,2, … ; 푟, 훼, 훽 > 0 (3.21푐) Γ 푟 푥! 퐵 훼,훽

Γ 푟 + 푥 Γ 훼 + 훽 Γ 푟 + 훼 Γ 푥 + 훽 = ; 푥 = 0,1,2, … ; 푟, 훼, 훽 > 0 (3.21푑) Γ 푟 푥! Γ훼Γ훽Γ 푟 + 훼 + 푥 + 훽

Z. Wang (2010). One mixed negative binomial distribution with application. Journal of Statistical Planning and Inference

Using Method of Moments 푥 Γ 푟 + 푥 푥 푓 푥 = −1 푘 퐸(푝 푟+푘 ) Γ 푟 푥! 푘 푘=0

퐸 푝푟+푘 푖푠 푡푕푒 푚표푚푒푛푡 표푓 표푟푑푒푟 푟 + 푘 푎푏표푢푡 푡푕푒 표푟푖푔푖푛 표푓 푡푕푒 푚푖푥푖푛푔 푑푖푠푡푟푖푏푢푡푖표푛

But

퐵 훼 + 푗, 훽 퐸 푃푖 = 퐵 훼, 훽

푥 Γ 푟 + 푥 푥 퐵 훼 + 푟 + 푘, 훽 ∴ 푓 푥 = −1 푘 ; 푓표푟 푥 = 0,1,2, … ; 푟, 훼, 훽 > 0 (3.22) Γ 푟 푥! 푘 퐵 훼, 훽 푘=0

Mixing using Recursive relation There are three ways of applying the recursive relation in this mixture, namely

1. Using ratio of the conditional distribution 2. Using ratio of the mixed distribution 3. Using a dummy function

1. Using ratio of the conditional distribution

1 푝푟 푥 = 푓 푥/푝 푔 푝 푑푝 (3.23) 0

푝(푥/푝)is a Negative Binomial distribution in this case

푟 + 푥 − 1 푓 푥/푝 = 푝푟 1 − 푝 푥 (3.24) 푥

Substituting 푥 푤푖푡푕 푥 − 1 in equation 3.24 we get

푟 + 푥 − 2 푓 (푥 − 1)/푝 = 푝푟 1 − 푝 푥−1 (3.25) 푥 − 1

Dividing equation 3.24 by equation 3.25 we get

푟 + 푥 − 1 푟 푥 푓(푥/푝) 푝 (1 − 푝) = 푥 푟 + 푥 − 2 푓 (푥 − 1)/푝 푝푟 1 − 푝 푥−1 푥 − 1

푟 + 푥 − 1 푟 + 푥 − 2 ! 푥 − 1 ! 푟 − 1 ! 푓 푥/푝 = (1 − 푝)푓 (푥 − 1)/푝 푥 푥 − 1 ! 푟 − 1 ! 푟 + 푥 − 2 !

푟 + 푥 − 1 푓 푥/푝 = 1 − 푝 푓 (푥 − 1)/푝 (3.26) 푥

Substituting equation (3.26) into equation (3.23)

1 푟 + 푥 − 1 푝푟 (푥) = 1 − 푝 푓 (푥 − 1)/푝 푔 푝 푑푝 0 푥

푟 + 푥 − 1 1 1 푝푟 푥 = 푓 (푥 − 1)/푝 푔 푝 푑푝 − 푝푓 (푥 − 1)/푝 푔 푝 푑푝 (3.27) 푥 0 0

Consider

1 푓 (푥 − 1)/푝 푔 푝 푑푝 (3.28) 0

This can be expressed as

1 푟 + 푥 − 2 푟 푥−1 푝 1 − 푝 푔 푝 푑푝 = 푝푟 푥 − 1 (3.29푎) 0 푥 − 1

Consider

1 푝푓 (푥 − 1)/푝 푔 푝 푑푝 (3.30) 0

1 푟 + 푥 − 2 = 푝푟+1 1 − 푝 푥−1푔 푝 푑푝 0 푥 − 1 푟 + 푥 − 2 1 푟 + 1 + 푥 − 2 = 푥 − 1 푝푟+1 1 − 푝 푥−1푔 푝 푑푝 푟 + 1 + 푥 − 2 푥 − 1 0 푥 − 1 푟 + 푥 − 2

= 푥 − 1 푝 (푥 − 1) 푟 + 1 + 푥 − 2 푟+1

푥 − 1 1 푟 푝푓 (푥 − 1)/푝 푔 푝 푑푝 = 푝푟+1 푥 − 1 (3.30푎) 0 푟 + 푥 − 1

Substituting equation 3.29a and 3.30a into 3.27 the Beta - Negative Binomial recursive mixture becomes

푟 + 푥 − 1 푟 푝 푥 = 푝 푥 − 1 − 푝 푥 − 1 (3.31) 푟 푥 푟 푟 + 푥 − 1 푟+1

풓 > 1, 푥 = 0,1,2 … . . ;

2. Using ratio of the mixed distribution

From equation 3.21b

푟 + 푥 − 1 ! 퐵 푟 + 훼, 푥 + 훽 푓 푥 = 푤푕푒푟푒 푥 = 0,1,2, … 푟 − 1 ! 푥! 퐵 훼, 훽

Considering the below ratio.

푓 푥 푟 + 푥 − 1 ! 퐵 푟 + 훼, 푥 + 훽 푟 − 1 ! 푥! 퐵 훼, 훽 = 푓 푥 − 1 푟 − 1 ! 푥! 퐵 훼, 훽 푟 + 푥 − 1 ! 퐵 푟 + 훼, 푥 − 1 + 훽

푓 푥 푟 + 푥 − 1 훽 + 푥 − 1 = 푓 푥 − 1 푥 푟 + 푥 + 훼 + 훽 − 1

훽 + 푥 − 1 푟 + 푥 − 1 푓 푥 = 푓 푥 − 1 3.32 푥 푟 + 푥 + 훼 + 훽 − 1

푓표푟 풓, 훼, 훽 > 0, 푥 = 0,1,2 … ;

3. Using a dummy function

Consider the mixture equation below

푟 + 푥 − 1 1 1 푓 푥 = 푝푟+훼−1 1 − 푝 푥+훽−1푑푝 3.33 푥 퐵 훼, 훽 0

Introducing the dummy function

1 퐵 훼, 훽 푓 푥 푟+훼−1 푥+훽−1 퐼푥 푟, 훼, 훽 = 푟+푥−1 = 푝 1 − 푝 푑푝 3.34 푥 0

Integrating the integral by parts.

1 1 푝푟+훼−1 1 − 푝 푥+훽−1푑푝 = 푢푑푣 0 0

1 1 푢푑푣 = 푢푣 − 푣푑푢 0 0

Let

푢 = 푝푟+훼−1 푎푛푑 푑푣 = 1 − 푝 푥+훽−1푑푝

Hence

1 − 푝 푥+훽 푑푢 = 푟 + 훼 − 1 푝푟+훼−2푑푝 푎푛푑 푣 = − 푥 + 훽

∴

1 1 1 − 푝 푥+훽 푝푟+훼−1 푟 + 훼 − 1 1 푝푟+훼−1 1 − 푝 푥+훽−1푑푝 = − + 1 − 푝 푥+훽 푝푟+훼−2푑푝 푥 + 훽 푥 + 훽 0 0 0 푟 + 훼 − 1 = 퐼 푟 − 1, 훼, 훽 3.35 푥 + 훽 푥+1

퐵 훼, 훽 푓 푥 푟 + 훼 − 1 퐼 푟 = = 퐼 푟 − 1 푥 푟+푥−1 푥 + 훽 푥+1 푥

퐵 훼, 훽 푓 푥 푟 + 훼 − 1 퐵 훼, 훽 푓 푥 + 1 = 3.36 푟+푥−1 푥 + 훽 푟+푥 푥 푥+1

This leads to

푟 + 푥 − 1 푥 + 훽 − 1 푓 푥 = 푓 푥 − 1 3.37 푥 푟 + 푥 − 2

푓표푟 푟, 훽, 훼 > 0 푎푛푑 푥 = 1,2,3, …

Identity

We can draw an identity based on the result from explicit mixing and that from method of moments as follows.

푥 푥 퐵 푟 + 훼, 푥 + 훽 = −1 푘 퐵 훼 + 푟 + 푘, 훽 ; 푓표푟 푥 = 0,1,2, … ; 푟, 훼, 훽 > 0 3.38 푘 푘=0

3.3.4 Properties of Beta – Negative Binomial Distribution

Mean Considering the result got from the recursive relation mixture i.e. equation 3.31

푟 + 푥 − 1 푟 푝 (푥) = 푝 (푥 − 1) − 푝 (푥 − 1) 푟 푥 푟 푟 + 푥 − 1 푟+1

∞

퐸 푋 = 푥 푝푟 (푥) 푥=0

∞ 푟 퐸 푋 = 푟 + 푥 − 1 푝 푥 − 1 − 푝 푥 − 1 (3.39) 푟 푟 + 푥 − 1 푟+1 푥=0

∞ ∞

= 푟 + 푥 − 1 푝푟 푥 − 1 − 푟 푝푟+1 푥 − 1 푥=0 푥=0

∞ ∞ 푟 + 푥 − 2 푟 + 푥 − 1 퐸 푋 = 푟 + 푥 − 1 푝푟 1 − 푝 푥−1 − 푟 푝푟+1 1 − 푝 푥−1 (3.40) 푥 − 1 푥 − 1 푥=0 푥=0

Consider the first part of the equation

∞ 푟 + 푥 − 2 푟 + 푥 − 1 푝푟 1 − 푝 푥−1 (3.40푎) 푥 − 1 푥=0

This can be expressed as follows

∞ 푟 + 푥 − 1 ! 푝푟 1 − 푝 푥−1 (3.40푏) 푟 − 1 ! 푥 − 1 ! 푥=0

But

푟 + 푥 − 1 ! 푟 + 푥 − 1 = 푟 푟 − 1 ! 푥 − 1 ! 푥 − 1

Hence

∞ ∞ 푟 + 푥 − 1 ! 푟 + 푥 − 1 푝푟 1 − 푝 푥−1 = 푟 푝푟 1 − 푝 푥−1 (3.41) 푟 − 1 ! 푥 − 1 ! 푥 − 1 푥=0 푥=0

∞ 푟 푟 + 푥 − 1 푟 = 푝푟+1 1 − 푝 푥−1 = 푝 푥 − 1 푝 푥=0

Since

∞ 푟 + 푥 − 1 푝푟+1 1 − 푝 푥−1 = 1 푥 − 1 푥=0

Consider the second part of the equation

∞ 푟 + 푥 − 1 푟 푝푟+1 1 − 푝 푥−1 (3.42) 푥 − 1 푥=0

This can be expressed as

∞ 푟 + 푥 − 1 푟 푝푟+1 1 − 푝 푥−1 = 푟 푥 − 1 푥=0

Since

∞ 푟 + 푥 − 1 푝푟+1 1 − 푝 푥−1 = 1 푥 − 1 푥=0

The mean therefore becomes

푟 1 − 푝 퐸 푋 = (3.43) 푝

Variance

∞ 2 2 퐸 푋 = 푋 푝푟 (푥) 푥=0

∞ ∞ 푟 + 푥 − 2 푟 + 푥 − 1 퐸 푋2 = 푥 푟 + 푥 − 1 푝푟 1 − 푝 푥−1 − 푟 푥 푝푟+1 1 − 푝 푥−1 (3.44) 푥 − 1 푥 − 1 푥=0 푥=0

Consider the first part of the equation above

∞ 푟 + 푥 − 2 푥 푟 + 푥 − 1 푝푟 1 − 푝 푥−1 (3.44푎) 푥 − 1 푥=0

It can be expressed or manipulated as follows

∞ 1 푥 푟 + 푥 − 1 ! 푝푟 1 − 푝 푥 1 − 푝 푟 − 1 ! (푥 − 1)! 푥=0

∞ 1 푥2 푟 + 푥 − 1 ! = 푝푟 1 − 푝 푥 1 − 푝 푟 − 1 ! 푥! 푥=0

∞ 1 푟 + 푥 − 1 = 푥2 푝푟 1 − 푝 푥 1 − 푝 푥 푥=0

But considering a Negative Binomial Distribution

∞ 푟 + 푥 − 1 퐸 푋2/푝 = 푥2 푝푟 1 − 푝 푥 푥 푥=0

푟 1 − 푝 1 + 푟 1 − 푝 퐸 푋2/푝 = 푝2

And hence

∞ 1 푥 푟 + 푥 − 1 ! 푟 1 + 푟 1 − 푝 푝푟 1 − 푝 푥 = 1 − 푝 푟 − 1 ! (푥 − 1)! 푝2 푥=0

Consider the second part of the equation

∞ 푟 + 푥 − 1 푟 푥 푝푟+1 1 − 푝 푥−1 (3.44푏) 푥 − 1 푥=0

This can be expressed as follows

∞ 푟푝 푥 푟 + 푥 − 1 ! = 푝푟 (1 − 푝)푥 1 − 푝 푟 푟 − 1 ! 푥 − 1 ! 푥=0

∞ 푟푝 푥2 푟 + 푥 − 1 ! = 푝푟 (1 − 푝)푥 1 − 푝 푟 푟 − 1 ! 푥! 푥=0

∞ 푝 푟 + 푥 − 1 = 푥2 푝푟 (1 − 푝)푥 1 − 푝 푥 푥=0

푝 푟 1 − 푝 1 + 푟 1 − 푝 = 푋 (1 − 푝) 푝2

푟 1 + 푟 1 − 푝 = 푝

The second moment therefore becomes

푟 1 + 푟 1 − 푝 푟 1 + 푟 1 − 푝 퐸 푋2 = − 푝2 푝

푟 1 + 푟 1 − 푝 1 퐸 푋2 = − 1 (3.45) 푝 푝

Hence

푟 1 + 푟 1 − 푝 1 푟2(1 − 푝)2 푣푎푟 푋 = − 1 − 푝 푝 푝2

푟(푟2 − 푝) 푣푎푟 푋 = (3.46) 푝2

3.3.5. Moment Generating Function Consider the explicit mixture of Beta distribution with the Negative Binomial distribution

ᴦ 푟 + 푥 퐵 푟 + 훼, 푥 + 훽 푓 푋 = 푥 = ; 푥 = 0,1,2, … ; 푟, 훼, 훽 > 0 ᴦ 푟 푥! 퐵 훼, 훽

푀퐺퐹 푔 푡 = 퐸 푡푛 ; 푛 = 0,1,2, ….

푔 푡 = 퐻 푟, 훽; 푟 + 훼 + 훽, 푡 푝푟 푍 = 0 (3.47)

Where

푎 푏 ∞ (푛 ) (푛 ) 푡푛 푐 퐻 푎, 푏; 푐, 푡 = (푛 ) 푖푠 푡푕푒 퐻푦푝푒푟푔푒표푚푒푡푟푖푐 푓푢푛푐푡푖표푛 푛! 푛=0 Γ 푥 + 푛 푥 = (푛) Γ푥

푎 = 푟; 푏 = 훽; 푐 = 푟 + 훼 + 훽; 푡 = 푡

∞ 푛 푟(푛)훽(푛) 푡 ∴ 푔 푡 = 푟 + 훼 + 훽 (푛) 푛! 푛=0

∞ Γ 푟 + 푛 Γ 훽 + 푛 Γ 푟 + 훼 + 훽 푡푛 푔 푡 = (3.48) Γ푟 Γ훽 Γ 푟 + 훼 + 훽 + 푛 푛! 푛=0

Differentiate 푔(푡) with respect to 푡

∞ Γ 푟 + 푛 Γ 훽 + 푛 Γ 푟 + 훼 + 훽 푡푛−1 푔′ 푡 = 푛 Γ푟 Γ훽 Γ 푟 + 훼 + 훽 + 푛 푛! 푛=0

∞ Γ 푟 + 푛 Γ 훽 + 푛 Γ 푟 + 훼 + 훽 푡푛−1 푔′ 푡 = (3.49) Γ푟 Γ훽 Γ 푟 + 훼 + 훽 + 푛 푛 − 1 ! 푛=0

Hence

∞ Γ 푟 + 푛 Γ 훽 + 푛 Γ 푟 + 훼 + 훽 푡푛−2 푔′′ 푡 = 푛 푛 − 1 (3.50) Γ푟 Γ훽 Γ 푟 + 훼 + 훽 + 푛 푛! 푛=0

∞ Γ 푟 + 푛 Γ 훽 + 푛 Γ 푟 + 훼 + 훽 푡푛−2 푔′′ 푡 = (3.51) Γ푟 Γ훽 Γ 푟 + 훼 + 훽 + 푛 푛 − 2 ! 푛=0

3.4. Special cases of beta – negative Binomial distribution

3.4.1. Uniform – Negative Binomial distribution

3.4.1.1. Uniform distribution Construction

Consider a beta distribution defined as follows

푝훼−1 1 − 푝 훽−1 푔 푝 = 0 < 푝 < 1; 훼, 훽 > 0 퐵 훼, 훽

If we let 훼 = 훽 = 1

We get the uniform distribution [0,1] given by

1 0 < 푝 < 1 푔 푝 = (3.52) 0 푒푙푠푒푤푕푒푟푒

Properties

Moment of order j about the origin of the uniform distribution g(p)

1 퐸 푃푗 = 푝푗 푑푝 0

1 푝푗 +1 = 푗 + 1 0 1 퐸 푝푗 = (3.53) 푗 + 1

Mean

1 퐸 푃 = (3.54) 2 Variance

1 푣푎푟 푃 = (3.55) 12

3.4.1.2. The mixtures Explicit mixing

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔(푝)푑푝 푥 0 where 푔(푝)is the uniform distribution.

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푑푝 푥 0

푟 + 푥 − 1 1 푝푟 1 − 푝 푥 푓 푥 = 퐵 푟 + 1, 푥 + 1 푑푝 푥 0 퐵 푟 + 1, 푥 + 1

푟 + 푥 − 1 푓 푥 = 퐵 푟 + 1, 푥 + 1 푥

푟 + 푥 − 1 ! ᴦ 푟 + 1 ᴦ 푥 + 1 푓 푥 = 푟 − 1 ! 푥! ᴦ 푟 + 푥 + 2

푟 푓 푥 = 푥 = 0,1,2, … ; 푟 > 0 (3.56) 푟 + 푥 + 1 (푟 + 푥)

Using recursive relation

The mixture between Negative Binomial and uniform distribution can be expressed in a recursive format in the following two ways.

1st Form

Here we consider the mixture from explicit mixing i.e. equation 3.56 푟 푓 푥 = 푟 + 푥 + 1 푟 + 푥

Formulation and working the below ratio.

푓 푥 푟 푟 + 푥 푟 + 푥 − 1 = 푓 푥 − 1 푟 + 푥 + 1 푟 + 푥 푟

Hence

푟 + 푥 − 1 푓 푥 = 푓 푥 − 1 (3.57) 푟 + 푥 + 1

2nd Form

Here we introduce a dummy function퐼푥 푟 that is of a Beta format as follows.

