**ABSTRACT**

In 2001, Surles & Padgett introduced Generalized Rayleigh Distribution (GRD). This

skewed distribution can be used quiet effectively in modeling life time data. In this work,

Bayesian estimates of the shape parameter of a GRD were determined under the

assumption of both informative (gamma) and non-informative (Extended Jeffery’s and

Uniform) priors. The Bayes estimates were obtained under both symmetric and asymmetric

loss functions. The performances of these estimates were compared to the Maximum

Likelihood Estimates (MLEs) using Monte Carlo simulation.

**TABLE OF CONTENTS**

Declaration …………………………………………………………………………………………………………….. iv

Certification ……………………………………………………………………………………………………………. v

Dedication ……………………………………………………………………………………………………………… vi

Acknowledgments ………………………………………………………………………………………………….vii

Abstract ………………………………………………………………………………………………………………. viii

Table of Contents ……………………………………………………………………………………………………. ix

List of Figures ………………………………………………………………………………………………………… xi

List of Tables ………………………………………………………………………………………………………. xiii

CHAPTER ONE: INTRODUCTION ……………………………………………………………………… 1

1.1 Background to the Study …………………………………………………………………………………….. 1

1.1.1 Theory of estimation ……………………………………………………………………………………………………………… 1

1.1.2 Generalized Rayleigh distribution (GRD) ………………………………………………………………………………… 2

1.2 Statement of the Problem ……………………………………………………………………………………. 4

1.3 Aim and Objectives of the Study ………………………………………………………………………….. 5

1.4 Significance of the Study …………………………………………………………………………………….. 5

1.5 Motivation …………………………………………………………………………………………………………. 6

1.6 Limitation …………………………………………………………………………………………………………. 6

1.7 Definition of Terms …………………………………………………………………………………………….. 6

1.9.1 Estimator ……………………………………………………………………………………………………………………………… 6

1.9.2 Prior distribution …………………………………………………………………………………………………………………… 7

1.9.3 Posterior distribution …………………………………………………………………………………………………………….. 7

1.9.4 Loss function ……………………………………………………………………………………………………………………….. 7

CHAPTER TWO: LITERATURE REVIEW ………………………………………………………….. 8

CHAPTER THREE: METHODOLOGY …………………………………………………………………. 15

3.1 Maximum Likelihood Method ………………………………………………………………………………… 15

3.2 Bayes Estimation of the Shape Parameter of GRD ………………………………………………. 16

3.2.1 Posterior risk and Bayes estimator …………………………………………………………………………………………. 16

3.2.2 Symmetric loss function ………………………………………………………………………………………………………. 18

3.2.3 Asymmetric loss function …………………………………………………………………………………………………….. 19

3.3 Bayesian Estimates under the Extended Jeffrey’s Prior ……………………………………….. 21

3.3.1 Transformation of the random variable M and its distribution …………………………………………………… 25

3.3.2 Convolution ……………………………………………………………………………………………………………………….. 27

3.3.3 Variance and relative efficiency of the estimates under extended Jeffrey’s prior using the various loss

functions ……………………………………………………………………………………………………………………………. 29

x

3.3.4 Posterior Risk …………………………………………………………………………………………………………………….. 32

3.4 Bayesian Estimates under the Uniform Prior ………………………………………………………. 33

3.5 Bayes Estimates under the Gamma Prior …………………………………………………………… 38

3.5.1 The distribution of the random variable H ………………………………………………………………………………. 42

3.5.2 Determination of variance, relative efficiency and posterior risk of the shape parameter under the

squares error, entropy and precautionary loss functions ……………………………………………………………. 44

CHAPTER FOUR: ANALYSIS AND DISCUSSION …………………………………………….. 47

CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMENDATIONS …… 58

5.1 Summary ………………………………………………………………………………………………………… 58

5.2 Conclusion ………………………………………………………………………………………………………. 58

5.3 Recommendations ……………………………………………………………………………………………. 59

5.4 Contribution to Knowledge ……………………………………………………………………………….. 59

5.5 Areas of Further Research ………………………………………………………………………………… 60

References ……………………………………………………………………………………………………………. 60

Appendix I …………………………………………………………………………………………………………… 64

**CHAPTER ONE**

INTRODUCTION

1.1 Background to the Study

Statistical Inference is the branch of statistics concerned with using probability concept to

deal with uncertainty in decision-making. It refers to the process of selecting a sample and

using a sample statistic to draw inference about a given population parameter. The field of

statistical inference is divided into the theory of estimation and hypothesis testing.

