Wrapped Log Kumaraswamy Distribution and Its Applications
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The Kumaraswamy Pareto IV Distribution
Austrian Journal of Statistics July 2021, Volume 50, 1{22. AJShttp://www.ajs.or.at/ doi:10.17713/ajs.v50i5.96 The Kumaraswamy Pareto IV Distribution M. H. Tahir Gauss M. Cordeiro M. Mansoor M. Zubair Ayman Alzaatreh Islamia University Federal University Govt. S.E. College Govt. S.E. College American University of Bahawapur of Pernambuco Bahawalpur Bahawalpur of Sharjah Abstract We introduce a new model named the Kumaraswamy Pareto IV distribution which extends the Pareto and Pareto IV distributions. The density function is very flexible and can be left-skewed, right-skewed and symmetrical shapes. It has increasing, decreasing, upside-down bathtub, bathtub, J and reversed-J shaped hazard rate shapes. Various structural properties are derived including explicit expressions for the quantile function, ordinary and incomplete moments, Bonferroni and Lorenz curves, mean deviations, mean residual life, mean waiting time, probability weighted moments and generating function. We provide the density function of the order statistics and their moments. The R´enyi and q entropies are also obtained. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. The usefulness of the new model is illustrated by means of three real-life data sets. In fact, our proposed model provides a better fit to these data than the gamma-Pareto IV, gamma-Pareto, beta-Pareto, exponentiated Pareto and Pareto IV models. Keywords: Arnold's Pareto, Kumaraswamy-G class, Pareto family, reliability. 1. Introduction The Pareto distribution and its generalizations are tractable statistical models for scien- tists, economists and engineers. These distributions cover a wide range of applications espe- cially in economics, finance, actuarial science, risk theory, reliability, telecommunications and medicine. -
The Kumaraswamy Generalized Gamma Distribution with Application in Survival Analysis
The Kumaraswamy Generalized Gamma Distribution with Application in Survival Analysis Marcelino A. R. de Pascoa and Edwin M. M. Ortega1 Department of Exact Sciences Universidade de São Paulo Gauss M. Cordeiro Department of Statistics and Informatics Universidade Federal Rural de Pernambuco Abstract We introduce and study the so-called Kumaraswamy generalized gamma distribution that is ca- pable of modeling bathtub-shaped hazard rate functions. The beauty and importance of this dis- tribution lies in its ability to model monotone and non-monotone failure rate functions, which are quite common in lifetime data analysis and reliability. The new distribution has a large number of well-known lifetime special sub-models such as the exponentiated generalized gamma, exponentiated Weibull, exponentiated generalized half-normal, exponentiated gamma, generalized Rayleigh, among others. Some structural properties of the new distribution are studied. We obtain two infinite sum representations for the moments and an expansion for the generating function. We calculate the den- sity function of the order statistics and an expansion for their moments. The method of maximum likelihood and a Bayesian procedure are adopted for estimating the model parameters. The usefulness of the new distribution is illustrated in two real data sets. Keywords: Censored data; Data analysis; Generalized gamma distribution; Maximum likelihood estimation; Moment; Order statistic. 1 Introduction The gamma distribution is the most popular model for analyzing skewed data. The generalized gamma distribution (GG) was introduced by Stacy (1962) and includes as special sub-models: the exponential, Weibull, gamma and Rayleigh distributions, among others. It is suitable for modeling data with different forms of hazard rate function: increasing, decreasing, in the form of bathtub and unimodal. -
C 2013 Johannes Traa
c 2013 Johannes Traa MULTICHANNEL SOURCE SEPARATION AND TRACKING WITH PHASE DIFFERENCES BY RANDOM SAMPLE CONSENSUS BY JOHANNES TRAA THESIS Submitted in partial fulfillment of the requirements for the degree of Master of Science in Electrical and Computer Engineering in the Graduate College of the University of Illinois at Urbana-Champaign, 2013 Urbana, Illinois Adviser: Assistant Professor Paris Smaragdis ABSTRACT Blind audio source separation (BASS) is a fascinating problem that has been tackled from many different angles. The use case of interest in this thesis is that of multiple moving and simultaneously-active speakers in a reverberant room. This is a common situation, for example, in social gatherings. We human beings have the remarkable ability to focus attention on a particular speaker while effectively ignoring the rest. This is referred to as the \cocktail party effect” and has been the holy grail of source separation for many decades. Replicating this feat in real-time with a machine is the goal of