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푑푝 푥 0 1 푓 푥 푟 푥 푟+푥−1 = 푝 1 − 푝 푑푝 푥 0

1 푓 푥 푟 푥 퐼푥 푟 = 푟+푥−1 = 푝 1 − 푝 푑푝 (3.58) 푥 0

Using integral by parts

Let

푢 = 1 − 푝 푥

푑푢 = −푥 1 − 푝 푥−1푑푝

푝푟+1 푣 = 푟 + 1

푑푣 = 푝푟 푑푝

푢 푑푣 = 푢푣 − 푣 푑푢

1 푝푟+1 푥 1 퐼 푟 = 1 − 푝 푥 + 푝푟+1 1 − 푝 푥−1푑푝 푥 푟 + 1 푟 + 1 0 0

1 푥 푟+1 푥−1 퐼푥 푟 = 푝 1 − 푝 푑푝 (3.59) 푟 + 1 0 푥 퐼 푟 = 퐼 푟 + 1 푥 푟 + 1 푥−1

푓 푥 푥 푓 푥 − 1 퐼 푟 = = 푥 푟+푥−1 푟 + 1 푟+푥−1 푥 푥−1

푥 푟+푥−1 푓 푥 = 푥 푓 푥 − 1 푟 + 1 푟+푥−1 푥−1

Hence 푟 푓 푥 = 푓 푥 − 1 3.60 푟 + 1

Using Method of Moments

푥 Γ 푟 + 푥 푥 푓 푥 = −1 푘 퐸(푝 푟+푘 ) Γ 푟 푥! 푘 푘=0

퐸 푝푟+푘 푖푠 푡푕푒 푚표푚푒푛푡 표푓 표푟푑푒푟 푟 + 푘 푎푏표푢푡 푡푕푒 표푟푖푔푖푛 표푓 푡푕푒 푚푖푥푖푛푔 푑푖푠푡푟푖푏푢푡푖표푛

1 퐸 푝푗 = 푗 + 1

푥 Γ 푟 + 푥 푥 1 푓 푥 = −1 푘 3.61 Γ 푟 푥! 푘 푟 + 푘 + 1 푘=0

푓표푟 푟 > 0; 푥 = 0,1,2, …

Identity

We can draw an identity based on the result from explicit mixing and that from method of moments as follows.

푥 푟 Γ 푟 + 푥 푥 1 = −1 푘 푟 + 푥 + 1 푟 + 푥 Γ 푟 푥! 푘 푟 + 푘 + 1 푘=0

푥 푟! 푥! 푥 1 = −1 푘 푟 + 푥 + 1 푟 + 푥 ! 푘 푟 + 푘 + 1 푘=0

푓표푟 푟 > 0; 푥 = 0,1,2, …

3.4.2. Power function - negative Binomial distribution

3.4.2.1. Power function distribution Construction

Consider the beta function

푝훼−1(1 − 푝)훽−1

푔 p = 퐵(훼, 훽) 0 < 푝 < 1; 훼, 훽 > 0 0 푒푙푠푒푤푕푒푟푒

Let 훽 =1 then

푝훼−1(1 − 푝)0 푔 푝 = 퐵(훼, 1)

훼푝훼−1 0 < 푝 < 1; 훼 > 0 푔 푝 = (3.62) 0 푒푙푠푒푤푕푒푟푒

This is the pdf of a power function distribution with parameter 훼

Properties

The moments of order j about the origin is

1 퐸 푃푗 = 훼푝훼−1푝푗 푑푝 0

1 = 훼 푝훼+푗 −1푑푝 0

1 푝훼+푗 −1 = 훼 훼 + 푗 0 훼 퐸 푃푗 = (3.63) 훼 + 푗

Mean 훼 퐸 푃 = (3.64) 훼 + 1

Variance

푣푎푟 푝 = 퐸 푝2 − 퐸 푝 2

훼 푎2 푣푎푟 푝 = − 훼 + 2 훼 + 1 2 훼 푣푎푟 푝 = (3.65) 훼 + 1 2 훼 + 2

3.4.2.2 The mixing Explicit mixing

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔(푝)푑푝 푥 0

푔(푝)is the power function distribution

푔 푝 = 훼푝훼−1

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 훼푝훼−1푑푝 푥 0

푟 + 푥 − 1 1 푓 푥 = 훼 푝푟+훼−1 1 − 푝 푥 푑푝 푥 0

푟 + 푥 − 1 1 푝푟+훼−1 1 − 푝 푥 푓 푥 = 훼 퐵(푟 + 훼, 푥 + 1) 푑푝 푥 0 퐵(푟 + 훼, 푥 + 1) 푟 + 푥 − 1 푓 푥 = 훼 퐵 푟 + 훼, 푥 + 1 (3.66) 푥

푓표푟 푥 = 0,12 …. ; 푎, 푟 > 0

This is the density function of the Negative Binomial – Power function expressed explicitly

Using Method of Moments

푥 Γ 푟 + 푥 푥 푓 푥 = −1 푘 퐸(푝 푟+푘 ) Γ 푟 푥! 푘 푘=0

퐸 푝푟+푘 푖푠 푡푕푒 푚표푚푒푛푡 표푓 표푟푑푒푟 푟 + 푘 푎푏표푢푡 푡푕푒 표푟푖푔푖푛 표푓 푡푕푒 푚푖푥푖푛푔 푑푖푠푡푟푖푏푢푡푖표푛 훼 퐸 푃푗 = 훼 + 푗

푥 Γ 푟 + 푥 푥 훼 푓 푥 = −1 푘 3.67 Γ 푟 푥! 푘 훼 + 푟 + 푘 푘=0

푓표푟 푟, 훼 > 0; 푥 = 0,1,2, …

Using the recursive relation expression

We can achieve this by the use of two different ways.

1st Form

Consider the result from explicit mixture as shown below.

푟 + 푥 − 1 푓 푥 = 훼 퐵 푟 + 훼, 푥 + 1 푥

Expressing this a ratio

푓 푥 푟+푥−1 퐵 푟 + 훼, 푥 + 1 = 푥 푓 푥 − 1 푟+푥−2 푥−1 퐵 푟 + 훼, 푥

Hence

푟 + 푥 − 1 푓 푥 = 푓 푥 − 1 3.68 푟 + 훼 + 푥

2nd Form

Consider the explicit expression below

푟 + 푥 − 1 1 푓 푥 = 훼 푝푟+훼−1 1 − 푝 푥 푑푝 푥 0

Manipulate this to a dummy function 퐼푥 푟, 훼 as shown below

1 푓 푥 푟+훼−1 푥 푟+푥−1 = 푝 1 − 푝 푑푝 푥 훼 0

Let

1 푓 푥 푟+훼−1 푥 퐼푥 푟, 훼 = 푟+푥−1 = 푝 1 − 푝 푑푝 (3.69) 푥 훼 0

Integrating the integral by parts

Let

푢 = 1 − 푝 푥

푑푢 = −푥 1 − 푝 푥−1푑푝

푑푣 = 푝푟+훼−1푑푝

푝푟+훼 푣 = 푟 + 훼

u dv = vu − v du

푝푟+훼 1 푥 1 퐼 푟, 훼 = 1 − 푝 푥 + 푝푟+훼 1 − 푝 푥−1푑푝 푥 푟 + 훼 푟 + 훼 0 0

1 푥 푟+훼 푥−1 퐼푥 푟, 훼 = 푝 1 − 푝 푑푝 (3.70) 푟 + 훼 0 푓 푥 푥 퐼 푟, 훼 = = 퐼 푟 − 1, 훼 푥 푟+푥−1 푟 + 훼 푥−1 푥 훼 푓 푥 푥 푓 푥 − 1 = 3.71 푟+푥−1 푟 + 훼 푟+푥−3 푥 훼 푥−1 훼

푥 푟+푥−1 푓 푥 = 푥 푓 푥 − 1 푟 + 훼 푟+푥−3 푥−1

Hence

푟 + 푥 − 1 푟 + 푥 − 2 푓 푥 = 푓 푥 − 1 3.72 푟 + 훼 푟 − 1

Identity

We can draw an identity based on the result from explicit mixing and that from method of moments as follows.

푥 푟 Γ 푟 + 푥 푥 훼 = −1 푘 푟 + 푥 + 1 (푟 + 푥) Γ 푟 푥! 푘 훼 + 푟 + 푘 푘=0

푥 푟! 푥! 푥 훼 = −1 푘 3.73 푟 + 푥 + 1 푟 + 푥 ! 푘 훼 + 푟 + 푘 푘=0

푓표푟 푥 = 0,1,2, … ; 푟, 훼 > 0

3.4.3. Arcsine – negative Binomial distribution 3.4.3.1 Arcsine distribution

Construction

1 The standard Arcsine distribution is a special case of the beta distribution with 훼 = 훽 = 2 Consider a beta distribution below

푝훼−1 1 − 푝 훽−1 푔 푝 = 0 ≤ 푝 ≤ 1; 훼, 훽 > 0 퐵 훼, 훽

1 If 훼 = 훽 = 2

Then

푝−1/2 1 − 푝 −1/2 푔 푝 = 1 1 0 ≤ 푝 ≤ 1 ; 훼, 훽 > 0 퐵( , ) 2 2

1 1 1 1 Γ Γ 퐵 , = 2 2 = 휋 2 2 Γ 1 1 푔 푝 = 0 ≤ 푝 ≤ 1; 푎, 푏 > 0 (3.74) 휋 푝 1 − 푝

This is the pdf of Arcsine distribution

Properties of Arcsine distribution

Jth moment about the origin

1 1 − − 푝 2 1 − 푝 2 푔 푝 = 0 ≤ 푝 ≤ 1 ; 훼, 훽 > 0 휋

1 1 1 1 푗 − − 퐸 푝푗 = 푝 2 1 − 푝 2 푑푝 휋 0

1 1 퐵 푗 + , ∴ 퐸 푝푗 = 2 2 휋

3 1 퐵 , 퐸 푃 = 2 2 휋

3 1 퐵 , 2 2 퐸 푃 = 1 1 퐵 , 2 2 1 ∴ 퐸 푃 = 3.75 2

푉푎푟 푃 = 퐸 푝2 − [퐸(푝)]2

1 ∴ 푉푎푟 푃 = 3.76 8

3.4.3.2 The mixture

a) Explicit mixing

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔 푝 푑푝 (3.77) 푥 0

Where 푔1(푝)is the arcsine distribution

푟 + 푥 − 1 1 1 푓 푥 = 푝푟 1 − 푝 푥 푑푝 푥 0 휋 푝 1 − 푝

1 1 푟 + 푥 − 1 퐵 푟 + , 푥 + 푓 푥 = 2 2 3.78 푥 휋 1 1 푓표푟 푟 > 0; 푥 = 0,1,2, … ; 푎푛푑 휋 = 퐵 , 2 2

b) Using recursive relation

We can achieve this by the use of two different ways.

1st Form

Consider the result from explicit mixture as shown below.

1 1 푟 + 푥 − 1 퐵 푟 + , 푥 + 푓 푥 = 2 2 푥 휋

Expressing this a ratio

1 1 푟+푥−1 퐵 푟 + , 푥 + 푓 푥 푥 2 2 = 1 1 푓 푥 − 1 푟+푥−2 퐵 푟 + , 푥 − 푥−1 2 2

Hence

푟 + 푥 − 1 2푥 − 1 푓 푥 = 푓 푥 − 1 3.79 2푥 푟 + 푥

푓표푟 푟 > 0; 푥 = 0,1,2, … ;

2nd Form

Consider the mixture equation

1 1 1 푟 + 푥 − 1 1 푟− 푥− 푓 푥 = 푝 2 1 − 푝 2푔 푝 푑푝 푥 휋 0

Manipulate this to a dummy function 퐼푥 푟 as shown below

1 1 1 휋푓 푥 푟− 푥− 2 2 퐼푥 푟, 푝 = 푟+푥−1 = 푝 1 − 푝 푔 푝 푑푝 (3.80) 푥 0

Using integration by parts to solve the integral

We let

1 푥− 푢 = 1 − 푝 2

1 푟− 푑푣 = 푝 2푑푝

3 1 푥− 푑푢 = − 푥 (1 − 푝) 2푑푝 2

1 푟+ 푝 2 푣 = 1 푟 + 2

1 1 푟+ 1 1 2 − 푥 1 1 3 푥− 푝 2 푟+ 푥− 2 2 2 퐼푥 푟, 푝 = 1 − 푝 1 + 1 푝 1 − 푝 푑푝 (3.81) 푟 + 푟 + 0 2 0 2

1 − 푥 1 1 3 2 푟+ 푥− 2 2 퐼푥 푟, 푝 = 1 푝 1 − 푝 푑푝 푟 + 0 2

1 1 3 1 − 2푥 푟+ 푥− = 푝 2 1 − 푝 2푑푝 2푟 + 1 0 1 − 2푥 퐼 푟, 푝 = 퐼 푟 + 1, 푝 3.82 푥 2푟 + 1 푥−1

휋푓 푥 1 − 2푥 휋푓 푥 − 1 퐼 푟, 푝 = = 푥 푟+푥−1 2푟 + 1 푟+푥−1 푥 푥−1

1 − 2푥 푟+푥−1 푓 푥 = 푥 푓 푥 − 1 2푟 + 1 푟+푥−1 푥−1 hence

푟 1 − 2푥 푓 푥 = 푓 푥 − 1 3.83 푥 2푟 + 1

a. Mixture from the method of moment 푥 Γ 푟 + 푥 푥 푓 푥 = −1 푘 퐸(푝 푟+푘 ) Γ 푟 푥! 푘 푘=0

퐸 푝푟+푘 푖푠 푡푕푒 푚표푚푒푛푡 표푓 표푟푑푒푟 푟 + 푘 푎푏표푢푡 푡푕푒 표푟푖푔푖푛 표푓 푡푕푒 푚푖푥푖푛푔 푑푖푠푡푟푖푏푢푡푖표푛

1 1 퐵 푗 + , 퐸 푃푗 = 2 2 휋 푥 1 1 Γ 푟 + 푥 푥 퐵 푟 + 푘 + , 푓 푥 = −1 푘 2 2 3.84 Γ 푟 푥! 푘 휋 푘=0 푓표푟 푟, 훼 > 0; 푥 = 0,1,2, …

Identity

We can draw an identity based on the result from explicit mixing and that from method of moments as follows.

푥 1 1 푥 1 1 퐵 푟 + , 푥 + = −1 푘 퐵 푟 + 푘 + , 3.85 2 2 푘 2 2 푘=0 푓표푟 푟, 훼 > 0; 푥 = 0,1,2, …

3.4.4. Negative Binomial – Truncated Beta distribution

3.4.4.1 Truncated Beta distribution

Construction Consider a two sided truncated function given by

훽 푝푎−1 1 − 푝 푏−1 푑푝 푤푖푡푕 1 < 훼 < 푝 < 훽 < 1; 푎, 푏 > 0 (3.86) 훼

This function can be expressed in terms of incomplete beta function as

훽 훽 훼 푝푎−1 1 − 푝 푏−1 푑푝 = 푝푎−1 1 − 푝 푏−1 푑푝 − 푝푎−1 1 − 푝 푏−1 푑푝 훼 0 0

= 퐵훽 푎, 푏 −퐵훼 (푎, 푏)

Thus

훽 푝푎−1 1 − 푝 푏−1 1 = 푑푝 훼 퐵훽 푎, 푏 −퐵훼 (푎, 푏)

This gives a distribution referred to as truncated beta distribution g(p)with parameters 훼, 훽, 푎, 푏

푝푎−1 1 − 푝 푏−1 푔 푝 = ; 푓표푟 1 < 훼 < 푝 < 훽 < 1; 푎, 푏 > 0 (3.87) 퐵훽 푎, 푏 −퐵훼 (푎, 푏)

Properties

Moments of order j about the origin of de distribution can be expressed as

훽 푝푎−1 1 − 푝 푏−1푃푗 퐸 푃푗 = 푑푝 훼 퐵훽 푎, 푏 −퐵훼 푎, 푏

훽 푝푎+푗 −1 1 − 푝 푏−1 퐸 푃푗 = 푑푝 훼 퐵훽 푎, 푏 − 퐵훼 푎, 푏

퐵훽 푎 + 푗, 푏 − 퐵훼 푎 + 푗, 푏 퐸 푃푗 = (3.88) 퐵훽 푎, 푏 −퐵훼 푎, 푏

3.4.4.2 The mixture

a) Explicit mixing

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔(푝)푑푝 푥 0

Where g(p) is the truncated beta distribution

1 푝푎+푗 −1 1−푝 푏−1 푓 푥 = 푟+푥−1 푝푟 1 − 푝 푥 푑푝 (3.89) 푥 0 퐵훽 푎,푏 −퐵훼 푎,푏

푟 + 푥 − 1 1 푝푎+푟+푗 −1 1 − 푝 푥+푏−1 푓(푥) = 푑푝 푥 0 퐵훽 푎, 푏 − 퐵훼 푎, 푏

푟 + 푥 − 1 퐵훽 푎 + 푟, 푏 + 푥 − 퐵훼 푎 + 푟, 푏 + 푥 푓(푥) = 푥 퐵훽 푎, 푏 − 퐵훼 푎, 푏

푟 + 푥 − 1 퐵훽 푎 + 푟, 푏 + 푥 − 퐵훼 푎 + 푟, 푏 + 푥 푓 푥 = 3.90 푥 퐵훽 푎, 푏 − 퐵훼 푎, 푏

푓표푟 푎, 푏 > 0; 1 < 훼 < 푝 < 훽 < 1; 푥 = 0,1,2 …

This is the pdf of the Negative Binomial – truncated beta mixture

b) Using method of moments

푥 Γ 푟 + 푥 푥 푓 푥 = −1 푘 퐸(푝 푟+푘 ) Γ 푟 푥! 푘 푘=0

퐸 푝푟+푘 푖푠 푡푕푒 푚표푚푒푛푡 표푓 표푟푑푒푟 푟 + 푘 푎푏표푢푡 푡푕푒 표푟푖푔푖푛 표푓 푡푕푒 푚푖푥푖푛푔 푑푖푠푡푟푖푏푢푡푖표푛

And

퐵훽 푎 + 푗, 푏 − 퐵훼 푎 + 푗, 푏 퐸 푃푖 = 퐵훽 푎, 푏 −퐵훼 푎, 푏

푥 Γ 푟 + 푥 푥 퐵훽 푎 + 푟 + 푘, 푏 − 퐵훼 푎 + 푟 + 푘, 푏 푓 푥 = −1 푘 3.91 Γ 푟 푥! 푘 퐵훽 푎, 푏 −퐵훼 푎, 푏 푘=0

푓표푟 푟 > 0; 1 < 훼 < 푝 < 훽 < 1; 푎, 푏 > 0; 푥 = 0,1,2, …

Identity

We can draw an identity based on the result from explicit mixing and that from method of moments as follows.