1.1.1 Theory of estimation

Statistical estimation or simply estimation is concerned with the methods by which

population characteristics are estimated based on information drawn from a sample. The

theory of estimation is further sub-divided into Point and Interval Estimation. A point

estimator is a random variable varying from sample to sample and its value is called point

estimate i.e. a point estimate is a single value estimate for the parameter. There are several

methods of finding a point estimator which can all be broadly classified into the Classical

Methods and Non-classical/ Bayesian Methods.

In classical approach, the unknown parameter is assumed to be fixed quantity. Inferences

in classical approaches are based on a random sample only i.e. if a random sample

,,⋯, is drawn from a population with probability function

; and based on

the sample knowledge the estimate of is obtained. The classical approach is based on the

concept of sampling distribution and it does not use any of the prior information available

as a result of familiarity with previous studies. There are different methods of point

estimation under the classical approach. These include: Maximum Likelihood Estimation

(MLE), Method of Moment Estimation, Percentile Estimation, Least Square Estimation,

Weighted Least Square Estimates, and so on. On the other hand, in Non-classical/ Bayesian

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approach, the parameter is assumed to be a random variable which can be described by a

probability distribution (known as prior distribution) that is, the unknown parameter θ

(being random) follows a prior distribution. Hence, Bayesian approach combines new

information that is available with prior information to form basis for inference.

The fundamental difference between the Bayesian and frequentist approaches to statistical

inference is characterized in the way they interpret probability, represent the unknown

parameters, acknowledge the use of prior information and make the final inferences. The

frequentist approach considers probability as a limiting long-run frequency, while the

Bayesian approach regards probability as a measure of the degree of personal belief about

the value of an unknown parameter θ.

1.1.2 Generalized Rayleigh distribution (GRD)

Twelve different families of cummulative distribution function used in modeling lifetime

data were suggested by Burr (1942). Burr Type X distribution is among the most popular

distributions that receives the most attention among these families of cummulative

distributions.

In 2001, two-parameter Burr Type X distribution was introduced by Surles & Padgett.

Kundu and Raqab (2005), Lio, et.al, (2011) and Abdel-Hady (2013) prefer to call this

distribution GRD which will be adopted in this work. For α>0 and λ>0, the Cumulative

Distribution Function (CDF) of the two-parameter GRD is given by:

F

x; α, λ = 1 −

for x, α , λ >0 (1.1)

Its probability density function (pdf) is given by:

3

f

x; α, λ = 21 −

, > 0

0, “#$ℎ&

‘ (1.2)

where α and λ are shape and scale parameters respectively.

Shape and scale parameters are used to determine the shape and location of a distribution.

Shape parameter allows a distribution to take on a variety of shapes depending on the value

of the shape parameter, while the scale parameter stretches or squeezes the graph of a

probability distribution. In general, the larger the scale parameter, the more spread out the

distribution and the smaller the parameter, the more compressed the distribution appears to

be.

GRD is widely used in modeling events that occur in different fields such as medicine,

social and natural sciences. In Physics for instance, the GRD is used in the study of various

types of radiation such as light and sound measurements. It is used as a model for wind

speed and is often applied to wind driven electrical generation. For details, see Samaila and

Cenac, (2006). It is also used in modeling strength and lifetime data (Surles and Padgett

(2001), Lio, et.al (2011), Kundu and Raqab, (2005)). Hence, the GRD has a survival and

hazard functions as shown in equations (1.3) and (1.4) respectively.

Survival function S

x; α, λ = 1 − F

x; α, λ = 1 − 1 −

(1.3)

Hazard function h

x; α, λ = *

; ,

+

; , =

,-.

/0 1

2.3

-.

/0

,-.

/0 1

2 (1.4)

Survival function is the probability that the survival time X takes a value greater than a

specific value x ie 4

= 5

> , while the hazard function is a measure of how likely

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an individual experiences an event as a function of his/her age. These two functions are

used to describe the distribution of survival time data.

