DOCSLIB.ORG

The Kumaraswamy-Rani Distribution and Its Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Kumaraswamy-Rani Distribution and Its Applications Research Article Journal of Volume 11:1, 2020 DOI: 10.37421/jbmbs.2020.11.436 Biometrics & Biostatistics ISSN: 2155-6180 Open Access The Kumaraswamy-Rani Distribution and Its Applications Zitouni Mouna* Laboratory of Probability and Statistics. Sfax University. B.P. 1171, Tunisia Abstract In this paper, a new distribution called the Kumaraswamy-Rani (KR) distribution, as a Special model from the class of Kumaraswamy Generalized (KW-G) distributions, is introduced. Its statistical properties are explored. Estimation parameters based on maximum likelihood are obtained. We have illustrated the performances of the proposed distribution by some simulation studies. Finally, the significance of the KR distribution in terms of modeling real data set has been highlighted before it has been compared to some one parameter lifetime distributions. Keywords: Fixed point method • Kumaraswamy distribution • Maximum likelihood estimation • Newton-raphson method • Rani distribution Introduction The KR Distribution Modeling and analysis of lifetime data is an important aspect of The Kumaraswamy-Generalized distribution statistical work in such wide variety of scientific and technological fields as medicine, engineering, insurance and finance... The cumulative density function (cdf) of the Kumaraswamy-Generalized (Kum-Generalized) distribution proposed by Cordeiro et al. [8] is given by Therefore, the construction of the distributions is the most important −− ab work from a probabilistic point of view and recently many researchers are Fx( ) = 1 (1 ( Gx ( )) ) , (1) concentrated in this area [1-3]. Where a>0, b>0 are shape parameters and G is the cdf of a continuous random variable X. Consquently, the Kumaraswamy-Generalized More that, varying lifetime distributions and their generalizations have distribution has the probability density function (pdf) specified by been created as exponential, Lindley, gamma, lognormal, Weibull, Sujatha, Akash, Rani. Further, A two-parameter family of distributions on called f()= x abg () x G () x(a−− 1) (1− G ()) x ab( 1) , (2) Kumaraswamy distribution is explored. It was introduced by Kumaraswamy ∂Gx() [4]. It has the flexibility in modeling lifetime data. where, gx( )= . ∂x The latter exhibits many similarities to the beta distribution and a According to Cordeiro et al. [8], using the binomial expansion for + number of advantages in terms of tractability which has been discussed by b ∈ , equation (2) can be rewritten as Jones [5]. Among these advantages, we mention: a simple explicit formula ∞ ai(+− 1) 1 for its distribution function, its quantile function. L-moments and moments f()=() x g x∑ wGi () x , (3) i=0 of order statistics [6]. In this direction, another new one parameter lifetime distribution has attracted the attention of statisticians and whetted then where the binomial coefficient wi is defined for all real numbers with, interest named Rani distribution. It was proposed by Rama Shanker and it b −1 has important statistical properties [7]. For the motivating properties of the − i wi = ( 1) ab i Kumaraswamy and Rani distributions, we shall introduce a new probability distribution based on the Kumaraswamy distribution The Rani distribution and the Rani distribution. The proposed distribution is called the The Rani distribution is proposed by Rama Shanker [7]. It is defined Kumaraswamy-Rani (KR) distribution with twos positive parameters over