푥 푥 퐵 푎 + 푟, 푏 + 푥 − 퐵 푎 + 푟, 푏 + 푥 = −1 푘 { 퐵 푎 + 푟 + 푘, 푏 − 퐵 푎 + 푟 + 푘, 푏 } 훽 훼 푘 훽 훼 푘=0

3.92

푓표푟 푟 > 0; 1 < 훼 < 푝 < 훽 < 1; 푎, 푏 > 0; 푥 = 0,1,2, …

3.5. Negative Binomial – Confluent Hypergeometric distribution

3.5.1. Confluent Hypergeometric Construction

Given the confluent Hypergeometric function

1 푎−1 푏 푝 (1 − 푝) −푝휇 1퐹1 푎, 푎 + 푏; −휇 = 푒 푑푝 0 퐵(푎, 푏)

Dividing both sides by1퐹1 푎, 푎 + 푏; −휇 we get

1 푝푎−1(1 − 푝)푏−1 푒−푝휇 푑푝 = 1 0 퐵(푎, 푏)1퐹1 푎, 푎 + 푏; −휇

Since the LHS of the above equation is equated to 1, then the LHS equation qualifies to be a pdf. This is the pdf of Confluent Hypergeometric and is expressed as follows.

푝푎−1 1 − 푝 푏−1 푔 푝 = 푒−푝휇 푓표푟 0 < 푝 < 1; 푎, 푏 > 0; −∞ < 휇 < ∞ (3.93) 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇

푟푒푓 푁푎푑푎푟푎푗푎푕 푎푛푑 퐾표푡푧 (2007)

Finding the jth moment of the Confluent Hypergeometric distribution

1 퐸 푃푗 = 푝푗 푔 푝 푑푝 0

1 푝푗 푝푎−1 1 − 푝 푏−1 퐸 푃푗 = 푒−푝휇 푑푝 0 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇

1 1 퐸 푃푗 = 푝푗 +푎−1 1 − 푝 푏−1푒−푝휇 푑푝 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇 0

퐵 푗 + 푎, 푏 1 푝푗 +푎−1 1 − 푝 푏−1푒−푝푥 퐸 푃푗 = 푑푝 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇 0 퐵 푗 + 푎, 푏

But

1 푝푗 +푎−1 1 − 푝 푏−1푒−푝휇 푑푝 = 1퐹1 푎 + 푗, 푎 + 푗 + 푏; −휇 0 퐵 푗 + 푎, 푏

Thus

퐵 푗 + 푎, 푏 1퐹1 푎 + 푗, 푎 + 푗 + 푏; −휇 퐸 푃푗 = (3.94) 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇

3.5.2 Confluent Hypergeometric – Negative Binomial mixing a) Explicit mixing

1 푓 푥 = 푓(푥/푝)푔 푝 푑푝 0

Where

푟 + 푥 − 1 푓 푥/푝 = 푝푟 1 − 푝 푥 0 < 푝 < 1; 푟 > 0; 푥 = 0,1,2, …. 푥

푝푎−1 1 − 푝 푏 푔 푝 = 푒−푝휇 푓표푟 0 < 푝 < 1; 푎, 푏 > 0; −∞ < 휇 < ∞ 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇

1 푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푝푎−1 1 − 푝 푏푒−푝흁 푑푝 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇 푥 0

1 푟 + 푥 − 1 1 = 푝푟+푎−1 1 − 푝 푥+푏 푒−푝흁 푑푝 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇 푥 0

퐵(푟 + 푎, 푥 + 푏) 푟 + 푥 − 1 1 푝푟+푎−1 1 − 푝 푥+푏−1푒−푝흁 = 푑푝 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇 푥 0 퐵(푟 + 푎, 푥 + 푏)

퐵(푟 + 푎, 푥 + 푏)1퐹 푎 + 푟, 푎 + 푏 + 푟 + 푥; −휇 푟 + 푥 − 1 푓 푥 = 1 (3.95) 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇 푥

For 푟, 푎, 푏 > 0; −∞ < 휇 < ∞; 푥 = 0,1,2, …

Properties of Confluent Hypergeometric – Negative Binomial Mixture

First 3 moments about the origin

푎1퐹1(푎 + 1, 푎 + 푏 + 1; −휇) 퐸 푋 = 푎 + 푏 1퐹1(푎, 푎 + 푏; −휇)

푎(푎 + 1) 1퐹1(푎 + 2, 푎 + 푏 + 2; −휇) 퐸 푋2 = (푎 + 푏)(푎 + 푏 + 1) 1퐹1(푎, 푎 + 푏; −휇)

푎 푎 + 1 푎 + 2 1퐹1 푎 + 3, 푎 + 푏 + 3; −휇 퐸 푋3 = (푎 + 푏) 푎 + 푏 + 1 푎 + 푏 + 2 1퐹1 푎, 푎 + 푏; −휇

푉푎푟 푋 = 퐸 푋2 − 퐸 푋 2

2 푎(푎 + 1)1퐹1(푎 + 2, 푎 + 푏 + 2; −휇) 푎1퐹1(푎 + 1, 푎 + 푏 + 1; −휇) 푉푎푟 푋 = – (푎 + 푏)(푎 + 푏 + 1)1퐹1(푎, 푎 + 푏; −휇) 푎 + 푏 1퐹1(푎, 푎 + 푏; −휇)

Ref; A paper by E.Gomez; J.M. Perez – Sanchez; F.J Vazquez – Polo and A. Hernandez – Bastida

b) Mixing using method of moments

푥 Γ 푟 + 푥 푥 푓 푥 = −1 푘 퐸(푝 푟+푘 ) Γ 푟 푥! 푘 푘=0

퐸 푝푟+푘 푖푠 푡푕푒 푚표푚푒푛푡 표푓 표푟푑푒푟 푟 + 푘 푎푏표푢푡 푡푕푒 표푟푖푔푖푛 표푓 푡푕푒 푚푖푥푖푛푔 푑푖푠푡푟푖푏푢푡푖표푛

퐵 푗 + 푎, 푏 1퐹1 푎 + 푗, 푎 + 푗 + 푏; −휇 퐸 푃푗 = 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇

Therefore

푥 Γ 푟 + 푥 푥 퐵 푟 + 푘 + 푎, 푏 1퐹1 푎 + 푟 + 푘, 푎 + 푟 + 푘 + 푏; −휇 푓 푥 = −1 푘 (3.96) Γ 푟 푥! 푘 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇 푘=0

푓표푟 푥 = 0,1,2, … ; 푎, 푏 > 표 ; 푟 > 0

Identity

We can draw an identity based on the result from explicit mixing and that from method of moments as follows.

푥 푥 퐵 푟 + 푎, 푥 + 푏 퐹 푎 + 푟, 푎 + 푏 + 푟 + 푥; −휇 = −1 푘 퐵 푟 + 푘 + 푎, 푏 퐹 푎 + 푟 + 푘, 푎 + 푟 + 푘 + 푏; −휇 1 1 푘 1 1 푘=0

푓표푟 푥 = 0,1,2, … ; 푎, 푏 > 표 ; 푟 > 0 3.97

Applications of Confluent Hypergeometric – Negative Binomial mixture

Confluent Hypergeometric – Negative Binomial mixture is a common distribution used in insurance to a variety of claims

E.Gomez; J.M. Perez – Sanchez; F.J Vazquez – Polo and A. Hernandez – Bastida found out that when you calculate the expected frequencies for automobile insurance claims and using the Confluent Hypergeometric – Negative Binomial mixture then the outcome came out satisfactorily fit.

The mixture was found to be a better alternative to the standard Negative Binomial distribution and other mixtures.

Suppose the number of claims in a portfolio of policies in a time period is denoted by N

Let 푋푖, 푖 = 1,2,3, , … be the amount of the ith claim

푆 = 푋1 + 푋2 + 푋3 + ⋯ … . . +푋푁 will be the aggregate or total claims generated by the potfolio.

Note

1. The random variables 푋푖 , 푖 = 1,2,3, , … 푁 are i.i.d with a CDF F(x) and pdf f(x) 2. The random variable N; 푋1, 푋2, 푋3 are mutually independent

Suppose we are using the Confluent Hypergeometric – Negative Binomial mixture model for N, the CDF of the distribution of total claims become

∞ ∗푘 퐹푠 푋 = 퐹 (X)Pr⁡(푁 = 푘) 푘=0

Where

퐹∗푘 is the kth fold convolution of F and Pr⁡(푁 = 푘) is defined above

3.6. Gauss Hypergeometric – Negative Binomial Distribution

3.6.1. Gauss Hypergeometric distribution Construction

Given the Gauss Hypergeometric function

1 푡푎−1 1 − 푡 푏−1 2퐹1 푎, 휀, 푎 + 푏; −푧 = 휀 0 퐵 푎, 푏 1 + 푧푡

푓표푟 0 < 푡 < 1; 푎, 푏 > 0; −∞ < 휀 < ∞

Divide both sides by2퐹1 푎, 휀, 푎 + 푏; −푧 to get

1 푡푎−1 1 − 푡 푏−1 휀 = 1 0 퐵 푎, 푏 1 + 푧푡 2퐹1 푎, 휀, 푎 + 푏; −푧

This forms the pdf of gauss Hypergeometric given by

푡푎−1 1 − 푡 푏−1 푔 푡 = 휀 3.98 퐵 푎, 푏 1 + 푧푡 2퐹1 푎, 휀, 푎 + 푏; −푧

푓표푟 0 < 푡 < 1; 푎, 푏 > 0; −∞ < 휀 < ∞

(see Armero and Bayarri (1994))

Finding the jth moment about the origin

1 푗 +푎−1 푏−1 푗 1 푡 1 − 푡 퐸 푇 = 휀 푑푡 퐵 푎, 푏 2퐹1 푎, 휀, 푎 + 푏; −푧 0 1 + 푧푡

퐵 푗 + 푎, 푏 1 푡푗 +푎−1 1 − 푡 푏−1 = 휀 푑푡 퐵 푎, 푏 2퐹1 푎, 휀, 푎 + 푏; −푧 0 퐵 푗 + 푎, 푏 1 + 푧푡

퐵 푗 + 푎, 푏 2퐹1 푗 + 푎, 휀, 푎 + 푏 + 푗; −푧 퐸 푇푗 = (3.99) 퐵 푎, 푏 2퐹1 푎, 휀, 푎 + 푏; −푧

3.6.2. Gauss Hypergeometric – Negative Binomial mixture a) Explicit mixing

∞ 푟 + 푥 − 1 푓 푥 = 푝푟 (1 − 푝)푥 푔(푝)푑푝 0 푥

Where g(p) is the pdf of the Gauss Hypergeometric distribution

푝푎−1 1 − 푝 푏−1 푔 푝 = 휀 퐵 푎, 푏 1 + 푧푡 2퐹1 푎, 휀, 푎 + 푏; −푧

푟 + 푥 − 1 1 ∞ 푝푟 (1 − 푝)푥 푝푎−1 1 − 푝 푏−1 푓 푥 = 휀 푑푝 푥 퐵 푎, 푏 2퐹1 푎, 휀, 푎 + 푏; −푧 0 1 + 푧푡

푟 + 푥 − 1 퐵(푎 + 푟, 푏 + 푥) ∞ 푝푟+푎−1(1 − 푝)푥+푏−1 = 휀 푑푝 푥 퐵 푎, 푏 2퐹1 푎, 휀, 푎 + 푏; −푧 0 퐵(푎 + 푟, 푏 + 푥) 1 + 푧푡

푟 + 푥 − 1 퐵 푎 + 푟, 푏 + 푥 2퐹1 푎 + 푟, 휀; 푎 + 푏 + 푟 + 푥; −푧 푓 푥 = (3.100) 푥 퐵 푎, 푏 2퐹1 푎, 휀, 푎 + 푏; −푧

For 푥 = 0,1,2 … ; 푟 = 0,1,2 … . ; 푎, 푏 > 0; −∞ < 휀 < ∞

b) Using method of moments

푥 Γ 푟 + 푥 푥 푓 푥 = −1 푘 퐸(푝 푟+푘 ) Γ 푟 푥! 푘 푘=0

퐸 푝푟+푘 푖푠 푡푕푒 푚표푚푒푛푡 표푓 표푟푑푒푟 푟 + 푘 푎푏표푢푡 푡푕푒 표푟푖푔푖푛 표푓 푡푕푒 푚푖푥푖푛푔 푑푖푠푡푟푖푏푢푡푖표푛

And the jth moment for the Gauss Hypergeometric distribution is given by

퐵 푗 + 푎, 푏 2퐹1 푗 + 푎, 휀, 푎 + 푏 + 푗; −푧 퐸(푃푗 ) = 퐵 푎, 푏 2퐹1 푎, 휀, 푎 + 푏; −푧

Hence

푥 Γ 푟 + 푥 푥 퐵 푗 + 푎, 푏 2퐹1 푟 + 푘 + 푎, 휀, 푎 + 푏 + 푟 + 푘; −푧 푓 푥 = −1 푘 (3.101) Γ 푟 푥! 푘 퐵 푎, 푏 2퐹1 푎, 휀, 푎 + 푏; −푧 푘=0

Identity

We can draw an identity based on the result from explicit mixing and that from method of moments as follows.

푥 푥 퐵 푎 + 푟, 푏 + 푥 퐹 푎 + 푟, 휀; 푎 + 푏 + 푟 + 푥; −푧 = −1 푘 퐵 푗 + 푎, 푏 퐹 푟 + 푘 + 푎, 휀, 푎 + 푏 + 푟 + 푘; −푧 2 1 푘 2 1 푘=0

For 푥 = 0,1,2 … ; 푟 = 0,1,2 … . ; 푎, 푏 > 0; −∞ < 휀 < ∞ 3.102

CHAPTER 4

NEGATIVE BINOMIAL MIXTURES BASED ON DISTRIBUTIONS BEYOND BETA 4.1 Introduction

All the distributions we are going to consider in this category are within the [0,1] domain and are not based on beta distribution. Some of them are listed below;

1. Kumaraswamy (I) and (II) distribution 2. Triangular distribution 3. Truncated Exponential distribution 4. Truncated Gamma distribution 5. Minus Log distribution 6. Two – Sided Ogive distribution 7. Ogive distribution 8. Two – sided power distribution

When carrying out the mixing we will use the following methods

a. Moments method b. Direct integration and substitution also referred to as explicit mixing c. Iteration method(where applicable)

4.2 Negative Binomial – Kumaraswamy (I) Distribution

4.2.1 Kumaraswamy (I) Distribution Construction

Given the Kumaraswamy (II) Distribution with a pdf

푔 푝 = 푎푏 1 − 푝푎 푏−1푝푎−1 0 < 푝 < 1; 푎, 푏 > 0 (4.01)

1 Let 푝 = 푈 푎 0 < 푢 < 1; 푎 > 0

Then

1 1− 푔 푢 = 푎푏 1 − 푢 푏−1푢 푎 퐽

But

1 −1 푑푝 푢푎 퐽 = = 푑푢 푎

Substituting 퐽

1 −1 1 푎 1− 푢 푔 푢 = 푎푏 1 − 푢 푏−1푢 푎 푎 Therefore

푏 1 − 푢 푏−1 푓표푟 0 < 푢 < 1; 푏 > 0 푔 푢 = (4.02) 0 푒푙푠푒푤푕푒푟푒 This is the pdf of Kumaraswamy (I) distribution with parameter 푏

It is always denoted by Kw (I) distribution

Properties of Kw (I) distribution

The jth moment about the origin of the Kw (I) distribution is given below

1 퐸 푈푗 = 푏 푢푗 1 − 푢 푏−1푑푝 0

퐸 푈푗 = 푏퐵 푗 + 1, 푏 (4.03)

1 퐸 푈 = 푏퐵 2, 푏 = (4.04) 푏 + 1 푏 + 2

2 푣푎푟 푈 = 퐸 푈2 − 퐸 푈

2 1 = − 푏 + 3 푏 + 2 푏 + 1 1 + 푏 2 2 + 푏 2

2푏2 + 5푏 + 1 푣푎푟 푈 = (4.05) 1 + 푏 2 2 + 푏 2 3 + 푏

4.2.2 Negative Binomial – Kumaraswamy (I) distribution mixing 4.2.2.1 Explicit mixing

This refers to the mixing on the direct substitution basis

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔 푝 푑푝 푥 0

Such that 푔(푝) is he pdf of Kw (I) distribution

푏 1 − 푝 푏−1 푓표푟 0 < 푝 < 1; 푏 > 0 푔 푝 = 0 푒푙푠푒푤푕푒푟푒

Replacing 푔(푝)

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푏 1 − 푝 푏−1푑푝 푥 0

푟 + 푥 − 1 1 푓(푥) = 푏 푝푟 1 − 푝 푥+푏−1 푑푝 푥 0 푟 + 푥 − 1 푓 푥 = 푏퐵(푟 + 1, 푥 + 푏) 푥

푟+푥−1 푓 푥 = 푥 푏퐵 푟 + 1, 푥 + 푏 푥 = 0,1,2, … ; 푏, 푟 > 0 (4.06)

See Li Xiaohuet al (2011) on Binomial mixture

4.2.2.2 Mixing using the moments method

푥 Γ 푟 + 푥 푥 푃푟표푏 푋 = 푥 = −1 푘 퐸(푝 푟+푘 ) Γ r 푥! 푘 푘=0

퐸 푝푟+푘 푖푠 푡푕푒 푚표푚푒푛푡 표푓 표푟푑푒푟 푟 + 푘 푎푏표푢푡 푡푕푒 표푟푖푔푖푛 표푓 푡푕푒 푚푖푥푖푛푔 푑푖푠푡푟푖푏푢푡푖표푛

푥 Γ 푟 + 푥 푥 푃푟표푏 푋 = 푥 = 푏 −1 푘 퐵 푟 + 푘 + 1, 푏 4.07 Γ r 푥! 푘 푘=0

푓표푟 푟, 푏 > 0; 푥 = 0,1,2, …

Identity

We can draw an identity based on the result from explicit mixing and that from method of moments as follows.

