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Bayesian Estimation of the Shape Parameter of Odd Generalized Exponential-Exponential Distribution

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Bayesian Estimation of the Shape Parameter of Odd Generalized Exponential-Exponential Distribution.

ABSTRACT

The Odd Generalized Exponential-Exponential Distribution (OGEED) could be used in various fields to model variables whose chances of success or survival decreases with time.

It was also discovered that the OGEED has higher positive skewness and has been found to have performed better than some existing distributions such as the Gamma, Exponentiated Exponential, Weibull and Pareto distributions in a real life applications.

The shape parameter of the Odd Generalized Exponential-Exponential Distribution using the Bayesian method of estimation and comparing the estimates with that of maximum Likelihood by assuming two non-informative prior distributions namely; Uniform and Jeffrey prior distributions.

These estimates were obtained using the squared error loss function (SELF), Quadratic loss function (QLF) and precautionary loss function (PLF).

The posterior distributions of the OGEED were derived and also the Estimates and risks were also obtained using the above mentioned priors and loss functions.

Furthermore, we carried out Monte-Carlo simulation using R software to assess the performance of the two methods by making use of the Biases and MSEs of the Estimates under the Bayesian approach and Maximum likelihood method.

TABLE OF CONTENTS

Title Page………………… i

Declaration……………………. ii

Certification…………………… iii

Dedication…………… iv

Acknowledgement…………………….. v

Table of Contents………………… vii

LIST OF TABLE………………… ix

List of Figures………………… x

List of Abbreviations……………. xi

Symbols………………… xii

Abstract…………………. xiii

CHAPTER ONE

INTRODUCTION…………………. 1

  • Background of the Study…………………… 1
  • Generalized Inverse Exponential Distribution………………… 2
  • Statement of Problem……………… 3
  • Aim and Objectives……………… 3
  • Significance and Justification of the study………… 4
  • Definition of Terms…………….. 4

CHAPTER TWO

LITERATURE REVIEW……………… 6

  • Baseline Distribution……….. 6
  • Bayesian Concept…………. 6

CHAPTER THREE

RESEARCH METHODOLOGY……………. 12

  • Likelihood Function…………… 12
  • Maximum Likelihood Estimation………….. 13
  • Prior Distribution………….. 13
    • Uniform prior…………. 13
    • Jeffrey’s prior……… 14
    • Extended Jeffrey’s prior……. 15
  • Posterior Distribution……….. 15
    • Posterior distribution under the uniform prior…….. 16
    • Posterior distribution under the Jeffrey’s prior…… 19
    • Posterior distribution under the extended Jeffrey’s prior……… 22
  • Loss Function………. 24
    • Squared error loss function….. 24
    • Precautionary loss function……. 31
  • Transformation of the random variable N and its ….. 37
  • Simulation Study………….. 54
  • Monte-Carlo test………. 55

CHAPTER FOUR

ANALYSIS AND DISCUSSION OF RESULT……….. 57

4.1. Result 0f Analysis…………… 58

4.3 Discussion of Result……….. 61

CHAPTER FIVE

SUMMARY, CONCLUSION AND RECOMMENDATION… 67

  • Summary…………… 67
  • Conclusion………………… 67
  • Recommendation………………… 68
  • Contribution to Knowledge……………. 68

REFERENCES………….. 54

APPENDIXES…………………. 56

INTRODUCTION

1.1 Background of the Study

In Bayesian approach, the prior information is combined with any new information that is available to form the basis for statistical inference.

Statistical approaches that use prior knowledge in addition to the sample evidence to estimate the population parameters are known as Bayesian methods.

The Bayesian approach seeks to optimally merge information from two sources: (1) knowledge that is known from theory or opinion formed at the beginning of the research in the form of a prior, and (2) information contained in the data in the form of likelihood functions.

Basically, the prior distribution represents our initial belief, whereas the information in the data is expressed by the likelihood function. Combining prior distribution and likelihood function, we can obtain the posterior distribution.

The one parameter Exponential distribution describes the time between events in a Poisson process. Its discrete analogue is the Geometric distribution.

Apart from its usage in Poisson processes, it has been used extensively in the literature for life testing.

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):153-159.

Azam, Z. and Ahmad, S. A. (2014a). Bayesian Analysis of Power Function Distribution Using Different Loss Functions. International Journal of Hybrid Information Technology, 7(6); 229-244.

Azam, Z. and Ahmad, S. A. (2014b). Bayesian Approach in Estimation of Scale Parameter of Nakagami Distribution. International Journal of Advanced Science and Technology , 65; 71-80.

Dey, S. (2010). Bayesian estimation of the shape parameter of the generalized exponential distribution under different loss functions. Pakistan Journal of Statistics and Operation Research, 6(2), 163-174.

Feroze, N. (2012). Estimation of scale parameter of inverse gausian distribution under a Bayesian framework using different loss functions. Scientific Journal of Review, 1(3); 39- 52.

Himanshu, P. and Arun, K. R. (2009). Bayesian Estimation of the Shape Parameter of Generalized Pareto Distribution  Under  Asymmetric  Loss  Functions.  Hacettepe Journal of Mathematics and Statistics, 38 (1), 69-83.

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