Abstract
Mixed Poisson distributions have been used in scientificfields for modelling non-homogenous populations, (Karlis and xekalaki, 2005). For example, in acturial applications, mixed Poisson distributions are used for modeling total claims in insurance. In this work, the concentration is mainly on the estimation of the parameters of Poisson- Lindley distribution using EM Algorithm which wasfirst introduced by Dempster (1977). One-parameter, two-parameter and three-parameter Lindley distributions are compounded by the Poisson distribution to form the Poisson-Lindley distributions and then the param- eters estimated. In order to carry out the EM Algorithm successfully, the posterior distribution is applied and the posterior expectation calculated.
Various properties of each distribution are determined, for example, ther th moment, the cumulative distribution function (cdf), the moment generating function (mgf), the prob- ability generating function (mgf) and the characteristic function.
Table of Contents
Abstract ii
Declaration and Approval iv
Dedication vii
Acknowledgments x
Chapter 1: General Introduction
1.1 Background Information 1
1.2 Research Problem 1
1.3 Objectives 2
1.4 Methodology 2
1.5 Literature Review 3
1.6 Significance of the study 4
Chapter 2: Poisson-One Parameter Lindley Distribution
2.1 Introduction 5
2.2 Construction 5
2.3 Posterior Distribution 6
2.4 Properties 7
2.4.1 The rth moment 7
2.5 Estimation 8
2.5.1 EM Algorithm for One-Parameter Poisson-Lindley Distribution 8
Iterative schemes 10
Iterative scheme 1 11
Iterative scheme 2 11
Chapter 3: Poisson-Two Parameter Lindley Distribution
3.1 Introduction 13
3.2 Construction 13
3.3 Posterior Distribution 14
3.4 Properties 15
3.4.1 The rth moment 15
3.4.2 Probability Generating Function 16
3.4.3 Moment Generating Function 17
3.5 Estimation 17
3.5.1 EM Algorithm for Two-Parameter Poisson-Lindley Distribution 17
Iterative schemes 19
Iterative scheme 1 19
Iterative scheme 2 19
Chapter 4: Poisson-Two Parameter Lindley Distribution
4.1 Introduction 21
4.2 Construction 21
4.3 Posterior Distribution 22
4.4 Properties 23
4.4.1 Cumulative Distribution Function 23
4.4.2 Ther th Moment 23
4.5 Estimation 23
4.5.1 EM Algorithm for Two-Parameter Poisson-Linldey Distribution 23
Iterative Schemes 26
Iterative Scheme 1 26
Chapter 5: Poisson Quasi-Lindley Distribution
5.1 Introduction 27
5.2 Construction 27
5.3 Posterior Distribution 28
5.4 Properties 29
5.4.1 The rth moment 29
5.4.2 Cumulative Distribution Function 29
5.4.3 Moment Generating Function 29
5.5 Estimation 30
5.5.1 EM Algorithm for Poisson Quasi-Lindley Distribution 30
Iterative Schemes 31
Chapter 6: Poisson -Three Parameter Lindley Distribution
6.1 Introduction 33
6.2 Construction 33
6.3 Posterior Distribution 34
6.4 Properties 35
6.4.1 Cumulative Distribution Function 35
6.4.2 Probability Generating Function 36
6.4.3 Moment Generating Function 36
6.4.4 Characteristic Function 37
6.4.5 The rth Moment 37
6.5 Estimation 37
6.5.1 EM Algorithm for Three Parameter Poisson-Lindley Distribution 38
Iterative Schemes 40
Iterative Scheme 1 40
Iterative Scheme 2 41
Chapter 7: Conclusion and Recommendations
7.1 Conclusion 42
7.2 Recommendations 42
Bibliography 43
Chapter 1
General Introduction
1.1 Background Information
Probability distribution can be constructed by combining two or more distributions to obtain a new distribution called called a mixture. There are three types of mixtures namely; finite mixtures, continous mixtures and discrete mixtures.
Lindley distribution is an example of a finite mixture of two gamma distributions. It was first introduced by D.V Lindley (1958) in the context of Bayesian analysis as a counter example of fiducial statistics. The Lindley distribution has been used in the modelling and analyzing of lifetime data that are crucial in many applied sciences including medicine, engineering, insurance and finance. The introduction of Poisson-Lindley distribution was as a result of comparing the Poisson distribution with the Poisson-Lindley distribution to see the most preferable.
This was achieved by Shanker and Fesshaye, (2015) who fiNed some data sets in ecology and genetics to the two distributions and they found that Poisson-Lindley distribution is more flexible for analyzing different types of count data than Poisson distribution. The EM Algorithm was first introduced by Dempster (1977) to obtain the MLEs of incom- plete data. Mclachlan (2004) states that the EM Algorithm is n applicable approach in the iterative computation of maximum likelihood estimates which is useful in a variety of incomplete data.