푥 푥 퐵 푟 + 1, 푥 + 푏 = −1 푘 퐵 푟 + 푘 + 1, 푏 푘 푘=0

푓표푟 푟, 푏 > 0; 푥 = 0,1,2,3, …

4.3 Negative Binomial – Truncated Exponential Distribution

4.3.1 Truncated Exponential Distribution (TEX(흀, 풃)) Construction

Let 푌 be a one sided truncated Exxponential randon variable, the pdf of 푌can be evaluated as follows

푏 푒−푦/휆 푑푦 = [1 − 푒−푏/휆 ] 0 휆

Dividing both sides by [1 − 푒−푏/휆]

1 푏 푒−푦/휆 휆 −푏/휆 푑푦 = 1 0 [1 − 푒 ]

Let 푦 = 푝푏

푑푦 = 푏푑푝

푏 1 푒−푏푝/휆 휆 −푏/휆 푑푦 = 1 0 [1 − 푒 ]

This is a pdf which can be expressed as

푏푝 푏 − 푒 휆 휆 푏 0 < 푝 < 1; 푏, 휆 > 0 푔 푝 = − (4.08) 1 − 푒 휆

0 푒푙푠푒푤푕푒푟푒

This the Truncated Exponential distribution with parameters 휆 푎푛푑 푏

4.3.2 Negative Binomial – Truncated Exponential distribution mixing 4.3.2.1 Explicit mixing

This is direct substitution and integration expressed as follows

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔 푝 푑푝 푥 0

푔 푝 is the pdf of truncated exponential distribution with parameters 휆 푎푛푑 푏

푏푝 푏 − 1 푒 휆 푟 + 푥 − 1 푟 푥 휆 푓 푥 = 푝 1 − 푝 푏 푑푝 푥 − 0 1 − 푒 휆

1 푏푝 푟 + 푥 − 1 푏 − 푟 푥 휆 푓 푥 = 푏 푝 1 − 푝 푒 푑푝 (4.09) 푥 − 휆 1 − 푒 휆 0

Consider a Confluent Hypergeometric function

1 푎−1 푏−1 −푝푥 1퐹1 푎, 푎 + 푏; −푥 퐵 푎, 푏 = 푝 1 − 푝 푒 푑푝 0

Therefore

1 푏푝 푟 푥 − 푏 푝 1 − 푝 푒 휆 푑푝 =1 퐹1 푟 + 1, 푟 + 푥 + 2; − 퐵 푟 + 1, 푥 + 1 (4.10) 0 휆

Thus equation 4.09 results to

푏 푏 퐹 푟 + 1, 푟 + 푥 + 2; − 퐵 푟 + 1, 푥 + 1 푟 + 푥 − 1 1 1 휆 푓(푥) = 푏 푥 − 휆 1 − 푒 휆

푏 푏 퐹 푟 + 1, 푟 + 푥 + 2; − 1 1 휆 (푟 + 푥 − 1)! 푟 푟 − 1 ! 푥! 푓(푥) = 푏 − 푟 − 1 ! 푥! (푟 + 푥 + 1) 푟 + 푥 푟 + 푥 − 1 ! 휆 1 − 푒 휆

푏 푏 퐹 푟 + 1, 푟 + 푥 + 2; − 1 1 휆 푟 푓 푥 = 푏 푓표푟 푥 = 0,1,2, . . ; 푏, 푟, 휆 > 0 (4.11) − 푟 + 푥 + 1 푟 + 푥 휆 1 − 푒 휆

This the mixture distribution of Negative Binomial distribution and Truncated Exponential distribution in its simplest form

4.4 Negative Binomial – Truncated Gamma Distribution

4.4.1 Truncated Gamma Distribution Construction

Consider an incomplete Gamma function given by

푎 훾 푏, 푎 = 푡푏−1푒−푡푑푡 0

Divide both sides by 훾(푎, 푏)

푎 푡푏−1푒−푡 푑푡 = 1 0 훾 푎, 푏

Then let 푡 = 푎푝 푤푕푖푐푕 푖푚푝푙푖푒푠 푑푡 = 푎 푑푝

Therefore

1 푎푏 푝푏−1푒−푎푝 푑푝 = 1 0 훾 푎, 푏

It forms a pdf which can be expressed as

푎푏 푝푏−1푒−푎푝 0 < 푝 < 1; 푎, 푏 > 0 푔 푝 = 훾 푎, 푏 (4.12) 0 푒푙푠푒푤푕푒푟푒 ith moment

1 푎푏 푝푏+푗 −1푒−푎푝 퐸 푃푗 = 푑푝 0 훾 푎, 푏

푏 1 푏+푗 푏+푗 −1 −푎푝 푗 푎 훾 푎, 푏 + 푗 푎 푝 푒 퐸 푃 = 푏+푗 푑푝 푎 훾 푎, 푏 0 훾 푎, 푏 + 푗

푎푏 훾 푎, 푏 + 푗 퐸 푃푗 = 푎푏+푗 훾 푎, 푏

훾 푎, 푏 + 푗 퐸 푃푗 = 푎푗 훾 푎, 푏

4.4.2 Negative Binomial – Truncated Gamma Distribution mixing 4.4.2.1 Explicit mixing

By substituting 푔(푝) and integrating the integral

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔 푝 푑푝 푥 0

Where 푔(푝) is he pdf of truncated gamma distribution 푟 + 푥 − 1 1 푎푏 푝푏−1푒−푎푝 푓 푥 = 푝푟 1 − 푝 푥 푑푝 푥 0 훾 푎, 푏

푟 + 푥 − 1 푎푏 1 푓 푥 = 푝푟 1 − 푝 푥 푝푏−1푒−푎푝 푑푝 푥 훾 푎, 푏 0

푟 + 푥 − 1 푎푏 1 푓 푥 = 푝푟+푏−1 1 − 푝 푥 푒−푎푝 푑푝 (4.13) 푥 훾 푎, 푏 0

Consider a Confluent Hypergeometric Function

1 푎−1 푏−1 −푝푥 1퐹1 푎, 푎 + 푏; −푥 퐵 푎, 푏 = 푝 1 − 푝 푒 푑푝 0

Therefore

1 푟+푏−1 푥 −푎푝 푝 1 − 푝 푒 푑푝 =1 퐹1 푟 + 푏, 푟 + 푏 + 푥 + 1; −푎 퐵 푟 + 푏, 푥 + 1 0

Thus

푏 푟 + 푥 − 1 푎 1퐹1 푟 + 푏, 푟 + 푏 + 푥 + 1; −푎 퐵 푟 + 푏, 푥 + 1 푓 푥 = (4.14) 푥 훾 푎, 푏

푓표푟 푥 = 0,1,2, … ; 푟, 푎, 푏 > 0

(see Bhattacharya (1968) on Binomial mixture)

4.4.2.2. Using the method of moments

푥 Γ 푟 + 푥 푥 푓 푥 = −1 푘 퐸(푝 푟+푘 ) Γ 푟 푥! 푘 푘=0

퐸 푝푟+푘 푖푠 푡푕푒 푚표푚푒푛푡 표푓 표푟푑푒푟 푟 + 푘 푎푏표푢푡 푡푕푒 표푟푖푔푖푛 표푓 푡푕푒 푚푖푥푖푛푔 푑푖푠푡푟푖푏푢푡푖표푛

푥 Γ 푟 + 푥 푥 훾 푎, 푏 + 푟 + 푘 푓(푥) = −1 푘 Γ 푟 푥! 푘 푎 푟+푘 훾 푎, 푏 푘=0

푓표푟 푥 = 0,1,2, … ; 푟, 푎, 푏 > 0

Identity

We can draw an identity based on the result from explicit mixing and that from method of moments as follows.

푥 푥 훾 푎, 푏 + 푟 + 푘 푎푏 퐹 푟 + 푏, 푟 + 푏 + 푥 + 1; −푎 퐵 푟 + 푏, 푥 + 1 = −1 푘 1 1 푘 푎 푟+푘 푘=0

푓표푟 푥 = 0,1,2, … ; 푟, 푎, 푏 > 0

4.5. Negative Binomial – Minus Log Distribution

4.5.1. Minus Log Distribution Construction

Let the Random variable X have the uniform pdf 푈[0,1], Let [푥1, 푥2]denote a random sample from the distribution.

The joint pdf 푥1 푎푛푑푥2is then

푓(푥 )푓 푥 0 < 푥 < 1; 0 < 푥 < 1 푕(푥 푥 ) = 1 2 1 2 1 2 0 푒푙푠푒푤푕푒푟푒

Consider the two random variables 푃 = 푥1푥2 푎푛푑 푌 = 푥2

The joint pdf of 푃 푎푛푑 푌 is

푔 푝, 푦 = 1 퐽

Where

푑푥1 푑푥1 푑푝 푑푦 퐽 = 푑푥2 푑푥2 푑푝 푑푦

1 푝 − 1 퐽 = 푦 푦2 = 푦 0 1

1 Thus 푔 푝, 푦 = 0 < 푝 < 푦 < 1 푦

The marginal pdf of p is

1 1 푔 푝 = 푑푦 푝 푦

1 푔 푝 = log 푦 푝 = 0 − log 푝

푔 푝 = − log 푝 0 < 푝 < 1 (4.16)

This is the pdf of minus log distribution

Properties of Minus log distribution

The jth moment of the minus log distribution is given below

1 퐸 푃푗 = 푃푗 (− log 푝) 푑푝 0

Let 푎 = − log 푝 푝 = 푒−푎 푑푝 = −푒−푎 푑푎

1 퐸 푃푗 = − 푎푒−푎 푗 +1 푑푎 0

Using the minus sign outside the integral to swap the limits, we will have.

∞ 퐸 푃푗 = 푎푒−푎 푗 +1 푑푎 0

1 퐸 푃푗 = (4.17) 푗 + 1 2

1 퐸 푃 = (4.18) 4 7 푣푎푟 푃 = (4.19) 144

4.5.2Negative Binomial – Minus Log Distribution mixing 4.5.2.1 Explicit mixing

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔 푝 푑푝 푥 0

푔(푝) is he pdf of minus log distribution.

Thus

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 − log 푝 푑푝 (4.20) 푥 0

Let 푎 = − log 푝 푝 = 푒−푎 푑푝 = −푒−푎

We have

푟 + 푥 − 1 1 푓 푥 = − 푎푒−푟푎 1 − 푒−푎 푥 푒−푎 푑푎 푥 0

푥 푟 + 푥 − 1 푥 1 푓(푥) = −1 푘 − 푎푒−푎푘 푒−푟푎 푒−푎 푑푎 푥 푘 푘=0 0

푥 푟 + 푥 − 1 푥 1 푓 푥 = −1 푘 − 푎푒−푎 푟+푘+1 푑푎 푥 푘 푘=0 0

Using the minus sign outside the integral to swap the limits, we will have.

푥 푟 + 푥 − 1 푥 ∞ 푓 푥 = −1 푘 푎푒−푎 푟+푘+1 푑푎 푥 푘 푘=0 0

푥 푟 + 푥 − 1 푥 1 푓 푥 = −1 푘 푥 푘 푟 + 푘 + 1 2 푘=0

The mixture can be expressed as

푥 푟 + 푥 − 1 푥 1 푓 푥 = −1 푘 푥 = 0,1,2, … 푟 > 0 (4.21) 푥 푘 푟 + 푘 + 1 2 푘=0

4.5.2.1 Mixing using method of moments

푥 Γ 푟 + 푥 푥 푓 푥 = −1 푘 퐸(푝 푟+푘 ) Γ 푟 푥 ! 푘 푘=0

퐸 푝푟+푘 푖푠 푡푕푒 푚표푚푒푛푡 표푓 표푟푑푒푟 푟 + 푘 푎푏표푢푡 푡푕푒 표푟푖푔푖푛 표푓 푡푕푒 푚푖푥푖푛푔 푑푖푠푡푟푖푏푢푡푖표푛

푥 Γ 푟 + 푥 푥 1 푓 푥 = −1 푘 (4.22) Γ 푟 푥 ! 푘 푟 + 푘 + 1 2 푘=0

푓표푟 푥 = 0,1,2, … 푟 > 0

4.6 Negative Binomial – Standard Two – Sided Power Distribution

4.6.1 Standard Two – Sided Power Distribution Construction

The standard II sided power distribution can be viewed as a particular case of the general two sided continuous family with support [0, 1] given by the density below 푝 푕 /휑 0 < 푝 < 휃 휃 푔 푝/휃, 푕(./휑) = 1 − 푝 (4.23) 푕 /휑 휃 < 푝 < 1 1 − 휃

Where 푕(./휑) is an appropriately selected continuous pdf on [0, 1] with parameter(s) φ

푕(./휑) is called a general density such that

푕 푦 = 푘푦푘−1 0 < 푦 < 1 ; 푘 > 0 which is a power function distribution

Then

푝 푘−1 푘 0 < 푝 < 휃 휃 푔 푝 = 푘−1 (4.24) 1 − 푝 푘 휃 < 푝 < 1 1 − 휃

And this is the pdf of a two – sided power distribution with parameters 푘 푎푛푑 휃

4.6.2 Negative Binomial – Standard Two – Sided Power Distribution 4.6.2.1 Explicit mixing

This entails direct substitution into the following equation

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔 푝 푑푝 푥 0

Where 푔(푝) is he pdf of Standard Two – Sided Power Distribution

휃 1 푟 + 푥 − 1 1 푟+푘−1 푥 1 푟 푥+푘−1 푓 푥 = 푘 푘−1 푝 1 − 푝 푑푝 + 푘−1 푝 1 − 푝 푑푝 푥 휃 0 1 − 휃 휃

푟 + 푥 − 1 퐵휃 푟 + 푘, 푥 + 1 퐵 푟 + 1, 푥 + 푘 − 퐵휃 푟 + 1, 푥 + 푘 푓 푥 = 푘 + (4.25) 푥 휃푘−1 1 − 휃 푘−1

This is the mixture between Negative Binomial distribution and the standard two – side power distribution.

4.7. Negative Binomial – Ogive Distribution

4.7.1. Ogive Distribution Construction

The general form of an Ogive distribution is given by

2푚(푚 + 1) 1 − 푚2 푔 푝 = 푝(푚−1)/2 + 푝푚 0 < 푝 > 1 ; 푚 > 0 (4.26) 3푚 + 1 3푚 + 1

From (퐷표푟푝 푎푛푑 퐾표푡푧 (2003))

4.7.2. Negative Binomial – Ogive Distribution mixing 4.7.2.1. Explicit mixing

This entails direct substitution into the following equation

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔 푝 푑푝 푥 0

Where 푔(푝) is he pdf of the Ogive distribution

푟 + 푥 − 1 1 2푚 푚 + 1 푚 −1 1 − 푚2 푓 푥 = 푝푟 1 − 푝 푥 푝 2 + 푝푚 푑푝 (4.26) 푥 0 3푚 + 1 3푚 + 1

1 푚 −1 1 푟 + 푥 − 1 푚 + 1 푟+ 푓(푥) = 2푚 푝 2 1 − 푝 푥 푑푝 + 1 − 푚 푝푟+푚 1 − 푝 푥 푑푝 푥 3푚 + 1 0 0

푟 + 푥 − 1 푚 + 1 1 2푟+푚 −1 1 = 2푚 푝 2 1 − 푝 푥 푑푝 + 1 − 푚 푝푟+푚 1 − 푝 푥 푑푝 푥 3푚 + 1 0 0

푟 + 푥 − 1 푚 + 1 2푟 + 푚 − 1 푓 푥 = 2푚퐵 + 1, 푥 + 1 + 1 − 푚 퐵 푟 + 푚 + 1, 푥 + 1 (4.27) 푥 3푚 + 1 2

푓표푟 푥 = 0,1,2, … . ; 푟 > 0; 푚 > 0

4.8. Negative Binomial – Standard Two – Sided Ogive Distribution

4.8.1. Standard Two – Sided Ogive Distribution Construction

The two sided ogive distribution can be viewed as a particular case of the general two sided continuous family with [0, 1] given by the density 푝 푕 /휑 0 < 푝 < 휃 휃 푔 푝/휃, 푕(./휑) = 1 − 푝 (4.28) 푕 /휑 휃 < 푝 < 1 1 − 휃

Where 푕(./휑) is an appropriately selected continuous pdf on [0, 1] with parameter(s) φ

푕(./휑) is called a general density such that

2푚 푚 + 1 푚 −1 1 − 푚2 푧 푦 = 푦 2 + 푦푚 (4.29) 3푚 + 1 3푚 + 1

0 < 푦 < 1; 푚 > 0

This is an Ogive distribution and so

푚 −1 2 푚 2푚 푚 + 1 푝 2 1 − 푚 푝 + 0 < 푝 < 휃 ; 푚 > 0 3푚 + 1 휃 3푚 + 1 휃 푔 푝 = 푚 −1 (4.30) 2 푚 2푚 푚 + 1 1 − 푝 2 1 − 푚 1 − 푝 + 휃 < 푝 < 1 ; 푚 > 0 3푚 + 1 1 − 휃 3푚 + 1 1 − 휃

And this is the pdf of the two sided Ogive Distribution with parameters 푚 푎푛푑 휃

When 푝 = 휃 then the two – sided Ogive distribution is smooth and this is the reflection point.

Although this contradicts the situation at the reflection point of the two – sided power function

4.8.2. Negative Binomial – Standard Two – Sided Ogive Distribution mixing This entails direct substitution into the following equation

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔 푝 푑푝 푥 0

푔(푝) is the pdf of the Standard Two – Sided Ogive Distribution

푚 −1 휃 1 2 푚 푟 + 푥 − 1 2푚 푚 + 1 푝 2 1 − 푚 푝 푓 푥 = 푝푟 1 − 푝 푥 + 푥 0 3푚 + 1 휃 3푚 + 1 휃 0 푚 −1 1 2 푚 2푚 푚 + 1 1 − 푝 2 1 − 푚 1 − 푝 + + 푑푝 3푚 + 1 1 − 휃 3푚 + 1 1 − 휃 휃

푟 + 푥 − 1 푚 + 1 2푚 휃 2푟+푚 −1 1 − 푚 휃 푓 푥 = 푝 2 1 − 푝 푥 푑푝 + 푝푟−푚 1 − 푝 푥 푑푝 푥 3푚 + 1 푚 −1 휃푚 휃 2 0 0 2푚 1 2푥+푚 −1 1 − 푚 1 + 푝푟 1 − 푝 2 푑푝 + 푝푟 1 − 푝 푥+푚 푑푝 푚 −1 1 − 휃 푚 1 − 휃 2 휃 휃

푟 + 푥 − 1 푚 + 1 2푚 2푟 + 푚 − 1 푓 푥 = 퐵 + 1, 푥 + 1 푥 3푚 + 1 푚 −1 휃 2 휃 2 1 − 푚 + 퐵 푟 − 푚 + 1, 푥 + 1 휃푚 휃 2푚 2푥 + 푚 − 1 2푥 + 푚 − 1 + 퐵 푟 + 1, + 1 − 퐵 (푟 + 1, 푚 −1 2 휃 2 1 − 휃 2 1 − 푚 + 1) 퐵 푟 + 1, 푥 + 푚 + 1 − 퐵 (푟 + 1, 푥 + 푚 + 1) (4.31) 1 − 휃 푚 휃

푓표푟 푥 = 0,1,2, …. ; 푚 > 0 ; 푟 > 0

4.9. Negative Binomial – Triangular Distribution

4.9.1. Triangular Distribution Construction

0 θ 1

Diagram 1

The Triangular distribution 푇(0,1, 휃) arise from the conjuction of two lines which share a common vertex.