the support x>0 and it has the pdf given by: denoted b and β. In this study some statistical properties of this distribution θ 5 are derived to show their applicability in the real lifetime data analysis. The fx( )= (θ + x4 ) e−θ x , (4) rest of the paper is organized as follows. In Section 2, the KR distribution θ 5 + 24 is identified in addition, some important mathematical properties of the new where θ is the scale parameter of the Rani distribution denoted R(θ). The distribution are presented. In Section 3, maximum likelihood estimation latter is a mixture of exponential (θ) and gamma (5,θ) with mixing proportion is discussed. In section 4, some simulation studies are reported. A real θ 5 . Furthermore, the corresponding cumulative distribution function lifetime data set are used in Section 5 to illustrate the usefulness of the KR θ 5 + 24 distribution. Finally, our work is crowned with conclusion. (cdf) is given by: θθxx(33+ 4 θ 22 x ++ 12 θ x 24) −+ −θ x (5) *Address for Correspondence: Zitouni Mouna, Laboratory of Probability and Fx( )=1 1 5 e . Statistics. Sfax University. B.P. 1171, Tunisia, Tel: +216 74 244 423; E-mail: θ + 24 [email protected] The KR distribution Copyright: © 2020 Mouna Z. This is an open-access article distributed under the In this subsection, we give the new distribution called KR distribution. terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author It is formulated using the Rani (θ>0) and the Kumaraswamy (ab = 1, > 0) and source are credited. distributions. Received 21 January 2020; Accepted 06 February 2020; Published 13 February The KR density for x>0 is expressed as 2020 Mouna Z. J Biom Biostat, Volume 11:1, 2020 (b− 1) 5 33 22 5 θ −−θθθθxx(+ 4 θ x ++ 12 θ x 24) θ ++4 xx θ 4 −θ x fx( )= b55 ( x ) e 1 e . (6) θ + θθ++ b5 () xe 24 24 fx() θ + 24 Hx( )= = 33 22 . The parameters vector of the KR distribution is Θ =(b ,β ). Moreover, 1− Fx () θθxx(+ 4 θ x ++ 12 θ x 24) −θ x 1+ 5 e using equation (3), equation (6) can be defined as: θ + 24 i θ5 ∞ θθ33+ θ 22 ++ θ 4 −−θθxxxx( 4 x 12 x 24) Now, we shall introduce the following Theorem about the inverse fx( )= (θ +x ) e wi 1−+ 1 e , (7) θθ55++∑ 24 i=0 24 cumulative KR distribution. The latter characterizes the proposed b −1 distribution. −1 − i Theorem 2.1: θ where wi = ( 1) bi. Let Y KR( b , ), then xF= () y is the zero of the polynomial equation given by Remark 1: Let X KR( b ,θ ). If b=1, then XR ()θ (Figure 1). θθ44x+4, 33 x + Ax 2 ++ Bx C (9) We can find easily the cumulative distribtion function (c.d.f) of the KR where, distribution θθ771 b A =− +−− (1 (1µθ )b )( + 122 ), θθxx(33+ 4 θ 22 x ++ 12 θ x 24) Fx( ) = 1−+ 1 e−θ x , x> 0. (8) 22 5 θ + 24 1 B = (1−− (1µ )b )( θ66 + 24 θθ ) − , Using the c.d.f in order to obtain the Survival function (Figure 2): b 1 33 22 5 b θθxx(+ 4 θ x ++ 12 θ x 24) −θ +θµ −− Sx( )=1−+ Fx ( )= 1 e x C = (24 )(1 (1 ) ) 5 θ + 24 and µ ∈]0,1[ . and the Hazard function which is an important quantity characterizing life phenomena is given by Proof: In order to get the zero of equation (9), we shall introduce the following equation: Fx( ) = 1−− (1 Gx ( ))b =µµ , ∈]0,1[. 0.5 So, b=0.4, theta=1.5 0.45 b=0.4, theta=2 θθxx(33+ 4 θ 22 x ++ 12 θ x 24) 1 b=0.6, theta=1.5 −+ −θ x −−µ b 1 1 5 e = 1 (1 ) . 0.4 θ + 24 0.35 We have also, 1 5 33 22 0.3 θ++24 θθxx ( + 4 θ x + 12 θ x + 24) −−µ b −θ x (1 ) e 5 = 0. f(x) θ + 24 0.25 In the following, we get, 0.2 1 θ5++24 θθxx (33 + 4 θ 22 x + 12 θ x + 24) θ x − µ b 0.15 e (1 ) = 5 . θ + 24 0.1 It is to be noted that the limited development to order 2 of exponential 0.05 function at 0 can be written as: 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 θ x ()θ x x ex=1++θ . 