Mclachlan (2004) explains that the EM Algorithm is simply a generic method for com- puting the MLE of an incomplete data by formulating an associated complete data and exploiting the simplicity of the MLE of the laNer to compute the MLE of the former.
1.2 Research Problem
Karlis(2005), discussed the EM Algorithm for mixed Poisson and other discrete distribu- tions and he considered the Poisson-Lindley distribution using the posterior expectation which was first introduced by Sapatinas (1995).
However, he did not show how we obtain the posterior expectation of the Poisson-Lindley distribution and how he arrived at the final solution. The problem is one would like to know how he came up with the final solution.
This project reconstructs the Poisson-Lindley distribution, that is, one parameter, two parameter and three parameter Poisson-Lindley distributions and then estimate the parameters of the distributions using the EM Algorithm method where the posterior distribution and the posterior expectation is applied.
1.3 Objectives
The main objective of this study is to estimate the parameters of Poisson-Lindley distribution with the help of the posterior distribution and the posterior expectation.
The specific objectives are:
1. To estimate the parameter of one-parameter Poisson-Lindley distribution using the EM Algorithm method.
2. Using the EM Algorithm method to estimate the parameters of two-parameter Poisson- Lindley distribution.
3. Estimating the parameters of three-parameter Poisson-Lindley distribution using the EM Algorithm method.
1.4 Methodology
The method used for the estimation of the parameters is the EM Algorithm method which is described below.
Expectation-Maximization Algorithm method EM Algorithm method was first introduced by Dempster et al. (1977) to obtain the maximum likelihood estimates for an incomplete data.
Assume that the true data are made of an observed part X and unobserved part λ.
This then ensures that the log likelihood function of the complete data(x i,λ i) for i= 1,2,3,...,n factorizes into two parts (Kostas 2007).
This then implies that the joint density function of X andΛis given by:
f(x,λ) =f(x/λ)g(λ)
The likelihood function is:
For optimization, l1 is differentiated with respect to parameters in the function f(xi / λi); then equated to zero.
Similarly, l2 is differentiated with respect to the parameters in g(λi) and then equated to zero.
1.5 Literature Review
Estimating the parameters of a mixing distribution was first introduced by Pearson (1984) who estimated the parameters of the mixture of two normal densities. Poisson-Lindley distribution was first introduced by Sankaran (1970) who introduced the one-parameter discrete Poisson-Lindley distribution by mixing Poisson distribution with the Lindley distribution.
Although there are up to five parameters Lindley distribution, Poisson-Lindley distribution have only been introduced upto three-parameter Poisson-Lindley distribution which include Poisson-Lindley (Sankaran 1970), Poisson Quasi-Lindley (Shanker and Mishra, 2013), another two parameter Poisson-Lindley (Shanker and Mishra, 2013), thre parametr Poisson-Lindley (Kishore et. al, 2018).
Under estimation, EM Algorithm is used to obtain the estimates of the parameters. EM Algorithm have not been considered in most of the articles as a way of estimating the parameters.
More concentration have been on the method of moments and the maximum likelihood estimation as a way of estimating the parameters of a distribution.
Karlis (2005) was the first to use EM Algorithm method to estimate the parameters of mixed Poisson and other discrete distributions. Shanker and Fesshaye (2015) used the method of moments and the maximum likelihood estimation method to estimate the parameter of one-parameter Poisson-Lindley distribution.
Shanker and Mishra (2015) used the method of moments and the maximum likelihood estimation method to estimate the parameters of a Poisson Quasi-Lindley distribution. Kishore et. al (2018) introduced a new three-parameter Poisson-Lindley distribution and used the method of moments and the maximum likelihood estimation method in estimating the parameters of the distribution.
1.6 Significance of the study
Mixture models have a wide variety of applications in statistics.
The number of applications have increased in the recent years mainly because of the availability of high speed computer resources which have removed any obstacles to apply such methods.
"Thus mixture models have found applications in fields as diverse as data modelling, discriminant analysis, cluster analysis, outlier-robustness studies, ANOVA models, kernel density estimation, latent structure models, empirical Bayes estimation, Bayesian statistics, random variable generation, approximation of the distribution of some statistic and others." (Karlis and Xekalaki, 2005).
Therefore, Poisson mixtures have also a variety of applications in statistical science since its an example of ixture models.
In this work, our main concern is on the applications of Poisson-Lindley distributions. Poisson-Lindley distribution was first introduced by Sankaran (1970) to model count data. In Ecology, Poisson-Linldey distribution has been used as an important tool for modelling count data to analyse the relationship between organisms and their environment.(Shanker nad Fesshaye, 2015)
In Genetics, which is a branch of biological science that deals with heredity and variation, Poisson-Lindley distribution has been considered as an important tool for modelling the genetics data.
In acturial science, Sankaran (1970) applied the Poisson-Lindley distribution to errors and accidents.
Ghitany (2009) applied the Poisson-Lindley distribution to determine service rate (how long a customer waits on the queue at the bank).
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