BASS. Single-channel methods attempt to identify the individual speakers from a single recording. However, with the advent of hand-held consumer electronics, techniques based on microphone array processing are becoming increasingly popular. Multichannel methods record a sound field from various locations to incorporate spatial information. If the speakers move over time, we need an algorithm capable of tracking their positions in the room. For compact arrays with 1-10 cm of separation between the microphones, this can be accomplished by applying a temporal filter on estimates of the directions-of-arrival (DOA) of the speakers. In this thesis, we review recent work on BSS with inter-channel phase difference (IPD) features and provide extensions to the case of moving speakers. -
Chapter 2 Parameter Estimation for Wrapped Distributions
ABSTRACT RAVINDRAN, PALANIKUMAR. Bayesian Analysis of Circular Data Using Wrapped Dis- tributions. (Under the direction of Associate Professor Sujit K. Ghosh). Circular data arise in a number of different areas such as geological, meteorolog- ical, biological and industrial sciences. We cannot use standard statistical techniques to model circular data, due to the circular geometry of the sample space. One of the com- mon methods used to analyze such data is the wrapping approach. Using the wrapping approach, we assume that, by wrapping a probability distribution from the real line onto the circle, we obtain the probability distribution for circular data. This approach creates a vast class of probability distributions that are flexible to account for different features of circular data. However, the likelihood-based inference for such distributions can be very complicated and computationally intensive. The EM algorithm used to compute the MLE is feasible, but is computationally unsatisfactory. Instead, we use Markov Chain Monte Carlo (MCMC) methods with a data augmentation step, to overcome such computational difficul- ties. Given a probability distribution on the circle, we assume that the original distribution was distributed on the real line, and then wrapped onto the circle. If we can unwrap the distribution off the circle and obtain a distribution on the real line, then the standard sta- tistical techniques for data on the real line can be used. Our proposed methods are flexible and computationally efficient to fit a wide class of wrapped distributions. Furthermore, we can easily compute the usual summary statistics. We present extensive simulation studies to validate the performance of our method. -
Arxiv:1711.00149V1 [Stat.ME] 1 Nov 2017 Rml Aoal,I Em Fsmlct,Wt H Eadsrbto.T Distribution
Statistical Inference of Kumaraswamy distribution under imprecise information Indranil Ghosh University of North Carolina, Wilmington, North Carolina e-mail: [email protected] Abstract Traditional statistical approaches for estimating the parameters of the Kumaraswamy distribution have dealt with precise information. However, in real world situations, some information about an underlying experimental process might be imprecise and might be represented in the form of fuzzy information. In this paper, we consider the problem of estimating the parameters of a univariate Ku- maraswamy distribution with two parameters when the available observations are described by means of fuzzy information. We derive the maximum likelihood estimate of the parameters by using Newton- Raphson as well as EM algorithm method. Furthermore, we provide an approximation namely, Tierney and Kadane’s approximation, to compute the Bayes estimates of the unknown parameters. The esti- mation procedures are discussed in details and compared via Markov Chain Monte Carlo simulations in terms of their average biases and mean squared errors. 1 Introduction Kumaraswamy (1980) introduced a two parameter absolutely continuous distribution which compares ex- tremely favorably, in terms of simplicity, with the beta distribution. The Kumaraswamy distribution on the interval (0, 1), has its probability density function (pdf) and its cumulative distribution function (cdf) with two shape parameters a> 0 and b> 0 defined by f(x)= a bxa−1(1 − xa)b−1I(0 <x< 1) and F (x)=1 − (1 − xa)b. (1) If a random variable X has pdf given in (1) then we will write X ∼ K(a,b). The density function in (1) has similar properties to those of the beta distribution but has some advantages in terms of tractability. -
A General Approach for Obtaining Wrapped Circular Distributions Via Mixtures