2 Figure 1: The KR densities of selected values of b and θ... Then, we obtain: ()θx2 1 θ5++24 θθxx (33 + 4 θ 22 x + 12 θ x + 24) ++θµ −b (10) 1 x ((1 ) ) = 5 . 2 θ + 24 1 Applying some mathematical treatments on equation (10) yields, b=8, theta=2 0.9 1 7 b=6, theta=8 44 33 2 b θ 2 b=5, theta=6 θθx+4 xx + −− ((1 µ ) )( + 12 θ ) 0.8 2 11 0.7 +(1 −µθ )bb (65 + 24 θ )x + (24 + θ )(1 −− (1 µ ) ) = 0. 0.6 0.5 Proving that 0.4 θθ44x+4 33 x + Ax 2 ++ Bx C = 0, 0.3 where, 0.2 1 7 b θ 2 0.1 A =−− ((1µθ ) )( + 12 ), 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 B = ((1−+µθ )b )(6 24 θ ) Figure 2: The cumulative distribution functions of selected values of b and θ.. and Page 2 of 4 Mouna Z. J Biom Biostat, Volume 11:1, 2020 1 5 C = (24+θµ )(1 −− (1 )b ) 4.5 which completes this proof. 4 3.5 Parameters Estimations 3 In this section, the maximum likelihood method for the parameters 2.5 estimation of the KR distribution is derived. For this reason, a brief f(x) description of the method for the fitting of the KR parameters is presented. 2 The likelihood function for the parameters is given by 1.5 (b− 1) n 5 33 22 θ −−θθxxθθxx(+ 4 θ x ++ 12 θ x 24) θ ++4 iiii i i L=∏ b55 () xei 1 e . θθ++24 24 (11) 1 i=1 The log-likelihood is formulated as: 0.5 n 5 θ −θ 0 ++θ 4 xi 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 log( L ) = n log( b )∑ log5 (xi ) e i=1 θ + 24 x n 33 22 Figure 3: Histogram of random samples from a KR distribution and a curve of θθxx(+ 4 θ x ++ 12 θ x 24) −θ x +−(be 1)log 1 +ii i i i .
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Wrapped Log Kumaraswamy Distribution and Its Applications
    International Journal of Mathematics and Computer Research ISSN: 2320-7167 Volume 06 Issue 10 October-2018, Page no. - 1924-1930 Index Copernicus ICV: 57.55 DOI: 10.31142/ijmcr/v6i10.01 Wrapped Log Kumaraswamy Distribution and its Applications K.K. Jose1, Jisha Varghese2 1,2Department of Statistics, St.Thomas College, Palai,Arunapuram Mahatma Gandhi University, Kottayam, Kerala- 686 574, India ARTICLE INFO ABSTRACT Published Online: A new circular distribution called Wrapped Log Kumaraswamy Distribution (WLKD) is introduced 10 October 2018 in this paper. We obtain explicit form for the probability density function and derive expressions for distribution function, characteristic function and trigonometric moments. Method of maximum likelihood estimation is used for estimation of parameters. The proposed model is also applied to a Corresponding Author: real data set on repair times and it is established that the WLKD is better than log Kumaraswamy K.K. Jose distribution for modeling the data. KEYWORDS: Log Kumaraswamy-Geometric distribution, Wrapped Log Kumaraswamy distribution, Trigonometric moments. 1. Introduction a test, atmospheric temperatures, hydrological data, etc. Also, Kumaraswamy (1980) introduced a two-parameter this distribution could be appropriate in situations where distribution over the support [0, 1], called Kumaraswamy scientists use probability distributions which have infinite distribution (KD), for double bounded random processes for lower and (or) upper bounds to fit data, where as in reality the hydrological applications. The Kumaraswamy distribution is bounds are finite. Gupta and Kirmani (1988) discussed the very similar to the Beta distribution, but has the important connection between non-homogeneous Poisson process advantage of an invertible closed form cumulative (NHPP) and record values.