The density of the triangular distribution is defined by

푔1 푝 푖푓 0 < 푝 < 휃

푔 푝 = 푔2 푝 푖푓 휃 < 푝 < 1 (4.32) 0 푒푙푠푒 푤푕푒푟푒

When

푔1(푝) is the equation of the line 0,0 , 휃, 2 computed as

푔 푝 2 1 = 푝 휃

Therefore

2푝 푔 푝 = (4.32푖) 1 휃

푔2(푝) is the equation of the line 휃, 2 , (1, 0) computed as

푔 푝 −2 2 = 푝 − 1 1 − 휃

Therefore

2(1 − 푝) 푔 푝 = (4.32푖푖) 2 1 − 휃

Thus the density of the triangular distribution becomes

2푝 0 < 푝 < 휃 휃 푔 푝 = 2(1 − 푝) (4.33) 휃 < 푝 < 1 1 − 휃 0 푒푙푠푒푤푕푒푟푒

Ref (Kotz S et al (2004) - Beyond Beta – Other Continuous families of Distributions with Bounded Support and applications page 1 – 31)

Properties of Triangular distribution

The moment of order 푗 about the origin of triangular distribution is worked out below

휃 2푝 1 2 1 − 푝 퐸 푃푗 = 푃푗 푑푝 + 푃푗 푑푝 0 휃 휃 1 − 휃

휃 1 2푝푗 +2 2 푝푗 +1 푝푗 +2 퐸 푃푗 = + − 휃 푗 + 2 1 − 휃 푗 + 1 푗 + 2 0 휃

2휃푗 +2 2 1 1 휃푗 +1 휃푗 +2 퐸 푃푗 = + − − + 휃 푗 + 2 1 − 휃 푗 + 1 푗 + 2 푗 + 1 푗 + 2

2휃푗 +2 2 1 푗 + 1 휃푗 +2 − 푗 + 2 휃푗 +1 퐸 푃푗 = + + 휃 푗 + 2 1 − 휃 푗 + 1 푗 + 2 푗 + 1 푗 + 2

2 휃푗 +2 1 − 휃 1 푗 + 1 휃푗 +2 − 푗 + 2 휃푗 +1 퐸 푃푗 = + + 1 − 휃 휃 푗 + 2 푗 + 1 푗 + 2 푗 + 1 푗 + 2

2 휃푗 +1 1 − 휃 1 푗 + 1 휃푗 +2 − 푗 + 2 휃푗 +1 퐸 푃푗 = + + 1 − 휃 푗 + 2 푗 + 1 푗 + 2 푗 + 1 푗 + 2

2 푗 + 1 휃푗 +1 1 − 휃 + 1 + 푗 + 1 휃푗 +2 − 푗 + 2 휃푗 +1 퐸 푃푗 = 1 − 휃 푗 + 1 푗 + 2

2 푗 + 1 휃푗 +1 − 푗 + 1 휃푗 +2 + 1 + 푗 + 1 휃푗 +2 − 푗 + 2 휃푗 +1 = 1 − 휃 푗 + 1 푗 + 2

2 1 − 휃푗 +1 퐸 푃푗 = (4.34) 1 − 휃 푗 + 1 푗 + 2

Mean is therefore

1 + 휃 퐸 푃 = (4.35) 3

Variance od 푝 is given by

2 푉푎푟 푃 = 퐸 푃2 − 퐸 푃

휃2 − 휃 + 1 푉푎푟 푃 = (4.36) 18

4.9.2. Negative Binomial – Triangular distribution mixing 4.9.2.1. Explicit mixing

This entails direct substitution into the following equation

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔 푝 푑푝 푥 0

Where 푔(푝) is he pdf of the Triangular distribution

푟 + 푥 − 1 2 휃 2 1 푓 푥 = 푝푟+1(1 − 푝)푥 푑푝 + 푝푟 1 − 푝 푥+1푑푝 (4.37) 푥 휃 0 1 − 휃 휃

푟 + 푥 − 1 2 2 푓(푥) = 퐵 푟 + 2, 푥 + 1 + 퐵 푟 + 1, 푥 + 2 − 퐵 푟 + 1, 푥 + 2 푥 휃 휃 1 − 휃 휃

2ᴦ 푟 + 푥 퐵휃 푟 + 2, 푥 + 1 퐵 푟 + 1, 푥 + 2 − 퐵휃 푟 + 1, 푥 + 2 푓 푥 = + (4.38) 푥! Γ푟 휃 1 − 휃

푓표푟 푥 = 0,1,2, … … ; 0 < 휃 < 1, 푟 > 0

And this is the mixture of Triangular distribution and Negative Binomial distribution expressed explicitly

4.9.2.2. Mixing using the method of moments

Here we have two ways of carrying out mixing using this method

Case 1

푟 + 푥 − 1 1 푓 푥 = 푝푟 1 − 푝 푥 푔 푝 푑푝 푥 0

Where 푔(푝) is he pdf of the triangular distribution

But we know that

푥 푥 1 − 푝 푥 = −1 푥−푘 푝푥−푘 푘 푘=0

Therefore

푥 푟 + 푥 − 1 1 푥 푓 푥 = 푝푟 −1 푥−푘 푝푥−푘 푔 푝 푑푝 푥 푘 0 푘=0

푥 푟 + 푥 − 1 푥 1 푓 푥 = −1 푥−푘 푝푟+푥−푘 푔 푝 푑푝 (4.39) 푥 푘 푘=0 0

푥 푟 + 푥 − 1 푥 푓 푥 −1 푥−푘 퐸 푝푟+푥−푘 (4.40) 푥 푘 푘=0

푥 푟 + 푥 − 1 푥 2 1 − 휃푟+푥−푘+1 푓 푥 = −1 푥−푘 (4.41) 푥 푘 푟 + 푥 − 푘 + 1 푟 + 푥 − 푘 + 2 1 − 휃 푘=0

Case 2

From the formula

푥 푟 + 푥 − 1 ! 푓 푥 = −1 푘 퐸(푝 푟+푘 ) 푟 − 1 ! 푥 − 푘 ! 푘! 푘=0

퐸 푝푟+푘 푖푠 푡푕푒 푚표푚푒푛푡 표푓 표푟푑푒푟 푟 + 푘 푎푏표푢푡 푡푕푒 표푟푖푔푖푛 표푓 푡푕푒 푚푖푥푖푛푔 푑푖푠푡푟푖푏푢푡푖표푛

푥 Γ 푟 + 푥 푥 2 1 − 휃푟+푘+1 푓 푥 = −1 푘 (4.42) Γ 푟 ! 푥 ! 푘 1 − 휃 푟 + 푘 + 1 푟 + 푘 + 2 푘=0

Properties of Negative Binomial – Triangular distribution mixture

The mean is given by

1 훼 1 − 푝 퐸 푋 = 푔 푝 (4.43) 0 푝

퐸(푋) = 훼퐸 푃−1 − 훼

−2 log 휃 퐸 푋 = 훼 − 1 (4.44) 1 − 휃

Since 0 < 휃 ≤ 1

Therefore

퐸 푋 > 훼 ∨ 휃

Variance

Variance of this mixture doesn’t exist since the second inverse moment of the triangular distribution doesn’t exist. It is important to note that this distribution has a very long tail.

Identity

We can draw an identity based on the result from explicit mixing and that from method of moments as follows.

푥 1 − 휃 4 푥 1 − 휃푟+푘+1 퐵 푟 + 2, 푥 + 1 + 퐵 푟 + 1, 푥 + 2 − 퐵 푟 + 1, 푥 + 2 = −1 푘 2 1 − 휃 휃 휃 휃 푘 푟 + 푘 + 1 푟 + 푘 + 2 푘=0

For 푥 = 0,1,2, … … ; 0 < 휃 < 1, 푟 > 0

CHAPTER 5

GEOMETRIC DISTRIBUTION MIXTURES WITH BETA GENERATED DISTRIBUTIONS IN THE [0, 1] DOMAIN

5.1. Introduction Geometric distribution is a special form of the Negative Binomial Distribution when 푟 = 1. It is therefore important to note that the results of Negative Binomial mixtures can be used to generate the geometric mixtures by substituting the value of 푟 = 1in the Negative Binomial mixtures. We are going to review the Geometric distribution mixtures with Beta generated distributions in the [0,1] domain in this chapter.

5.2. Geometric – Classical Beta Distribution Classical Beta Distribution

푥훼−1 1 − 푥 훽−1 푓 푋 = 푥; 훼, 훽 = 퐵 훼, 훽

푓표푟 0 < 푥 < 1; 훼, 훽 > 0

5.2.1. Classical Beta – Geometric distribution from explicit mixing Classical Beta – Negative Binomial distribution from explicit mixing

푟 + 푥 − 1 ! 퐵 푟 + 훼, 푥 + 훽 푓 푥 = 푟 − 1 ! 푥! 퐵 훼, 훽

푓표푟 훼, 훽 > 0; 푥 = 0,1,2, , …

From the above expression of the BND, when r=1, we attain the following distribution which is the Beta - geometric mixture

퐵 1 + 훼, 푥 + 훽 푓 푥 = (5.01) 퐵 훼, 훽

푓표푟 훼, 훽 > 0; 푥 = 0,1,2, , …

5.2.2. Classical Beta – Geometric distribution from method of moments mixing Classical Beta – Negative Binomial distribution from method of moments mixing

푥 Γ 푟 + 푥 ! 푥 퐵 훼 + 푟 + 푘, 훽 푓 푥 = −1 푘 Γ 푟 ! 푥 ! 푘 퐵 훼, 훽 푘=0

푓표푟 훼, 훽 > 0; 푥 = 0,1,2, , …

From the above expression of the BND we substitute r with 1 to attain the following distribution which is the Beta - geometric mixtures

푥 푥 퐵 훼 + 푘 + 1, 훽 푓 푥 = −1 푘 (5.02) 푘 퐵 훼, 훽 푘=0

푓표푟 훼, 훽 > 0; 푥 = 0,1,2, , …

5.2.3. Classical Beta – Geometric distribution from recursive relation There are three ways of applying the recursive relation in this mixture, namely

a. Using ratio of the conditional distribution b. Using ration of the mixed distribution c. Using a dummy function

a. Using ratio of the conditional distribution

1 푃푟표푏 푥 = 푓 푥/푝 푔 푝 푑푝 (5.03) 0

푓(푥/푝)is a Geometric distribution in this case

푓 푥/푝 = 푝 1 − 푝 푥 (5.04)

Substituting 푥 푤푖푡푕 푥 − 1 in equation 5.04 we get

푓 (푥 − 1)/푝 = 푝 1 − 푝 푥−1 (5.05)

Dividing equation 5.04 by equation 5.05 we get

푓(푥/푝) 푝(1 − 푝)푥 = 푓 (푥 − 1)/푝 푝 1 − 푝 푥−1

푓 푥/푝 = (1 − 푝)푓 (푥 − 1)/푝

푓 푥/푝 = 1 − 푝 푓 (푥 − 1)/푝 (5.06)

Substituting equation (5.06) into equation (5.03)

1 푃푟표푏(푥) = 1 − 푝 푓 (푥 − 1)/푝 푔 푝 푑푝 0

1 1 푃푟표푏 푥 = 푓 (푥 − 1)/푝 푔 푝 푑푝 − 푝푓 (푥 − 1)/푝 푔 푝 푑푝 (5.07) 0 0

Consider

1 푓 (푥 − 1)/푝 푔 푝 푑푝 (5.08) 0

This can be expressed as

1 푥−1 푝 1 − 푝 푔 푝 푑푝 = 푝1 푥 − 1 (5.08푎) 0

Consider

1 푝푓 (푥 − 1)/푝 푔 푝 푑푝 (5.09) 0

1 = 푝2 1 − 푝 푥−1푔 푝 푑푝 0

1 푝2 1 − 푝 푥−1 = 퐵 3, 푥 푔 푝 푑푝 = 퐵 3, 푥 0 퐵 3, 푥

1 푝푓 (푥 − 1)/푝 푔 푝 푑푝 = 퐵 3, 푥 (5.09푎) 0

Substituting equation 5.08a and 5.09a into 5.07 the Beta - Geometric recursive mixture becomes

푃푟표푏 푥 = 푃푟표푏 푥 − 1 − 퐵 3, 푥 ; 푓표푟 푥 = 0,1,2 … . . ; (5.10)

b. Using ratio of the mixed distribution

From equation 5.01

퐵 1 + 훼, 푥 + 훽 푓 푥 = 푤푕푒푟푒 푥 = 0,1,2, … 퐵 훼, 훽

Considering the below ratio.

푓 푥 퐵 1 + 훼, 푥 + 훽 퐵 훼, 훽 = 푓 푥 − 1 퐵 훼, 훽 퐵 1 + 훼, 푥 + 훽 − 1

푓 푥 푥 + 훽 − 1 = 푓 푥 − 1 푥 + 훽 + 훼

푥 + 훽 − 1 푓 푥 = 푓 푥 − 1 5.11 푥 + 훽 + 훼

푓표푟 풓, 훼, 훽 > 0, 푥 = 0,1,2 … . . ;

c. Using a dummy function

Consider the mixture equation below

1 1 푓 푥 = 푝훼 1 − 푝 푥+훽−1푑푝 (5.11) 퐵 훼, 훽 0

Introducing the dummy function

1 훼 푥+훽−1 퐼푥 훼, 훽 = 퐵 훼, 훽 푓 푥 = 푝 1 − 푝 푑푝 (5.13) 0

Integrating the integral by parts.

1 1 푝훼 1 − 푝 푥+훽−1푑푝 = 푢푑푣 0 0

1 1 푢푑푣 = 푢푣 − 푣푑푢 0 0

Let

푢 = 푝훼 푎푛푑 푑푣 = 1 − 푝 푥+훽−1푑푝

Hence

1 − 푝 푥+훽 푑푢 = 훼푝훼−1푑푝 푎푛푑 푣 = − 푥 + 훽

∴

1 1 1 − 푝 푥+훽 푝훼 훼 1 푝훼 1 − 푝 푥+훽−1푑푝 = − + 1 − 푝 푥+훽 푝훼−1푑푝 5.14 푥 + 훽 푥 + 훽 0 0 0 훼 = 퐼 훼 − 1, 훽 푥 + 훽 푥+1 훼 퐼 훼, 훽 = 퐵 훼, 훽 푓 푥 = 퐼 훼 − 1, 훽 5.15 푥 푥 + 훽 푥+1

훼 퐵 훼, 훽 푓 푥 = 퐵 훼 − 1, 훽 푓 푥 + 1 푥 + 훽

∴

훼 − 1 푓 푥 = 푓 푥 + 1 (5.16) 훼

푓표푟 훼 > 0 푎푛푑 푥 = 1,2,3, …

This is the beta geometric distribution expressed iteratively

5.2.4. Properties of beta – Geometric distribution Moment generating function

Consider the moment generating function from the beta – Negative Binomial distribution as expressed in chapter 2

∞ 푛 푟(푛)훽(푛) 푡 푔 푡 = 푟 + 훼 + 훽 (푛) 푛! 푛=0

∞ Γ 푟 + 푛 Γ 훽 + 푛 Γ 푟 + 훼 + 훽 푡푛 푔 푡 = (4.05) Γ푟 Γ훽 Γ 푟 + 훼 + 훽 + 푛 푛! 푛=0

When 푟 = 1 we will end up with the moment generating function for the beta geometric distribution as stipulated below.

∞ Γ 1 + 푛 Γ 훽 + 푛 Γ 1 + 훼 + 훽 푡푛 푔 푡 = (5.06) Γ훽 Γ 1 + 훼 + 훽 + 푛 푛! 푛=0

Differentiate 푔(푡) with respect to 푡

∞ Γ 1 + 푛 Γ 훽 + 푛 Γ 1 + 훼 + 훽 푡푛−1 푔′ 푡 = 푛 (5.07) Γ훽 Γ 1 + 훼 + 훽 + 푛 푛! 푛=0

∞ Γ 1 + 푛 Γ 훽 + 푛 Γ 1 + 훼 + 훽 푡푛−1 푔′ 푡 = (5.08) Γ훽 Γ 1 + 훼 + 훽 + 푛 푛 − 1 ! 푛=0

∞ Γ 1 + 푛 Γ 훽 + 푛 Γ 1 + 훼 + 훽 푡푛−2 푔′′ 푡 = 푛 푛 − 1 (5.09) Γ훽 Γ 1 + 훼 + 훽 + 푛 푛! 푛=0

∞ Γ 1 + 푛 Γ 훽 + 푛 Γ 1 + 훼 + 훽 푡푛−2 푔′′ 푡 = (5.10) Γ훽 Γ 1 + 훼 + 훽 + 푛 푛 − 2 ! 푛=0

5.3. Geometric – Uniform distribution The uniform distribution [0,1] given by

1 0 < 푝 < 1 푔 푝 = (5.11) 0 푒푙푠푒푤푕푒푟푒

The Negative Binomial – uniform distribution have the following formats from the respective methods used in mixing the two distributions

Negative Binomial – uniform distribution from explicit mixing 푟 푓 푥 = ; 푓표푟 푟 > 0 푎푛푑 푥 = 0,1,2, … (5.12) 푟 + 푥 + 1 (푟 + 푥)

Negative Binomial – uniform distribution from recursive relation

1st form

푟 + 푥 − 1 푓 푥 = 푓 푥 − 1 푓표푟 푟 > 0 푎푛푑 푥 = 0,1,2, … (5.13푎) 푟 + 푥 + 1

2nd form 푟 푓 푥 = 푓 푥 − 1 푓표푟 푟 > 0 푎푛푑 푥 = 0,1,2, … (5.13푎) 푟 + 1

Negative Binomial – uniform distribution from method of Moments

푥 Γ 푟 + 푥 푥 1 푓 푥 = −1 푘 5.14 Γ 푟 푥! 푘 푟 + 푘 + 1 푘=0

In the above Negative Binomial mixtures it is important to note that if r=1 then they become geometric - uniform distribution mixtures from the respective mixing methods as shown below.

5.3.1. Geometric – uniform distribution from explicit mixing 1 푓 푥 = 푓표푟 푥 = 0,1,2, … (5.15푎) 2 + 푥 (1 + 푥)

푓 푥 = 퐵 2, 푥 + 1 푓표푟 푥 = 0,1,2, … (5.15푏)

5.3.2. Geometric – uniform distribution from recursive relation 1st form

Here we consider the mixture from explicit mixing method. 푥 푓 푥 = 푓 푥 − 1 ; 푓표푟 푥 = 0,1,2, … (5.16푎) 푥 + 2

2nd form

1 푓 푥 = 푓 푥 − 1 ; 푓표푟 푥 = 0,1,2, … (5.16푏) 2

5.3.3. Geometric – uniform distribution from Method of moments 푥 푥 1 푓 푥 = −1 푘 ; 푓표푟 푥 = 0,1,2, … 5.16 푘 푘 + 2 푘=0

5.4. Negative Binomial – power function distribution

5.4.1. Power function distribution 푔 푝 = 푎푝푎−1 0 < 푝 < 1; 훼 > 0 (5.17)

This is the pdf of a power function distribution with parameter 훼

The following are the results of from chapter 3 in relation to the distributions emanating from the Negative Binomial mixture with the power function.

a. Negative Binomial – power function mixing from explicit format

푟 + 푥 − 1 푓 푥 = 훼 퐵 푟 + 훼, 푥 + 1 (5.18) 푥

푓표푟 푟, 푎 > 0; 푥 = 0,12 ….

This is the density function of the Negative Binomial – power function expressed explicitly

b. Negative Binomial – power function from method of moments

푥 Γ(푟 + 푥) 푥 훼 푓 푥 = −1 푘 (5.19) Γ 푟 푥 ! 푘 훼 + 푟 + 푘 푘=0

푓표푟 푟, 훼 > 0 푎푛푑 푥 = 0,1,2, …

c. Negative Binomial – power distribution in a recursive expression

1st form (from explicit mixture result)

푟 + 푥 − 1 푓 푥 = 푓 푥 − 1 (5.20푎) 푥 + 푎 + 1

2nd form (using a dummy function)

푟 + 푥 − 1 푟 + 푥 − 2 푓 푥 = 푓 푥 − 1 (5.20푏) 푟 + 푎 푟 − 1

When r=1 the above mixtures become geometric – power mixtures

5.4.2. Geometric – Power distribution from explicit mixture

푓 푥 = 훼 퐵 1 + 훼, 푥 + 1 (5.21)

푓표푟 푎 > 0; 푥 = 0,12 ….