UC Santa Barbara UC Santa Barbara Previously Published Works Title A General Approach for obtaining wrapped circular distributions via mixtures Permalink https://escholarship.org/uc/item/5r7521z5 Journal Sankhya, 79 Author Jammalamadaka, Sreenivasa Rao Publication Date 2017 Peer reviewed eScholarship.org Powered by the California Digital Library University of California 1 23 Your article is protected by copyright and all rights are held exclusively by Indian Statistical Institute. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Sankhy¯a:TheIndianJournalofStatistics 2017, Volume 79-A, Part 1, pp. 133-157 c 2017, Indian Statistical Institute ! AGeneralApproachforObtainingWrappedCircular Distributions via Mixtures S. Rao Jammalamadaka University of California, Santa Barbara, USA Tomasz J. Kozubowski University of Nevada, Reno, USA Abstract We show that the operations of mixing and wrapping linear distributions around a unit circle commute, and can produce a wide variety of circular models. In particular, we show that many wrapped circular models studied in the literature can be obtained as scale mixtures of just the wrapped Gaussian and the wrapped exponential distributions, and inherit many properties from these two basic models. -
The Kumaraswamy Transmuted Pareto Distribution Sher B
Chhetri et al. Journal of Statistical Distributions and Applications (2017) 4:11 DOI 10.1186/s40488-017-0065-4 RESEARCH Open Access The Kumaraswamy transmuted Pareto distribution Sher B. Chhetri1, Alfred A. Akinsete2* , Gokarna Aryal3 and Hongwei Long1 *Correspondence: [email protected] Abstract 2Department of Mathematics, In this work, a new five-parameter Kumaraswamy transmuted Pareto (KwTP) Marshall University, Huntington, WV, USA distribution is introduced and studied. We discuss various mathematical and statistical Full list of author information is properties of the distribution including obtaining expressions for the moments, available at the end of the article quantiles, mean deviations, skewness, kurtosis, reliability and order statistics. The estimation of the model parameters is performed by the method of maximum likelihood. We compare the distribution with few other distributions to show its versatility in modeling data with heavy tail. Keywords: Kumaraswamy distribution, Quadratic rank transmutation map (QRTM), Pareto distribution, Hazard function, Maximum likelihood estimation AMS Subject Classification: 60E05, 62E15, 62H12 1 Introduction Furtherance to the work by Eugene et al. (2002), who proposed and defined the beta- generated class of distributions for a continuous random variable, derived from the logit of the beta random variable, many statistical distributions have been proposed and studied by numerous authors. According to Eugene et al. (2002), suppose X is a random variable with cumulative distribution function (CDF) F(x),theCDFforthe beta-generated family is obtained by applying the inverse probability transformation to the beta density function. The CDF for the beta-generated family of distributions is given by F(x) 1 − − G(x) = tu 1(1 − t)v 1 dt,0< u, v < ∞,(1) B(u, v) 0 where B(u, v) = (u)(v)/ (u + v). -
Newdistns: an R Package for New Families of Distributions
JSS Journal of Statistical Software March 2016, Volume 69, Issue 10. doi: 10.18637/jss.v069.i10 Newdistns: An R Package for New Families of Distributions Saralees Nadarajah Ricardo Rocha University of Manchester Universidade Federal de São Carlos Abstract The contributed R package Newdistns written by the authors is introduced. This pack- age computes the probability density function, cumulative distribution function, quantile function, random numbers and some measures of inference for nineteen families of distri- butions. Each family is flexible enough to encompass a large number of structures. The use of the package is illustrated using a real data set. Also robustness of random number generation is checked by simulation. Keywords: cumulative distribution function, probability density function, quantile function, random numbers. 1. Introduction Let G be any valid cumulative distribution function defined on the real line. The last decade or so has seen many approaches proposed for generating new distributions based on G. All of these approaches can be put in the form F (x) = B (G(x)) , (1) where B : [0, 1] → [0, 1] and F is a valid cumulative distribution function. So, for every G one can use (1) to generate a new distribution. The first approach of the kind of (1) proposed in recent years was that due to Marshall and Olkin(1997). In Marshall and Olkin(1997), B was taken to be B(p) = βp/ {1 − (1 − β)p} for β > 0. The distributions generated in this way using (1) will be referred to as Marshall Olkin G distributions. Since Marshall and Olkin(1997), many other approaches have been proposed. -
University of Nevada, Reno Generalized Univariate
