Bayesian Estimation of the Shape Parameter of Generalized Rayleigh Distribution Under Symmetric and Asymmetric Loss Functions

Bayesian Estimation of the Shape Parameter of Generalized Rayleigh Distribution Under Symmetric and Asymmetric Loss Functions.

ABSTRACT

In 2001, Surles & Padgett introduced Generalized Rayleigh Distribution (GRD). This skewed distribution can be used quite effectively in modeling lifetime 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.

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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

• Background to the Study……. 1
  • Theory of estimation……………1
  • Generalized Rayleigh distribution (GRD)…………..2
• Statement of the Problem…………. 4
• Aim and Objectives of the Study……………. 5
• Significance of the Study………….. 5
• Motivation……….. 6
• Limitation……………. 6
• Definition of Terms…………….. 6
  • Estimator……………. 6
  • Prior distribution……….. 7
  • Posterior distribution…………. 7
  • Loss function……………7

CHAPTER TWO: LITERATURE REVIEW

CHAPTER THREE: METHODOLOGY

• Maximum Likelihood Method………….. 15
• Bayes Estimation of the Shape Parameter of GRD………… 16
  • Posterior risk and Bayes estimator………. 16
  • Symmetric loss function………….. 18
  • Asymmetric loss function…………… 19
• Bayesian Estimates under the Extended Jeffrey’s Prior……….. 21
  • Transformation of the random variable M and its distribution………… 25
  • Convolution….. 27
  • Variance and relative efficiency of the estimates under extended Jeffrey’s prior using the various loss functions 29
  • Posterior Risk………. 32
• Bayesian Estimates under the Uniform Prior………. 33
• Bayes Estimates under the Gamma Prior……………… 38
  • The distribution of the random variable H…………. 42
  • 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

CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMENDATIONS

• Summary…………………. 58
• Conclusion……………… 58
• Recommendations………….. 59
• Contribution to Knowledge……… 59
• Areas of Further Research………… 60

References…….. 60

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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.

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REFERENCES

Almutairi, A. O., and Heng, C. L. (2012). Bayesian Estimate for Shape Parameter from Generalized Power Function Distribution. Mathematical Theory and Modeling , II (12).
Burr, I. W. (1942). Cumulative Frequency Distribution. Annual of Mathematical Statistics , 13, 215- 232.
Dey, D. K., and Liu Pei-San L. (1992). On comparison of estimators in a generalized life model. Micro electron. Reliab. 32 (1) 207-221.
Dey, D. K., Gosh M., and Srinivasan C. (1987). Simultaneous estimation of parameters  under entropy loss. Journal of Statist. Plan. and Infer. 347-363.
George, C., and Roger, L. B. (2002). Statistical Inference (2nd Editio  ed.).  USA:  Wadsworth  Group.
Guure, B. C., and Ibrahim, A. N. (2014). Approximate Bayesian Estimates of Weibull parameters with Lindley’s method. Sains malasiana , 43 (9), 1433-1437.