5.4.3. Geometric – Power distribution from moments method

푥 푥 훼 푓 푥 = −1 푘 (5.22) 푘 훼 + 푘 + 1 푘=0

푓표푟 훼 > 0 푎푛푑 푥 = 0,1,2, …

5.4.4. Geometric – Power distribution in a recursive format 1st form (from explicit mixture result) 푥 푓 푥 = 푓 푥 − 1 (5.23푎) 푥 + 푎 + 1

2nd form (using a dummy function)

Replacing r by 1 in the denominator of the below equation will nullify the result.

푟 + 푥 − 1 푟 + 푥 − 2 푓 푥 = 푓 푥 − 1 푟 + 푎 푟 − 1

We can however calculate this from scratch to achieve the below result

Consider the explicit expression below

1 푓 푥 = 훼 푝훼 1 − 푝 푥 푑푝 0

Manipulate this to a dummy function 퐼 훼 푥 as shown below

1 푓 푥 훼 푥 퐼 훼 푥 = = 푝 1 − 푝 푑푝 5.24 훼 0

Integrating the integral by parts

Let

푢 = 1 − 푝 푥

푑푢 = −푥 1 − 푝 푥−1푑푝

푑푣 = 푝훼 푑푝

푝훼+1 푣 = 훼 + 1

u dv = vu − v du

1 푝훼+1 푥 1 퐼 푥 = 1 − 푝 푥 + 푝훼+1 1 − 푝 푥−1푑푝 5.25 훼 훼 + 1 훼 + 1 0 0

1 푥 훼+1 푥−1 퐼훼 푥 = 푝 1 − 푝 푑푝 (5.26) 훼 + 1 0

푓 푥 푥 퐼 푥 = = 퐼 푥 − 1 훼 훼 훼 + 1 훼+1

푓 푥 푥 푓 푥 − 1 = 5.27 훼 푟 + 훼 훼 − 1

Hence 훼푥 푓 푥 = 푓 푥 − 1 5.23푏 푟 + 훼

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5.5. Geometric – Arcsine distribution 5.5.1. Arcsine distribution

1 1 1 푔 푝 = 0 < 푝 < 1; 푎, 푏 > 0 푎푛푑 휋 = 퐵 , (5.28) 휋 푝 1 − 푝 2 2

This is the pdf of arcsine distribution

The following are some of the expressions of the Negative Binomial – arcsine distributions

a. Negative Binomial – Arcsine from Explicit mixing

1 1 푟 + 푥 − 1 퐵 푟 + , 푥 + 푓 푥 = 2 2 (5.29) 푥 휋

b. Negative Binomial– Arcsine distribution in a recursive format

1st form (from explicit mixture result)

푟 + 푥 − 1 2푥 − 1 푓 푥 = 푓 푥 − 1 (5.30푎) 2푥 푟 + 푥

2nd form (using a dummy function)

푟 푟 − 2푥 푓 푥 = 푓 푥 − 1 (5.30푏) 푥 2푟 − 1 c. Negative Binomial– Arcsine distribution from method of moments 푥 1 1 Γ 푟 + 푥 푥 퐵 푟 + 푘 + , 푓 푥 = −1 푘 2 2 5.31 Γ 푟 푥! 푘 휋 푘=0 푓표푟 푟, 훼 > 0; 푥 = 0,1,2, …

When r=1 the above mixtures become geometric – arcsine distributions mixtures

5.5.2 Geometric – Arcsine from explicit mixing 3 1 퐵 , 푥 + 푓 푥 = 2 2 (5.32) 휋

101

5.5.3. Geometric – Arcsine distribution in a recursive format 1st form (from explicit mixture result)

2푥 − 1 푓 푥 = 푓 푥 − 1 푓표푟 훼 > 0; 푥 = 0,1,2, … (5.33푎) 2 1 + 푥

2nd form (using a dummy function)

1 − 2푥 푓 푥 = 푓 푥 − 1 푓표푟 푥 = 0,1,2, … (5.33푏) 푥

5.5.4. Geometric – Arcsine distribution from method of moments 푥 3 1 푥 퐵 푘 + , 푓 푥 = −1 푘 2 2 5.34 푘 휋 푘=0 푓표푟 훼 > 0; 푥 = 0,1,2, …

5.6 Geometric – Truncated beta distribution

5.6.1. Truncated beta This below distribution is referred to as truncated beta distribution g(p)with parameters 훼, 훽, 푎, 푏

푝푎−1 1 − 푝 푏−1 푔 푝 = 푓표푟 1 < 훼 < 푝 < 훽 < 1; 푎, 푏 > 0 (5.35) 퐵훽 푎, 푏 −퐵훼 (푎, 푏)

The following are some of the expressions of the Negative Binomial – arcsine distributions

a. Negative Binomial – Truncated beta distribution explicitly mixed

퐵 푎+푟,푏+푥 −퐵 푎+푟,푏+푥 푓 푥 = 푟+푥−1 훽 훼 (5. 36) 푥 퐵훽 푎,푏 −퐵훼 푎,푏

푓표푟 푟, 푎, 푏 > 0; 1 < 훼 < 푝 < 훽 < 1; 푥 = 0,1,2 …

This is the pdf of the Negative Binomial – truncated beta mixture

102

b. Negative Binomial – Truncated beta distribution from method of moments

푥 Γ 푟 + 푥 푥 퐵훽 푎 + 푘 + 푟, 푏 − 퐵훼 푎 + 푘 + 푟, 푏 푓 푥 = −1 푘 5.37 Γ 푟 ! 푥! 푘 퐵훽 푎, 푏 −퐵훼 푎, 푏 푘=0

푓표푟 푟, 푎, 푏 > 0; 1 < 훼 < 푝 < 훽 < 1; 푥 = 0,1,2 …

When r=1 we will have truncated beta – geometric as follows

5.6.2 Geometric – Truncated beta from explicit mixing 퐵 푎+1,푏+푥 −퐵 푎+1,푏+푥 푓 푥 = 훽 훼 (5. 38) 퐵훽 푎,푏 −퐵훼 푎,푏

푓표푟 푎, 푏 > 0; 1 < 훼 < 푝 < 훽 < 1; 푥 = 0,1,2 …

5.6.3. Geometric – Truncated beta distribution from method of moments 푥 푥 퐵훽 푎 + 푘 + 1, 푏 − 퐵훼 푎 + 푘 + 1, 푏 푓 푥 = −1 푘 (5.39) 푘 퐵훽 푎, 푏 −퐵훼 푎, 푏 푘=0

푓표푟 푎, 푏 > 0; 1 < 훼 < 푝 < 훽 < 1; 푥 = 0,1,2 …

5.7 Geometric – Confluent Hypergeometric

5.7.1. Confluent Hypergeometric The pdf of confluent Hypergeometric expressed as follows

푝푎−1 1 − 푝 푏−1 푔 푝 = 푒−푝휇 푓표푟 0 < 푝 < 1; 푎, 푏 > 0; −∞ < 휇 < ∞ (5.40) 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇

푟푒푓 푁푎푑푎푟푎푗푎푕 푎푛푑 퐾표푡푧 (2007)

The Negative Binomial – confluent Hypergeometric distribution have the following formats from the respective methods used in mixing the two distributions

a. Confluent Hypergeometric – Negative Binomial distribution from explicit mixing

퐵(푟 + 푎, 푥 + 푏)1퐹 푎 + 푟, 푎 + 푏 + 푟 + 푥; −휇 푟 + 푥 − 1 푓 푥 = 1 (5.41) 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇 푥

푓표푟 푟, 푎, 푏 > 0; −∞ < 휇 < ∞; 푥 = 0,1,2, …

103

Properties of confluent Hypergeometric – Negative Binomial Mixture

First 3 moments about the origin

푎 1퐹1(푎 + 1, 푎 + 푏 + 1; −휇) 퐸 푋 = (5.42) 푎 + 푏 1퐹1(푎, 푎 + 푏; −휇)

푎 푎 + 1 1퐹1 푎 + 2, 푎 + 푏 + 2; −휇 퐸 푋2 = (5.43) 푎 + 푏 푎 + 푏 + 1 1퐹1 푎, 푎 + 푏; −휇

푎 푎 + 1 푎 + 2 1퐹1 푎 + 3, 푎 + 푏 + 3; −휇 퐸 푋3 = (5.44) (푎 + 푏) 푎 + 푏 + 1 푎 + 푏 + 2 1퐹1 푎, 푎 + 푏; −휇

푉푎푟 푋 = 퐸 푋2 − 퐸 푋 2

2 푎(푎 + 1)1퐹1(푎 + 2, 푎 + 푏 + 2; −흁) 푎 푎 + 1 1퐹1 푎 + 2, 푎 + 푏 + 2; −흁 푉푎푟 푋 = – 5.45 (푎 + 푏)(푎 + 푏 + 1)1퐹1(푎, 푎 + 푏; −흁) 푎 + 푏 푎 + 푏 + 1 1퐹1 푎, 푎 + 푏; −흁

b. Confluent Hypergeometric – Negative Binomial distribution from Method of Moments

푥 Γ(푟 + 푥) 푥 퐵 푟 + 푘 + 푎, 푏 1퐹 푎 + 푟 + 푘, 푎 + 푟 + 푘 + 푏; −휇 푓 푥 = −1 푘 1 (5.46) Γ 푟 푥 ! 푘 퐵 푎, 푏 1퐹 푎, 푎 + 푏; −휇 푘=0 1

푓표푟 푥 = 0,1,2, … ; 푎, 푏 > 표 ; 푟 > 0

In the above negative Binomial mixtures it is important to note that if r=1 then they become geometric Hypergeometric mixtures from the respective mixing methods as shown below.

104

5.7.2 Geometric – Confluent Hypergeometric distribution from explicit mixing

퐵(1 + 푎, 푥 + 푏)1퐹 푎 + 1, 푎 + 푏 + 1 + 푥; −휇 푓 푥 = 1 (5.47) 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇

푓표푟 푎, 푏 > 0; −∞ < 휇 < ∞; 푥 = 0,1,2, …

Properties of Geometric – Confluent Hypergeometric Mixture

First 3 moments about the origin

푎 1퐹1(푎 + 1, 푎 + 푏 + 1; −휇) 퐸 푋 = (5.48) 푎 + 푏 1퐹1(푎, 푎 + 푏; −휇)

푎(푎 + 1) 1퐹1(푎 + 2, 푎 + 푏 + 2; −휇) 퐸 푋2 = (5.49) (푎 + 푏)(푎 + 푏 + 1) 1퐹1(푎, 푎 + 푏; −휇)

푎 푎 + 1 푎 + 2 1퐹1 푎 + 3, 푎 + 푏 + 3; −휇 퐸 푋3 = (5.50) (푎 + 푏) 푎 + 푏 + 1 푎 + 푏 + 2 1퐹1 푎, 푎 + 푏; −휇

푉푎푟 푋 = 퐸 푋2 − 퐸 푋 2

푎(푎+1) 퐹 (푎+2,푎+푏+2;−휇) 푎(푎+1) 퐹 (푎+2,푎+푏+2;−휇) 2 푉푎푟 푋 = 1 1 – 1 1 (5.51) (푎+푏)(푎+푏+1)1퐹1(푎,푎+푏;−휇) (푎+푏)(푎+푏+1) 1퐹1(푎,푎+푏;−휇)

5.7.2 Geometric – Hypergeometric distribution from Method of Moments

푥 푥 퐵 1 + 푘 + 푎, 푏 1퐹 푎 + 푘 + 1, 푎 + 푘 + 푏 + 1; −휇 푓 푥 = −1 푘 1 (5.52) 푘 퐵 푎, 푏 1퐹 푎, 푎 + 푏; −휇 푘=0 1

푓표푟 푥 = 0,1,2, … ; 푎, 푏 > 표 ; 푟 > 0

105

5.8. Geometric – Gauss Hypergeometric distribution

5.8.1. Gauss Hypergeometric distribution The pdf of gauss Hypergeometric given by

푡푎−1 1 − 푡 푏−1 푔 푡 = 휀 (5.53) 퐵 푎, 푏 1 + 푧푡 2퐹1 푎, 휀, 푎 + 푏; −푧

푓표푟 0 < 푡 < 1; 푎, 푏 > 0; −∞ < 휀 < ∞

(see Armero and Bayarri (1994))

a. Gauss Hypergeometric – Negative Binomial distribution from Explicit mixing

푟 + 푥 − 1 퐵 푎 + 푟, 푏 + 푥 2퐹1 푎 + 푟, 휀; 푎 + 푏 + 푟 + 푥; −푧 푓 푥 = (5.54) 푥 퐵 푎, 푏 2퐹1 푎, 휀, 푎 + 푏; −푧

For 푟, 푎, 푏, 푧 > 0; −∞ < 휀 < ∞ ; 푥 = 0,1,2 … ;

b. Gauss Hypergeometric – Negative Binomial distribution from method of moments mixing

푥 Γ(푟 + 푥) 푥 퐵 푟 + 푘 + 푎, 푏 2퐹1 푟 + 푘 + 푎, 휀, 푎 + 푏 + 푟 + 푘; −푧 푓 푥 = −1 푘 (5.55) Γ 푟 푥 ! 푘 퐵 푎, 푏 2퐹1 푎, 휀, 푎 + 푏; −푧 푘=0

For 푟, 푎, 푏, 푧 > 0; −∞ < 휀 < ∞ ; 푥 = 0,1,2 … ;

5.8.3. Gauss Hypergeometric – Geometric distribution from Explicit mixing 퐵 푎 + 1, 푏 + 푥 2퐹1 푎 + 1, 휀; 푎 + 푏 + 푥 + 1; −푧 푓 푥 = (5.56) 퐵 푎, 푏 2퐹1 푎, 휀, 푎 + 푏; −푧

For 푎, 푏, 푧 > 0; −∞ < 휀 < ∞ ; 푥 = 0,1,2 … ;

5.8.4. Gauss Hypergeometric – Geometric distribution from method of moments mixing 푥 푥 퐵 푟 + 푘 + 푎, 푏 2퐹1 푘 + 푎 + 1, 휀, 푎 + 푏 + 푘 + 1; −푧 푓 푥 = −1 푘 (5.57) 푘 퐵 푎, 푏 2퐹1 푎, 휀, 푎 + 푏; −푧 푘=0

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CHAPTER 6

GEOMETRIC MIXTURES BASED ON DISTRIBUTIONS BEYOND BETA FROM NEGATIVE BINOMIAL MIXTURES

6.1. Introduction All the distributions we are going to consider in this category are within the [0,1] domain and are not based on beta distribution. Below distributions will be studied;

1. Kumaraswamy (I) and (II) distribution 2. Triangular distribution 3. Truncated exponential distribution 4. Truncated gamma distribution 5. Minus log distribution 6. Two – sided ogive distribution 7. Ogive distribution 8. Two – sided power distribution

5.2. Geometric – Kumaraswamy (I) Distribution

5.2.1. Kumaraswamy (I) Distribution 푔 푢 = 푏 1 − 푢 푏−1 푓표푟 0 < 푢 < 1; 푏 > 0 (6.01)

This is the pdf of Kumaraswamy (I) distribution with parameter 푏`

The Negative Binomial – Kumaraswamy (I) Distribution have the following formats from the respective methods used in mixing the two distributions

a. Negative Binomial – Kumaraswamy (I) distribution from explicit mixing

푟 + 푥 − 1 푓 푥 = 푏퐵 푟 + 1, 푥 + 푏 푥 = 0,1,2, … ; 푏, 푟 > 0 (6.02) 푥

See Li Xiaohuet et al (2011) on Binomial mixture

b. Negative Binomial – Kumaraswamy (I) distribution mixing from the method moments

푥 Γ 푟 + 푥 푥 푃 푋 = 푥 = −1 푘 푏퐵 푟 + 푘 + 1, 푏 (6.03) Γ 푟 푥 ! 푘 푘=1

푓표푟 푟, 푏 > 0 푎푛푑 푥 = 0,1,2, …

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When r=1 we will have the following geometric – Kumaraswamy (I) distribution as obtained from explicit mixing of the geometric distribution and the Kumaraswamy (I) distribution

5.2.2. Geometric – Kumaraswamy (I) distribution from explicit mixing

푓 푥 = 푏퐵 2, 푥 + 푏 ; 푥 = 0,1,2, … ; 푏 > 0 (6.04)

5.2.2. Geometric – Kumaraswamy (I) distribution from method of moments mixing When r=1 we will have geometric – KW (I) mixtures from the above negative Binomial mixtures as follows.