University of Nevada, Reno Generalized Univariate Distributions and a New Asymmetric Laplace Model A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics By Palash Sharma Dr. Tomasz Kozubowski/Thesis Advisor August, 2017 c 2017 Palash Sharma ALL RIGHTS RESERVED THE GRADUATE SCHOOL We recommend that the thesis prepared under our supervision by Palash Sharma entitled Generalized Univariate Distributions and a New Asymmetric Laplace Model be accepted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Tomasz J. Kozubowski, Ph.D., Advisor Anna Panorska, Ph.D., Committee Member Minggen Lu, Ph.D., Graduate School Representative David Zeh, Ph.D., Dean, Graduate School August, 2017 i ABSTRACT Generalized Univariate Distributions and a New Asymmetric Laplace Model By Palash Sharma This work provides a survey of general class of distributions generated from a mixture of beta random variables. We provide an extensive review of the literature, concern- ing generating new distributions via the inverse CDF transformation. In particular, we account for beta generated and Kumaraswamy generated families of distributions. We provide a brief summary of each of their families of distributions. We also propose a new asymmetric mixture distribution, which is an alternative to beta generated dis- tributions. We provide basic properties of this new class of distributions generated from the Laplace model. We also address the issue of parameter estimation of this new skew generalized Laplace model. ii ACKNOWLEDGMENTS At first, I would like to thank my honorable thesis advisor, Professor Tomasz J. Kozubowski, who showed me a great interest in the field of statistics and probability theory. -
On Families of Generalized Pareto Distributions: Properties and Applications
Journal of Data Science 377-396 , DOI: 10.6339/JDS.201804_16(2).0008 On Families of Generalized Pareto Distributions: Properties and Applications D. Hamed1 and F. Famoye2 and C. Lee2 1 Department of Mathematics, Winthrop University, Rock Hill, SC, USA; 2Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan, USA In this paper, we introduce some new families of generalized Pareto distributions using the T-R{Y} framework. These families of distributions are named T-Pareto{Y} families, and they arise from the quantile functions of exponential, log-logistic, logistic, extreme value, Cauchy and Weibull distributions. The shapes of these T-Pareto families can be unimodal or bimodal, skewed to the left or skewed to the right with heavy tail. Some general properties of the T-Pareto{Y} family are investigated and these include the moments, modes, mean deviations from the mean and from the median, and Shannon entropy. Several new generalized Pareto distributions are also discussed. Four real data sets from engineering, biomedical and social science are analyzed to demonstrate the flexibility and usefulness of the T-Pareto{Y} families of distributions. Key words: Shannon entropy; quantile function; moment; T-X family. 1. Introduction The Pareto distribution is named after the well-known Italian-born Swiss sociologist and economist Vilfredo Pareto (1848-1923). Pareto [1] defined Pareto’s Law, which can be stated as N Axa , where N represents the number of persons having income x in a population. Pareto distribution is commonly used in modelling heavy tailed distributions, including but not limited to income, insurance and city size populations. -
On Wrapped Binomial Model Characteristics
Mathematics and Statistics 2(7): 231-234, 2014 http://www.hrpub.org DOI: 10.13189/ms.2014.020701 On Wrapped Binomial Model Characteristics S.V.S.Girija1,*, A.V. Dattatreya Rao2, G.V.L.N. Srihari 3 1Mathematics, Hindu College, Guntur, India 2Acharya Nagarjuna University, Guntur, India 3Mathematics, Aurora Engineering College, Bhongir *Corresponding Author: [email protected] Copyright © 2014 Horizon Research Publishing All rights reserved. Abstract In this paper, a new discrete circular model, the latter case, the probability density function f (θ ) the Wrapped Binomial model is constructed by applying the method of wrapping a discrete linear model. The exists and has the following basic properties. characteristic function of the Wrapped Binomial Distribution f (θ ) ≥ 0 is also derived and the population characteristics are studied. 2π Keywords Characteristic Function, Circular Models, ∫ fd(θθ) =1 Trigonometric Moments, Wrapped Models 0 ff(θ) =( θπ + 2 k) , for any integer k, That is , f (θ ) is periodic with period 2.π 1. Introduction 2.1. Wrapped Discrete Circular Random Variables Circular or directional data arise in many diverse fields, such as Biology, Geology, Physics, Meteorology, and If X is a discrete random variable on the set of integers, + psychology, Medicine, Image Processing, Political Science, then reduction modulo 2π mm( ∈ ) wraps the integers Economics and Astronomy (Mardia and Jupp (2000). on to the group of mth roots of unity which is a sub group of Wrapped circular models (Girija (2010)), stereographic unit circle. circular models (Phani (2013)) and Offset circular models i.e. Φ=2ππxm( mod 2 ) (Radhika (2014)) provide a rich and very useful class of models for circular as well as l-axial data. -
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 BY 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 i 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. ii 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.