Characterization of Discrete Distributions

Based on Conditionality and Damage Models

John Panaretos - Ph.D. Thesis

FIRST PAGES
       

        Preface

        Acknowledgements

        Table of Contents
 

CHAPTER 1
NOTATION, TERMINOLOGY AND REVIEW OF THE LITERATURE

        1.0 General Introduction

        1.1 Notation and Terminology

        1.2 Probability Distributions

              1.2.1 Univariate Probability Distributions

              1.2.2 Truncated Univariate Distributions

              1.2.3 Bivariate Distributions

              1.2.4 Multivariate Distributions

        1.3 Literature Review

              1.3.1 The Damage Model and its Applications

              1.3.2 The R-R Characterization and its Variants

              1.3.3 Conditionality Characterizations

CHAPTER 2
THE RAO-RUBIN CHARACTERIZATION

        2.0 Introduction

        2.1 The Rao-Rubin Theorem. An Elementary Proof

        2.2 The Rao-Rubin Theorem. The Truncated Case

        2.3 Characterization of the Survival Distribution

        2.4 The Bivariate Extension

        2.5 The Truncated Bivariate Extension

        2.6 The Multivariate Extension

CHAPTER 3
SHANBHAG' S EXTENSION OF THE R-R CHARACTERIZATION. THE UNIVARIATE CASE

        3.0 Introduction

        3.1 Shanbhag' s Extension

        3.2 Shanbhag' s Extension in Relation to a Theorem by Patil and Seshadri

        3.3 Some Characterizations Based on the Extension

        3.4 An Interesting Limiting Case

        3.5 The Truncated Case of the Extension

        3.6 A Remark on Shanbgag' s Extension

CHAPTER 4
SHANBHAG' S EXTENSION OF THE R-R CHARACTERIZATION. THE BIVARIATE AND MULTIVARIATE CASE

        4.0 Introduction

        4.1 Shanbhag' s Extension. The Bivariate Case.

        4.2 Characterizations of Bivariate Distributions Based on the Extension

        4.3 The Truncated Bivariate Extension

        4.4 A Remark on the Truncated Bivariate Extension

        4.5 The Multivariate Extension

        4.6 The Multivariate Truncated Extension and its Variants

CHAPTER 5
CHARACTERIZATIONS OF FINITE DISCRETE DISTRIBUTIONS

        5.0 Introduction

        5.1 The Univariate Case

        5.2 Characterizations of Some Univariate Distributions Based on Theorem 5.1.1

        5.3 The Extension of Theorem 5.1.1 to the Truncated Case

        5.4 Characterizations when the Distribution of Y|X is Truncated

CHAPTER 6
CHARACTERIZATIONS OF BIVARIATE AND MULTIVARIATE FINITE DISTRIBUTIONS

        6.0 Introduction

        6.1 The Bivariate Extension of Theorem 5.1.1

        6.2 Characterizations of Some Known Bivariate Distributions

        6.3 The Extension of Theorem 6.1.1 to the Truncated Case

        6.4 The Multivariate Extension

        6.5 Characterizations of Some Multivariate Distributions

        6.6 The Extension of Theorem 6.4.1 to the Truncated Case

CHAPTER 7
THE EFFECT OF MIXING TO THE DAMAGE MODEL

        7.0 Introduction

        7.1 Damage Model with Original Distribution, Poisson and Survival Distribution Mixed Binomial

              7.1.1 (Y|X) ~ Binomial ^ Beta (Negative Hypergeometric)

              7.1.2 (Y|X) ~ Binomial ^ Right Truncated Beta

              7.1.3 (Y|X) ~ Binomial ^ Right Truncated Exponential

              7.1.4 (Y|X) ~ Binomial ^ Right Truncated Gamma

              7.1.5 An Interesting Relation Between GY (t) and GY|X=Y (t) in the Case where (Y|X=Y) ~ Binomial ^ Right Truncated Gamma

              7.1.6 Some Examples in the Case where the Distribution of Y|X is Binomial Mixed with a Discrete Distribution

        7.2 General Relations Between GY (t) and GY|X=Y (t) when X is Poisson and Y|X is Mixed Binomial

        7.3 Damage Model with Original Distribution Mixed Poisson and Survival Distribution Binomial

              7.3.1 X ~ Poisson ^ Beta

              7.3.2 X ~ Poisson ^ Right Truncated Beta

              7.3.3 X ~ Poisson ^ Exponential Truncated to the Right

              7.3.4 X ~ Poisson ^ Gamma Truncated at 1

              7.3.5 X ~ Geometric (Poisson ^ Exponential)

              7.3.6 X ~ Negative Binomial (Poisson ^ Gamma)

              7.3.7 An Example with a Discrete Mixing Distribution

        7.4 A General Relation Between G*Y (t) and G*Y|X=Y (t) when X is Mixed Poisson and Y|X is Binomial

CHAPTER 8
SOME INTERESTING PROPERTIES AND CHARACTERIZATIONS BASED ON THE DAMAGE MODEL

        8.0 Introduction

        8.1 The Effect of the Convolution

        8.2 Some Characterizations of the Distribution of X when the Distribution of Y|X is Given

        8.3 Characterization of the Distribution of Y|X

        8.4 Another Extension of the R-R Characterization

CHAPTER 9
CONCLUSIONS

        9.1 Discussion

        9.2 Statistical Importance of the Results

              9.2.1 Introcuction

              9.2.2 The R-R Condition in Relation to the Poissom Model

              9.2.3 On the Utility of the Characterizations Concerning the Poisson Distribution

              9.2.4 Shanghag' s Extension of the R-R Characterization

              9.2.5 The R-R Condition in Relation to Truncated Distributions

              9.2.6 Finite Distributions and the Damage Model.

              9.2.7 Bivariate and Multivariate Distributions

              9.2.8 The Damage Model with a Mixed Poisson as the Original Distribution

              9.2.9 The Damage Model with a Mixed Binomial as the Survival Distribution

              9.2.10 Concluding Remarks

        9.3 Scope for further Research.

APPENDIX
SUMMARIZING TABLE

REFERENCES