푥 푥 푃 푋 = 푥 = −1 푘 푏퐵 푘 + 2, 푏 (6.05) 푘 푘=1

푓표푟 푟, 푏 > 0 푎푛푑 푥 = 0,1,2, …

6.3. Geometric – Truncated Exponential Distribution (TEX(흀, 풃))

6.3.1. Truncated Exponential Distribution (TEX(흀, 풃)) This is a pdf which can be expressed as

푏 푒−푏푝/휆 푔 푝 = 휆 0 < 푝 < 1; 푏, 휆 > 0 (6.06) [1 − 푒−푏/휆 ]

This the truncated exponential distribution with parameters 휆 푎푛푑 푏

Negative Binomial – Truncated Exponential distribution from explicit mixing

푏 푏 퐹 푟+1,푟+푥+2;− 1 1 휆 푟 푓 푥 = 푏 푓표푟 푥 = 0,1,2, . . ; 푏, 푟, 휆 > 0 (6. 07) − 푟+푥+1 푟+푥 휆 1−푒 휆

6.3.1. Geometric – Truncated Exponential distribution from explicit maxing When r=1 the above mixture attains the status of a geometrical - truncated exponential distribution from explicit mixing of the geometric distribution with the truncated exponential distribution

푏 푏 퐹 2,1 + 푥 + 2; − 1 1 휆 1 푓 푥 = 푏 푓표푟 푥 = 0,1,2, . . ; 푏, 휆 > 0 (6.08) − 푥 + 2 1 + 푥 휆 1 − 푒 휆

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6.4. Geometric – Truncated Gamma Distribution

6.4.1. Truncated Gamma Distribution Below is the pdf of Truncated Gamma Distribution

푎푏 푝푏−1푒−푎푝 0 < 푝 < 1; 푎, 푏 > 0 푔 푝 = 훾 푎, 푏 (6.09) 0 푒푙푠푒푤푕푒푟푒

a. Negative Binomial – Truncated Gamma Distribution from explicit mixing

푏 푟 + 푥 − 1 푎 1퐹1 푟 + 푏, 푟 + 푏 + 푥 + 1; −푎 퐵 푟 + 푏, 푥 + 1 푓 푥 = 푥 = 0,1,2, … ; 푟, 푎, 푏 > 0 (6.10) 푥 훾 푎, 푏

(see Bhattacharya, S.K. (1968) on Binomial mixture)

b. Negative Binomial – Truncated Gamma Distribution from moments method mixing

푥 Γ 푟 + 푥 푥 훾 푎, 푏 + 푟 + 푘 푓(푥) = −1 푘 Γ 푟 푥 ! 푘 푎푟+푘 훾 푎, 푏 푘=0

푓표푟 푎, 푏, 푟 > 0; 푥 = 0,1,2, …

6.4.2. Geometric – Truncated Gamma distribution from explicit mixing When r=1 the above mixture attains the status of a Geometric – Truncated Gamma distribution stated below

푏 푎 1퐹1 1 + 푏, 푏 + 푥 + 2; −푎 퐵 1 + 푏, 푥 + 1 푓 푥 = 푥 = 0,1,2, … ; 푎, 푏 > 0 (6.11) 훾 푎, 푏

6.4.1. Geometric – Truncated Gamma distribution from moments method mixing When r=1 the above mixture attains the status of a geometric – truncated gamma distribution stated below

푥 푥 훾 푎, 푏 + 푘 + 1 푓 푥 = −1 푘 6.12 푘 푎1+푘 훾 푎, 푏 푘=0

푓표푟 푎, 푏, 푟 > 0; 푥 = 0,1,2, …

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6.5. Geometric – Minus Log Distribution

6.5.1. Minus Log Distribution 푔 푝 = − log 푝 푓표푟 1 ≤ 푝 ≤ 0 6.13

This is the pdf of minus log distribution

a. Negative Binomial – Minus Log Distribution from method of moments mixing

푥 푟 + 푥 − 1 푥 1 푓 푥 = −1 푘 푥 = 0,1,2, … 푟 > 0 (6.14) 푥 푘 푟 + 푘 + 1 2 푘=0

6.5.1Geometric – Minus Log distribution from method of moments mixing

When r=1 the above mixtures attains the status of a geometric – minus log distribution which is expressed in the equation below

푥 푥 1 푓 푥 = −1 푘 푥 = 0,1,2, … …. (6.15) 푘 푘 + 2 2 푘=0

6.6. Geometric – Standard Two – Sided Power Distribution

6.6.1. Standard Two – Sided Power Distribution 푝 푘−1 푘 0 < 푝 < 휃 휃 푔 푝 = 푘−1 (6.16) 1 − 푝 푘 휃 < 푝 < 1 1 − 휃

This is the pdf of a two – sided power distribution with parameters 푘 푎푛푑 휃

a. Negative Binomial – Standard Two – Sided Power Distribution from explicit mixing

푟 + 푥 − 1 1 1 풇 풙 = 푘 퐵 푟 + 푘, 푥 + 1 + 퐵 푟 + 1, 푥 + 푘 − 퐵 푟 + 1, 푥 + 푘 6.17 푥 휃푘−1 휃 1 − 휃 푘−1 휃

This is the distribution obtained from the mixture of Negative Binomial distribution and the standard two – side power distribution.

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6.6.1. Geometric – Standard Two – Sided Power distribution from explicit mixing When r=1, the above mixture reduces to geometric – standard two – sided power distribution which is stated below.

1 1 풇 풙 = 푘 퐵 1 + 푘, 푥 + 1 + 퐵 2, 푥 + 푘 − 퐵 2, 푥 + 푘 (6.18) 휃푘−1 휃 1 − 휃 푘−1 휃

6.7. Geometric – Ogive Distribution

6.7.1. Ogive Distribution The general form of an Ogive distribution is given by

2푚(푚 + 1) 1 − 푚2 푔 푝 = 푝(푚−1)/2 + 푝푚 0 < 푝 > 1 ; 푚 > 0 (6.19) 3푚 + 1 3푚 + 1

From (퐷표푟푝 푎푛푑 퐾표푡푧 (2003))

a. Negative Binomial – Ogive Distribution from explicit mixing

푟 + 푥 − 1 푚 + 1 2푟 + 푚 − 1 푓 푥 = 2푚퐵 + 1, 푥 + 1 + 1 − 푚 퐵 푟 + 푚 + 1, 푥 + 1 (6.20) 푥 3푚 + 1 2

푓표푟 푥 = 0,1,2, … . ; 푟 > 0; 푚 > 0

6.7.2. Geometric – Ogive distribution from explicit mixing When r=1, the above mixture reduces to geometric – ogive distribution which is stated below.

푚 + 1 푚 + 1 푓 푥 = 2푚퐵 + 1, 푥 + 1 + 1 − 푚 퐵 푚 + 2, 푥 + 1 (6.21) 3푚 + 1 2

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6.8. Geometric – Standard Two – Sided Ogive Distribution

6.8.1. Standard Two – Sided Ogive Distribution 푚 −1 2 푚 2푚 푚 + 1 푝 2 1 − 푚 푝 + 0 < 푝 < 휃 ; 푚 > 0 3푚 + 1 휃 3푚 + 1 휃 푔 푝 = 푚 −1 (6.22) 2 푚 2푚 푚 + 1 1 − 푝 2 1 − 푚 1 − 푝 + 휃 < 푝 < 1 ; 푚 > 0 3푚 + 1 1 − 휃 3푚 + 1 1 − 휃

This is the pdf of the two sided Ogive Distribution with parameters 푚 푎푛푑 휃

When 푝 = 휃 then the two – sided Ogive distribution is smooth and this is the reflection point.

Although this contradicts the situation at the reflection point of the two – sided power function

a. Negative Binomial – Standard Two – Sided Ogive Distribution from explicit mixing

푟 + 푥 − 1 푚 + 1 2푚 2푟 + 푚 − 1 푓 푥 = 퐵 + 1, 푥 + 1 푥 3푚 + 1 푚 −1 휃 2 휃 2 1 − 푚 + 퐵 푟 − 푚 + 1, 푥 + 1 휃푚 휃 2푚 2푥 + 푚 − 1 2푥 + 푚 − 1 + 퐵 푟 + 1, + 1 – 퐵 (푟 + 1, 푚 −1 2 휃 2 1 − 휃 2 1 − 푚 + 1) 퐵 푟 + 1, 푥 + 푚 + 1 − 퐵 (푟 + 1, 푥 + 푚 + 1) (6.23) 1 − 휃 푚 휃

푓표푟 푥 = 0,1,2, …. ; 푚 > 0 ; 푟 > 0

6.8.1. Geometric – Standard Two – Sided Ogive Distribution from explicit mixing When r=1, the above mixture reduces to geometric – Two Sided Ogive distribution which is stated below.

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푚 + 1 2푚 푚 + 3 1 − 푚 푓 푥 = 퐵 , 푥 + 1 + 퐵 2 − 푚, 푥 + 1 3푚 + 1 푚 −1 휃 2 휃푚 휃 휃 2 2푚 2푥 + 푚 + 1 2푥 + 푚 + 1 1 − 푚 + 퐵 2, – 퐵 푟 + 1, 퐵 2, 푥 푚 −1 2 휃 2 1 − 휃 푚 1 − 휃 2

+ 푚 + 1 − 퐵휃 2, 푥 + 푚 + 1 6.24

6.9. Geometric – Triangular Distribution

6.9.1. Triangular Distribution Z

0 θ 1

Thus the density of the triangular distribution is

2푝 0 < 푝 < 휃 휃 푔 푝 = 2(1 − 푝) (6.25) 휃 < 푝 < 1 1 − 휃 0 푒푙푠푒푤푕푒푟푒

a. Negative Binomial – triangular distribution mixing

2ᴦ 푟 + 푥 퐵휃 푟 + 2, 푥 + 1 퐵 푟 + 1, 푥 + 2 − 퐵휃 푟 + 1, 푥 + 2 푓 푥 = + (6.26) 푥! ᴦ푟 휃 1 − 휃

푓표푟 푥 = 0,1,2, … … ; 0 < 휃 < 1, 푟 > 0

This is the mixture of triangular distribution and Negative Binomial distribution expressed explicitly

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b. Negative Binomial – Triangular distribution from method of moments mixing

Case 1

푥 Γ 푟 + 푥 ! 푥 2 1 − 휃푟+푥−푘+1 푝 푥 = −1 푥−푘 (4.27) Γ 푟 ! 푥 ! 푘 푟 + 푥 − 푘 + 1 푟 + 푥 − 푘 + 2 1 − 휃 푘=0

푓표푟 푟 > 0; 0 < 푝 < 휃 < 1 푎푛푑 푥 = 0,1,2, …

Case 2

푥 푥 2 1 − 휃푟+푘+1 푓 푥 = −1 푘 (6.28) 푘 1 − 휃 푟 + 푘 + 1 푟 + 푘 + 2 푘=0

푓표푟 푟 > 0; 0 < 푝 < 휃 < 1 푎푛푑 푥 = 0,1,2, …

6.9.2. Geometric – Triangular distribution from explicit mixing When r=1, the above mixture reduces to geometric – Triangular distribution which is stated below.

퐵휃 3, 푥 + 1 퐵 2, 푥 + 2 − 퐵휃 2, 푥 + 2 푓 푥 = 2 + (6.29) 휃 1 − 휃

푓표푟 푥 = 0,1,2, … … ; 0 < 휃 < 1

6.9.2. Geometric – Triangular distribution from method of moments mixing Case 1

When r=1, equation 4.27 above reduces to Geometric – Triangular distribution which is stated below.

푥 푥 2 1 − 휃푥−푘+2 푝 푥 = −1 푥−푘 (6.30) 푘 푥 − 푘 + 2 푥 − 푘 + 3 1 − 휃 푘=0

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Case 2

When r=1, equation 2.28 above reduces to geometric – Triangular distribution which is stated below.

푥 푥 2 1 − 휃푘+2 푓 푥 = −1 푘 (6.31) 푘 1 − 휃 푘 + 2 푘 + 3 푘=0

푓표푟 푟 > 0; 0 < 푝 < 휃 < 1 푎푛푑 푥 = 0,1,2, …

Properties of geometric – Triangular distribution mixture

The mean is given by

−2 log 휃 퐸 푋 = 훼 − 1 (6.32) 1 − 휃

Since 0 < 휃 ≤ 1

Therefore

퐸 푋 > 훼 ∨ 휃

Variance

Variance of this mixture doesn’t exist since the second inverse moment of the triangular distribution doesn’t exist. It is important to note that this distribution has a very long tail.

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CHAPTER 7

NEGATIVE BINOMIAL MIXTURES WITH P AS A CONSTANT IN THE MIXING DISTRIBUTION

7.1. Introduction This chapter talks about mixing of the negative Binomial with other distributions having r as a variable and holding p constant.

We are going to consider four mixing distributions, namely

1. Logarithmic distribution 2. Exponential distribution 3. Binomial distribution

To get the mean and variance of the mixture, we use the method of probability generation function by getting the moments i.e.

∞ 푘 퐺 푠 = 푝푘 푠 (7.01) 푘=0

7.2. Logarithmic distribution/ logarithmic series distribution/ log series distribution Construction

Consider a Maclaurin Series expansion

푝2 푝3 − log 1 − 푝 = 푝 + + + ⋯ 2 3 −1 푝2 푝3 1 = 푝 + + + ⋯ (7.02) log 1 − 푝 2 3

∞ −1 푝푘 1 = (7.03) log 1 − 푝 푘 푖=1

This qualifies to be a PMF since it cumulates to 1.

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The probability mass function is a log(p) distribution expressed as

−1 푝푘 푓 푋 = 푘 = ; 푓표푟 푘 = 0,1,2 … 푎푛푑 0 ≤ 푝 ≤ 1 (7.04) log 1 − 푝 푘

Its Cumulative Distribution Function is

푘 −1 푝푥 퐹 푘 = 푑푥 (7.05) 0 log 1 − 푝 푥

−1 푘 푝푥 퐹 푘 = 푑푥 log 1 − 푝 0 푥

퐵 푝;푘+1.0 퐹 푘 = 1 + (7.06) log 1−푝

푤푕푒푟푒 퐵 푖푠 푎푛 푖푛푐표푚푝푙푒푡푒 퐵푒푡푎 푓푢푛푐푡푖표푛

Negative Binomial – logarithmic mixture

∞ 푟 + 푘 − 1 푟 푘 푝푘 = 푝 푞 푔 푟 푑푟 (7.07) 0 푘 −1 푝푟 푔 푟 = log 1 − 푝 푟

∞ 푟 푟 + 푘 − 1 푟 푘 −1 푝 푝푘 = 푝 푞 푑푟 (7.08) 0 푘 log 1 − 푝 푟

The PGF is given by the following expression

∞ 푘 퐺 푠 = 푝푘 푠 푘=0

∞ 2푟 ∞ −1 푝 푟 + 푘 − 1 퐺 푠 = (푠푞)푘 푑푟 log 1 − 푝 푟 푘 0 푘=0

117

∞ −1 ∞ 푝2푟 −푟 퐺 푠 = (−푠푞)푘 푑푟 log 1 − 푝 푟 푘 0 푘=0

−1 ∞ 푝2푟 1 − 푠푞 −푟 퐺 푠 = 푑푟 (7.09) log 1 − 푝 0 푟

푟 −1 ∞ 푝2 1 퐺 푠 = 푑푟 (7.10) log 1 − 푝 0 1 − 푠푞 푟

7.3. Negative Binomial – exponential distribution 7.3.1. Exponential distribution

푔 푟 = 훽푒−훽푟 (7.11)

7.3.2. Negative Binomial – exponential mixture

∞ 푟 + 푘 − 1 푟 푘 푝푘 = 푝 푞 푔 푟 푑푟 0 푘

∞ 푟 + 푘 − 1 푟 푘 −훽푟 푝푘 = 훽 푝 푞 푒 푑푟 6.12 0 푘

The PGF is given by the following expression

∞ 푘 퐺 푠 = 푝푘 푠 푘=0

∞ ∞ 푟 + 푘 − 1 퐺 푠 = 훽 푝푟 푒−훽푟 (푞푠)푘 푑푟 (7.13) 푘 0 푘=0

∞ ∞ −푟 퐺 푠 = 훽 푝푟 푒−훽푟 (−푞푠)푘 푑푟 푘 0 푘=0

∞ 퐺 푠 = 훽 푝푟 푒−훽푟 (1 − 푞푠)−푟 푑푟 0

∞ 1 − 푞푠 −푟 퐺 푠 = 훽 푒−훽푟 푑푟 0 푝

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∞ 1−푞푠 −푟푙푛 퐺 푠 = 훽 푒−훽푟 푒 푝 푑푟 0

∞ 1 1−푞푠 −푟훽 1+ 푙푛 퐺 푠 = 훽 푒 훽 푝 푑푟 (7.15) 0

Let

1 1 − 푞푠 푦 = 푟훽 1 + 푙푛 훽 푝 푦 푟 = 1 1−푞푠 훽 1 + 푙푛 훽 푝

푑푦 푑푟 = 1 1−푞푠 훽 1 + 푙푛 훽 푝

Hence

훽 ∞ −푦 퐺 푠 = 1 1−푞푠 푒 푑푦 (7.16) 훽 1+ 푙푛 0 훽 푝

퐺 푠 = 1 1−푞푠 (7.17) 1 + 푙푛 훽 푝 or

훽

퐺 푠 = 1−푞푠 (7.18) 훽 + 푙푛 푝

Mean

Differentiate the PGF with respect to s

훽

퐺 푠 = 1−푞푠 훽 + 푙푛 푝

119

1−푞푠 훽푝 푞 훽 + 푙푛 . 0 − . − 푝 1−푞푠 푝 퐺′ 푠 = 2 1−푞푠 훽 + 푙푛 푝

′ 훽푞 퐺 푠 = 2 (7.19) 1−푞푠 1 − 푞푠 훽 + 푙푛 푝

훽푞 1 퐺′ 1 = 2 (1 − 푞) 1−푞 훽 + 푙푛 푝

푞 퐺′ 1 = (7.20) 훽 1 − 푞

푞 1 − 푝 퐸 푋 = 퐺′ 1 = = (6.21) 훽 1 − 푞 훽푝

푝 = 1 − 푞

Variance

Let

2 1 − 푞푠 ퟏ − 풒풔 훽 + 푙푛 = 푉 푝

2 풅푽 1 − 푞푠 풑 풒 1 − 푞푠 = 2 ퟏ − 풒풔 훽 + 푙푛 − + 훽 + 푙푛 [−푞] 풅풔 푝 ퟏ − 풒풔 풑 푝

2 풅푽 1 − 푞푠 풒 1 − 푞푠 = −2 ퟏ − 풒풔 훽 + 푙푛 − 풒 훽 + 푙푛 풅풔 푝 ퟏ − 풒풔 푝

Now

2 2 1−푞푠 1−푞푠 풒 1−푞푠 1 − 푞푠 훽 + 푙푛 . 0 − 훽푞 −2 ퟏ − 풒풔 훽 + 푙푛 − 풒 훽 + 푙푛 푝 푝 ퟏ−풒풔 푝 ′′ 퐺 푠 = 4 1−푞푠 1 − 푞푠 2 훽 + 푙푛 푝

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2 1−푞푠 풒 1−푞푠 훽푞 2 ퟏ − 풒풔 훽 + 푙푛 + 풒 훽 + 푙푛 푝 ퟏ−풒풔 푝 ′′ 퐺 푠 = 4 (7.22) 1−푞푠 1 − 푞푠 2 훽 + 푙푛 푝

2 1−푞 풒 1−푞 훽푞 2 ퟏ − 풒 훽 + 푙푛 + 풒 훽 + 푙푛 푝 ퟏ−풒 푝 ′′ 퐺 1 = 4 1−푞 1 − 푞 2 훽 + 푙푛 푝

훽푞 2훽푞 + 푞훽2 퐺′′ 1 = (7.23) 1 − 푞 2훽4

푞2 2 + 훽 퐺′′ 1 = 1 − 푞 2훽2

(1 − 푝)2 2 + 훽 퐺′′ 1 = 푝2훽2

푉푎푟 푋 = 퐺′′ 1 − 퐺′ (1) 2 − 퐺′ (1)

(1 − 푝)2 2 + 훽 1 − 푝 2 1 − 푝 푉푎푟 푋 = − − 푝2훽2 훽푝 훽푝

(1 − 푝)2 2 + 훽 − (1 − 푝)2 + 훽푝 1 − 푝 푉푎푟 푋 = (7.24) 푝2훽2

(1 − 푝)2[2 + 훽 − 1] + 훽푝 1 − 푝 푉푎푟 푋 = 푝2훽2

(1 − 푝)2[훽 + 1] + 훽푝 1 − 푝 푉푎푟 푋 = 푝2훽2

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Negative Binomial – Binomial

Binomial distribution 푛 푝푟 1 − 푝 푛−푟 0 < 푝 < 1; 푟 = 0,1.2 … 푛 푔 푟 = 푟 0 푒푙푠푒푤푕푒푟푒

∞ 푟 + 푘 − 1 푟 푘 푝푘 = 푝 푞 푔 푟 푑푟 0 푘

∞ 푟 + 푘 − 1 푟 푘 푛 푟 푛−푟 푝푘 = 푝 푞 푝 1 − 푝 푑푟 0 푘 푟

The PGF is given by the following expression

∞ 푘 퐺 푠 = 푝푘 푠 푘=0

∞ ∞ 푛 푟 + 푘 − 1 퐺 푠 = 푝2푟 1 − 푝 푛−푟 (푠푞)푘 푑푟 푟 푘 0 푘=0

∞ ∞ 푛 −푟 퐺 푠 = 푝2푟 1 − 푝 푛−푟 (−푠푞)푘 푑푟 푟 푘 0 푘=0

∞ 푛 퐺 푠 = 푝2푟 푞푛−푟 (1 − 푠푞)−푟 푑푟 0 푟

∞ 푛 퐺 푠 = 푞푛 푝2푟 푞 1 − 푠푞 −푟 푑푟 (7.24) 0 푟

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CHAPTER 8

SUMMARY AND CONCLUSION New distributions were generated as anticipated in the project statement and objectives.