Figure 1. 1: The Graph of Generalized Rayleigh Distribution for different values of shape parameters when the

scale parameter takes the value one

The graph of the distribution is shown in Figure 1.1 for different shape parameter values. It

is clear from the Figure that the pdf of a GRD is a decreasing function if α ≤ ½ and it is

right skewed uni-modal when α > ½. (See also Kundu and Raqab, 2005).

1.2 Statement of the Problem

Bayesian inference requires appropriate choice of priors for parameters. But there is no way

to conclude that one prior is better than another. In a situation where one have enough

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information about the parameter(s) then using informative prior(s) will be the best practice

for choosing a prior, otherwise, a non-informative prior suffices.

1.3 Aim and Objectives of the Study

The aim of this work is to estimate the shape parameter of GRD using Bayesian approach.

We wish to achieve the stated aim through the following objectives

i. By estimating the shape parameter (α) when the scale parameter (λ) is known using

both informative and non-informative priors under symmetric loss function

ii. By estimating the shape parameter (α) when the scale parameter (λ) is known using

both informative and non-informative priors under asymmetric loss functions

iii. To compare the performances of the proposed estimators with that of Maximum

Likelihood Estimators in terms of Mean Square Error

1.4 Significance of the Study

In Bayesian approach, the parameter is viewed as random variable behaving according to a

subjective (prior) probability distribution that describes our confidence about the actual

behavior of the parameter, whereas in classical approach, the parameter is assumed to be

fixed but unknown. In Bayesian inference, conclusions are made conditional on the

observed data i.e. there is no need to discuss sampling distribution using this method. While

in the classical approach one needs not be concerned about any prior knowledge other than

the available information observed. Bayesian inference also provides a convenient model

for implementing scientific method. The prior distribution is used to state the prior

knowledge we have about the parameter of interest, while the posterior distribution reflects

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the updated knowledge about the population parameter in line with the new information

collected from data.

1.5 Motivation

Statistically, modeling of real life scenario help us to better understand and explain

unforeseen eventualities when they take place, thereby enabling us to reproduce such a

scenario either on a large and/ or on a simplified scale aimed at describing only critical

parts of the phenomenon. These real life phenomena are captured by means of statistical

distribution models, which are extracted or learned directly from data gathered about them.

Every distribution model has a set of parameters that needs to be estimated. These

parameters specify any constant appearing in the model and provide a mechanism for

efficient and accurate use of data.

1.6 Limitation

The study will focus only on estimating the shape (α) parameter when the scale (λ)

parameter is known under the symmetric and asymmetric loss functions assuming

informative and non-informative priors.

1.7 Definition of Terms

1.9.1 Estimator

Let X be a random variable that follows a probability distribution function

; indexed

by a parameter . Let , ,⋯, be a random sample from the given population. Any

statistic that can be used to estimate the parameter is called an estimator of . The

numerical value of this statistic is called an estimate of and is denoted by 6.

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1.9.2 Prior distribution

A prior distribution is a probability distribution that captures the information about a

parameter(s) before data are taken into account. Prior distributions are sub-divided into two

classes: Informative and non-informative priors.

Let ,,⋯ be a random sample from a distribution with density

/ , where

(assumed random) is the unknown parameter to be estimated. The probability distribution

of is called the prior distribution of and is usually denoted as 8

.

1.9.3 Posterior distribution

Let ,,⋯ be a random sample from a distribution with density

/ , where is

an unknown parameter to be estimated. The conditional density 8

/ , ⋯, is

called the posterior distribution of and is given by

( ) ( ) ( )

1 2 ( )

/

/ , , n

f x

x x x

g x

q q

q = Õ ⋯ Õ

(1.5)

where 9

is the marginal distribution of and is given by

9

=

Σ;

/ 8

$ℎ< =# >=#?&@

A

/ 8

, $ℎ< =# ?B<@=<BC# ∞

D

‘ (1.6)

where 8

is the prior distribution of .

1.9.4 Loss function

Let ,,⋯ be a random sample from a distribution with density

/ , where is

an unknown parameter to be estimated. Let 6 be an estimator of . The function ℒ6,

represents the loss incurred when 6

is used in place of .

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