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • On the Kumaraswamy Fisher Snedecor Distribution
    Mathematics and Statistics 4(1): 1-14, 2016 http://www.hrpub.org DOI: 10.13189/ms.2016.040101 On the Kumaraswamy Fisher Snedecor Distribution Adepoju, K.A*, Chukwu, A.U, Shittu, O.I Department of Statistics, University of Ibadan, Nigeria Copyright©2016 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract We propose the Kumaraswamy-F (KUMAF) population means relies heavily on a good number of distribution which is a generalization of the conventional assumptions of the Analysis of Variance (ANOVA). Fisher Snedecor (F-distribution). The new distribution can For the F-ratio statistic there are two fundamental be used even when one or more of the regular assumptions assumptions: the variances of the compared populations are are violated. It is obtained with the addition of two shape the same; the estimates of the population variance are parameters to a continuous F-distribution which is independent. commonly used to test the null hypothesis in the Analysis of Therefore before we proceed with an analysis of the data Variance (ANOVA test). The statistical properties of the we have collected we have to make sure that these proposed distribution such as moments, moment generating assumptions have been met. function, the asymptotic behavior among others were These assumptions include independent of k populations investigated. The method of maximum likelihood is used to being tested, equality of the population variances, and estimate the model parameters and the observed information absence of outlier among others. When a number of the matrix is derived.
    [Show full text]
  • An Advanced Count Data Model with Applications in Genetics
    Appl. Math. Inf. Sci. Lett. 6, No. 3, 113-121 (2018) 113 Applied Mathematics & Information Sciences Letters An International Journal http://dx.doi.org/10.18576/amisl/060303 An Advanced Count Data Model with Applications in Genetics Zahoor Ahmad*, Adil Rahsid and T. R. Jan. P.G Department of Statistics, University of Kashmir, Srinagar, India Received: 19 Jan. 2018, Revised: 10 Mar.2018, Accepted: 15 Mar.2018. Published online: 1 Sep. 2018. Abstract:The present paper introduces an advanced count model which is obtained by compounding generalized negative binomial distribution with Kumaraswamy distribution. The proposed model has several properties such as it can be nested to different existing compound distributions on specific parameter setting. Similarity of the proposed model with existing compound distribution has been shown by means of reparameterization. The properties of the new model are discussed and explicit expressions are derived for the factorial moments. Further method of moments and maximum likelihood estimation is used to evaluate the moments. The potentiality of the proposed model has been tested by chi-square goodness of fit test by modeling the real world count data sets from genetics. Keywords: Generalized negative binomial distribution, Kumaraswamy distribution, compound distribution, factorial moment, count data. 1 Introduction distribution. In several research papers it has been found that compound Distributions are very flexible and can be used efficiently to Statistical distributions are commonly applied to describe model different types of data sets. With this in mind many real world phenomenon. Dueto the usefulness of statistical compound probability distributions have been constructed. distributions their theory is widely studied and new Sankaran (1970) obtained a compound of Poisson distributions are developed.
    [Show full text]
  • Cubic Transmuted Uniform Distribution: an Alternative to Beta and Kumaraswamy Distributions Md
    EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 3, 2019, 1106-1121 ISSN 1307-5543 { www.ejpam.com Published by New York Business Global Cubic Transmuted Uniform Distribution: An Alternative to Beta and Kumaraswamy Distributions Md. Mahabubur Rahman1;2, Bander Al-Zahrani1, Saman Hanif Shahbaz1, Muhammad Qaiser Shahbaz1,∗ 1 Department of Statistics, King Abdulaziz University, Jeddah, Saudi Arabia 2 Department of Statistics, Islamic University, Kushtia, Bangladesh Abstract. In this article, a new cubic transmuted (CT ) family of distributions has been proposed by adding one more parameter. We have introduced cubic transmuted uniform (CTU) distribution by using the proposed class. We have also provided a detail description of the statistical properties of the proposed CTU distribution along with its estimation and real-life application. 2010 Mathematics Subject Classifications: 33B20, 46N30, 60B15 Key Words and Phrases: Transmuted Distributions, Uniform Distribution, Maximum Likeli- hood Estimation 1. Introduction In statistical data analysis, quality of the procedures mainly depends upon the assumed probability model of the phenomenon. Large numbers of probability models are, therefore, being developed by the researcher. Enormous practical problems are on hand where the standard and extended probability distributions does not work. The generalization of standard probability distributions has been an area of interest by several authors. Azzalini [4] proposed a new method of generalizing the probability distributions by adding skewness parameter to symmetric distributions and this class is referred to as the skew-symmetric distributions. Eugene et al. [9] developed the Beta − G family of distributions by using the logit of beta distribution. The Beta − G family of dis- tributions extends the distribution of order statistics.
    [Show full text]