Chapter 1 briefly explained the concept around the Negative Binomial distribution. It goes to the extent of explaining the evolution and the reason why negative Binomial distribution is more efficient or more preferred as compared to other discrete distributions like Binomial and Poisson. The importance of probability mixing was highlighted as well. Applications areas were also briefly studied in this chapter

In the second chapter, we have shown various methods constructing Negative Binomial distribution. The main aim of this section was to not only identify various ways of developing Negative Binomial distribution, but to also verify the various forms of the Negative Binomial distribution. Some of the properties of the Negative Binomial distribution were identified in this chapter. Eg. Moments, mean, Variance, Kurtosis and Skewness

Chapter three majored in the construction of Negative Binomial Mixtures with prior distributions that are within the range of [0,1] and are related to beta distribution. The approach was to construct the mixing priors before subjecting them to the mixture. A few properties of these mixing distributions were discussed e.g. the jth moment, mean and variance.

Mixing distributions that are beyond Beta and within the [0,1] domain were considered as the mixing priors in chapter 4. The basic properties of the mixing distributions such as the jth moment, mean and variance were highlighted. The jth moment was key to mixing using the method of moments.

Chapter 5 and 6 were special cases of chapter 3 and 4 respectively by the fact that geometric distribution is also a special case of the Negative Binomial distribution. These chapters talks about the Geometric distribution mixtures with beta generated priors and Beyond Beta priors respectively. This was achieved by considering the first success (r=1).

A new way to conduct the mixing is through letting r be the varying variable instead of x. the parameter p is held constant in the respective mixtures. This was done in chapter 7 with few mixing distributions.

In conclusion, the project actually met its main objective of construction of Negative Binomial mixtures when the mixing distributions come from the probability of success and the number of success as random variable.

The following table is a summary of the project

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TABLED SUMMARY OF THE MIXTURES OF NEGATIVE BINOMIAL

Table 1: Negative Binomial mixtures with [0,1] domain distribution priors based on Classical Beta DISTRIBUTIONS BASED ON CLASSICAL BETA AND [0, 1] DOMAIN GENERATED BETA DISTRIBUTIONS Mixing Distribution Methods Mixture of mixing Classical beta Explicit ᴦ 푟 + 푥 퐵 푟 + 훼, 푥 + 훽 푓 푥 = ᴦ 푟 푥! 퐵 훼, 훽

푝훼−1 1 − 푝 훽−1 푥 Method Γ 푟 + 푥 푥 퐵 훼 + 푟 + 푘, 훽 푔 푝 = 푓 푥 = −1 푘 ; 퐵 훼, 훽 of Γ 푟 푥! 푘 퐵 훼, 훽 푓표푟 0 < 푝 < 1; 훼, 훽 > 0 moments 푘=0 푓표푟 푥 = 0,1,2, … ; 푟, 훼, 훽 > 0

Recursive a. Using ratio of the conditional distribution relation 푟 + 푥 − 1 푟 푝 푥 = 푝 푥 − 1 − 푝 푥 − 1 푟 푥 푟 푟 + 푥 − 1 푟+1

b. Using ration of the mixed distribution 푟 + 푥 − 1 2 푓 푥 = 푓 푥 − 1 푥 푟 + 푥 + 훼 + 훽 푓표푟 풓, 훼, 훽 > 0, 푥 = 1,2,3 … ;

c. Using a dummy function 푟 + 푥 − 1 푥 + 훽 − 1 푓 푥 = 푓 푥 = 푓 푥 − 1 푥 푟 + 푥 − 2 푓표푟 푟, 훽, 훼 > 0 푎푛푑 푥 = 1,2,3, …

Truncated Beta distribution Explicit 푟 + 푥 − 1 퐵훽 푎 + 푟, 푏 + 푥 − 퐵훼 푎 + 푟, 푏 + 푥 푓 푥 = 푥 퐵훽 푎, 푏 − 퐵훼 푎, 푏 푓표푟 푎, 푏; 1 < 훼 < 푝 < 훽 < 1; 푥 = 0,1,2 … > 0; 푎−1 푏−1 푝 1 − 푝 푔 푝 = 푥 퐵훽 푎, 푏 −퐵훼(푎, 푏) Method 퐵 푎 + 푟 + 푘, 푏 − 퐵 푎 + 푟 + 푘, 푏 Γ 푟 + 푥 푥 푘 훽 훼 푓표푟 1 < 훼 < 푝 < 훽 < 1; 푎, 푏 > 0 of 푓 푥 = −1 Γ 푟 푥! 푘 퐵훽 푎, 푏 −퐵훼 푎, 푏 moments 푘=0 푓표푟 푟 > 0; 1 < 훼 < 푝 < 훽 < 1; 푎, 푏 > 0; 푥 = 0,1,2, …

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1 1 Arcsine distribution Explicit 푟 + 푥 − 1 퐵(푟 + , 푥 + ) 1 푓(푥) = 2 2 푔1 푝 = 푥 휋 휋 푝 1 − 푝 1 1 푓표푟 푟 > 0; 푥 = 0,1,2, … ; 푎푛푑 휋 = 퐵 , 2 2 푓표푟 0 < 푝 < 1; 푎, 푏 > 0푎푛푑 휋 1 1 Method 푥 1 1 = 퐵 , Γ 푟 + 푥 푥 퐵 푟 + 푘 + , 2 2 of 푓 푥 = −1 푘 2 2 Γ 푟 푥! 푘 휋 moments 푘=0 1 1 푓표푟 푟, 훼 > 0; 푥 = 0,1,2, … , ; 푎푛푑 휋 = 퐵 , 2 2 Recursive 1st form relation 푟 + 푥 − 1 2푥 − 1 푓 푥 = 푓 푥 − 1 2푥 푟 + 푥 푓표푟 푟 > 0; 푥 = 1,2, … ;

2nd form 푟 1 − 2푥 푓 푥 = 푓 푥 − 1 푥 2푟 + 1 푓표푟 푟 > 0; 푥 = 1,2, … ; Power function distribution Explicit 푟 + 푥 − 1 푓 푥 = 훼 퐵(푟 + 훼, 푥 + 1) 푥 푓표푟 푥 = 0,12 …. ; 푎, 푟 > 0 푔 푝 = 푎푝푎−1 푓표푟 0 < 푝 < 1; 훼 > 0 Method 푥 of Γ 푟 + 푥 푥 훼 푓 푥 = −1 푘 moments Γ 푟 푥! 푘 훼 + 푟 + 푘 푘=0 푓표푟 푟, 훼 > 0; 푥 = 0,1,2, …

Recursive 1st form relation 푟 + 푥 − 1 푓 푥 = 푓 푥 − 1 푟 + 훼 + 푥 푓표푟 푟 > 0; 푥 = 1,2, … ; 2nd form 푟 + 푥 − 1 푟 + 푥 − 2 푓 푥 = 푓 푥 − 1 푟 + 훼 푟 − 1 푓표푟 푟 > 0; 푥 = 1,2, … ;

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Uniform distri2bution Explicit 푟 푥 = 0,1,2, … 푟 − 1; 푟 > 0 ퟏ ퟎ < 푝 < 1 푓 푥 = 푟 + 푥 + 1 푟 + 푥 품 풑 = ퟎ 풆풍풔풆풘풉풆풓풆 0 푒푙푠푒푤푕푒푟푒

푥 Γ 푟 + 푥 푥 1 푓 푥 = −1 푘 Γ 푟 푥! 푘 푟 + 푘 + 1 푘=0 푓표푟 푟 > 0; 푥 = 0,1,2, …

Recursive 1st form relation 푟 + 푥 − 1 푓 푥 = 푓 푥 − 1 푟 + 푥 + 1 2nd form 푟 푓 푥 = 푓 푥 − 1 푟 + 1 푓표푟 푟 > 0; 푥 = 1,2, … ; Explicit Confluent Hypergeometric distribution 퐵(푟 + 푎, 푥 + 푏)1퐹1 푎 + 푟, 푎 + 푏 + 푟 + 푥; −휇 푟 + 푥 − 1 푓 푥 = 푝푎−1 1 − 푝 푏−1 퐵 푎, 푏 퐹 푎, 푎 + 푏; −휇 푥 푔 푝 = 푒−푝휇 1 1 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇 푓표푟 푟, 푎, 푏 > 0; −∞ < 휇 < ∞; 푥 = 0,1,2, …

푥 푓표푟 0 < 푝 < 1; 푎, 푏 > 0; −∞ < 휇 < ∞ Method Γ 푟 + 푥 푥 퐵 푟 + 푘 + 푎, 푏 1퐹1 푎 + 푟 + 푘, 푎 + 푟 + 푘 + 푏; −휇 푓 푥 = −1 푘 of Γ 푟 푥! 푘 퐵 푎, 푏 1퐹1 푎, 푎 + 푏; −휇 moments 푘=0 푓표푟 푥 = 0,1,2, … ; 푎, 푏 > 표 ; 푟 > 0 Gauss Hypergeometric distribution Explicit 푔 푡 푟 + 푥 − 1 퐵 푎 + 푟, 푏 + 푥 2퐹1(푎 + 푟, 휀; 푎 + 푏 + 푟 + 푥; −푧) 푓(푥) = 푡푎−1 1 − 푡 푏−1 푥 퐵 푎, 푏 퐹 푎, 휀, 푎 + 푏; −푧 = 2 1 휀 푓표푟 푥 = 0,1,2 … ; 푟 = 0,1,2 … . ; 푎, 푏 > 0; −∞ < 휀 < ∞ 퐵 푎, 푏 1 + 푧푡 2퐹1 푎, 휀, 푎 + 푏; −푧 푓표푟 0 < 푡 < 1; 푎, 푏 > 0; −∞ < 휀 < ∞ 푥 Method Γ 푟 + 푥 푥 퐵 푗 + 푎, 푏 2퐹1 푟 + 푘 + 푎, 휀, 푎 + 푏 + 푟 + 푘; −푧 푓 푥 = −1 푘 of Γ 푟 푥! 푘 퐵 푎, 푏 2퐹1 푎, 휀, 푎 + 푏; −푧 푘=0 moments

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Table 2: Negative Binomial mixtures with [0,1] domain Beyond Beta distribution priors DISTRIBUTIONS BEYOND BETA Mixing Distribution Methods Mixture of mixing Kumaraswamy (I) Distribution Explicit 푟 + 푥 − 1 푓 푥 = 푏퐵 푟 + 1, 푥 + 푏 푓표푟 푥 = 0,1,2, … ; 푏, 푟 > 0 푥 푔 푢 푏−1 푥 푏 1 − 푢 푓표푟 0 < 푢 < 1; 푏 > 0 Method Γ 푟 + 푥 푥 = 푃푟표푏 푋 = 푥 = 푏 −1 푘 퐵 푟 + 푘 + 1, 푏 0 푒푙푠푒푤푕푒푟푒 of Γ r 푥! 푘 moments 푘=0 푓표푟 푟, 푏 > 0; 푥 = 0,1,2,3, …

푏 Truncated Exponential Distribution Explicit 푏 퐹 푟 + 1, 푟 + 푥 + 2; − 1 1 휆 푟 (TEX(휆, 푏)) 푓 푥 = 푏 − 푟 + 푥 + 1 푟 + 푥 푔 푝 휆 1 − 푒 휆 푏 −푏푝 /휆 푒 푓표푟 푥 = 0,1,2, . . ; 푏, 푟, 휆 > 0 휆 0 < 푝 < 1; 푏, 휆 > 0 = [1 − 푒−푏/휆 ] 0 푒푙푠푒푤푕푒푟푒

푏 Explicit 푟 + 푥 − 1 푎 1퐹1 푟 + 푏, 푟 + 푏 + 푥 + 1; −푎 퐵 푟 + 푏, 푥 + 1 푓 푥 = Truncated Gamma Distribution 푥 훾 푎, 푏 푔 푝 푓표푟 푥 = 0,1,2, … ; 푟, 푎, 푏 > 0 푎푏 푝푏−1푒−푎푝 = 0 < 푝 < 1; 푎, 푏 > 0 푥 훾 푎, 푏 Method Γ 푟 + 푥 푥 훾 푎, 푏 + 푟 + 푘 0 푒푙푠푒푤푕푒푟푒 of 푓(푥) = −1 푘 Γ 푟 푥! 푘 푎 푟+푘 훾 푎, 푏 moments 푘=0 푓표푟 푥 = 0,1,2, … ; 푟, 푎, 푏 > 0

푥 Minus Log Distribution Explicit 푟 + 푥 − 1 푥 1 − log 푝 0 < 푝 < 1 푓 푥 = −1 푘 푔 푝 = 푥 푘 푟 + 푘 + 1 2 0 푒푙푠푒푤푕푒푟푒 푘=0 푓표푟 푥 = 0,1,2, … 푟 > 0

푥 Method Γ 푟 + 푥 푥 1 푓 푥 = −1 푘 of Γ 푟 푥 ! 푘 푟 + 푘 + 1 2 moments 푘=0 푓표푟 푥 = 0,1,2, … 푟 > 0

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Standard Two – Sided Power Distribution Explicit 푘−1 푟 + 푥 − 1 퐵 푟 + 푘, 푥 + 1 퐵 푟 + 1, 푥 + 푘 − 퐵 푟 + 1, 푥 + 푘 푝 푓 푥 = 푘 휃 + 휃 푘 0 < 푝 < 휃 푥 휃푘−1 1 − 휃 푘−1 휃 푔 푝 = 푘−1 푓표푟 푥 = 0,1,2, … 푟 > 0 1 − 푝 푘 휃 < 푝 < 1 1 − 휃 Ogive Distribution Explicit 푟 + 푥 − 1 푚 + 1 2푟 + 푚 − 1 푓 푥 = 2푚퐵 + 1, 푥 + 1 + 1 − 푚 퐵 푟 + 푚 + 1, 푥 + 1 (4.27) 2푚 푚 + 1 푚 −1 1 − 푚2 푥 3푚 + 1 2 푔 푝 = 푝 2 + 푝푚 ; 푓표푟 푥 = 0,1,2, … . ; 푟 > 0; 푚 > 0 3푚 + 1 3푚 + 1 0 < 푝 > 1 ; 푚 > 0 Standard Two – Sided Ogive Distribution Explicit 푟 + 푥 − 1 푚 + 1 2푚 2푟 + 푚 − 1 1 − 푚 푔 푝 푓 푥 = 퐵 + 1, 푥 + 1 + 퐵 푟 − 푚 + 1, 푥 + 1 푚 −1 푚 −1 휃 푚 휃 2 푚 푥 3푚 + 1 2 휃 2푚 푚 + 1 푝 2 1 − 푚 푝 휃 2 + 0 < 푝 < 휃 ; 푚 > 0 3푚 + 1 휃 3푚 + 1 휃 2푚 2푥 + 푚 − 1 2푥 + 푚 − 1 = 푚 −1 2 푚 + 퐵 푟 + 1, + 1 − 퐵 (푟 + 1, 2푚 푚 + 1 1 − 푝 2 1 − 푚 1 − 푝 푚 −1 2 휃 2 + 휃 < 푝 < 1 ; 푚 > 0 2 1 − 휃 3푚 + 1 1 − 휃 3푚 + 1 1 − 휃 1 − 푚 + 1) 퐵 푟 + 1, 푥 + 푚 + 1 − 퐵 (푟 + 1, 푥 + 푚 + 1) 1 − 휃 푚 휃 푓표푟 푥 = 0,1,2, …. ; 푚 > 0 ; 푟 > 0 Triangular distribution Explicit 2푝 2ᴦ 푟 + 푥 퐵휃 푟 + 2, 푥 + 1 퐵 푟 + 1, 푥 + 2 − 퐵휃 푟 + 1, 푥 + 2 0 < 푝 < 휃 푓 푥 = + 휃 푥! Γ푟 휃 1 − 휃 푔 푝 = 2(1 − 푝) 푓표푟 푥 = 0,1,2, … … ; 0 < 휃 < 1, 푟 > 0 휃 < 푝 < 1 1 − 휃 Methods 1st Form 0 푒푙푠푒푤푕푒푟푒 푥 of Γ 푟 + 푥 푥 2 1 − 휃푟+푥−푘+1 푓 푥 = −1 푥−푘 moments Γ 푟 ! 푥 ! 푘 푟 + 푥 − 푘 + 1 푟 + 푥 − 푘 + 2 1 − 휃 푘=0 2nd Form 푥 Γ 푟 + 푥 푥 2 1 − 휃푟+푘+1 푓 푥 = −1 푘 Γ 푟 ! 푥 ! 푘 1 − 휃 푟 + 푘 + 1 푟 + 푘 + 2 푘=0

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A framework for the mixtures

Negative Binomial

distribution

Mixing distribution

Continuous distribution Discrete distribution

Finite negative Binomial Negative Binomial mixture mixture

Construction

Properties

Estimation

Applications

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This is the framework applied in the project. The mixing distributions are classified into two categories, namely, the continuous and the discrete mixing distributions. Constructions of these mixing distributions were first carried out before being incorporated in the mixture with the Negative Binomial distribution. Some of the properties, estimations and applications of the new distributions were highlighted.

Recommendations Negative Binomial mixtures can take other dimensions that were not actually covered in this project. More work can be done on these other scopes. They include:

a. Developing negative Binomial mixtures from the second definition of the Negative Binomial distribution such that

1 푥 − 1 푓 푥 = 푝푟 1 − 푝 푥−푟 푔 푝 푑푝 푥 = 푟, 푟 + 1, 푟 + 2, … 푟 − 1 0

b. Transforming the parameter푝 = 푒−휆so that the new distribution take the format

∞ 푟 + 푥 − 1 푥 푓 푥 = 푒−휆푟 1 − 푒−휆 푔 휆 푑휆 0 푥

푥 = 0,1,2,3 … . ; 푟, 휆 > 0

1 푥 − 1 푥−푟 푓 푥 = 푒−휆푟 1 − 푒−휆 푔 휆 푑휆 푟 − 1 0

푓표푟 푥 = 푟, 푟 + 1, 푟 + 2, … 휆 > 0

c. Transforming the parameter 푝 = 1 − 푒−휆such that the new distribution takes the form

∞ 푟 + 푥 − 1 푟 푓 푥 = 푒−휆푥 1 − 푒−휆 푔 휆 푑휆 0 푥

푥 = 0,1,2,3 … . ; 푟, 휆 > 0

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1 푥 − 1 푟 푓 푥 = 푒−휆 푥−푟 1 − 푒−휆 푔 휆 푑휆 푟 − 1 0

푓표푟 푥 = 푟, 푟 + 1, 푟 + 2, … 휆 > 0

We can generate the properties, estimate parameters and verify the identities of the new distributions.

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