Record values and their applications for exponential and Rayleigh distributions : a handbook / Mohammad Ahsanullah and Mohammad Shakil, editors.

Format:

Book

Contributor:

Shakil, Mohammad (Professor of mathematics), editor.

Ahsanullah, Mohammad, editor.

Series:

Mathematics research developments series.

Mathematics research developments

Language:

English

Subjects (All):

Exponential families (Statistics).

Order statistics.

Rayleigh model.

Physical Description:

1 online resource (156 pages)

Edition:

1st ed.

Place of Publication:

New York : Nova Science Publishers, [2023]

Summary:

"Records arise naturally in many fields of study such as climatology, sports, science, engineering, medicine, traffic, and industry, among others. An observation is called a record if its value is greater than (or analogously, less than) all the preceding observations. The exponential and Rayleigh distribution play pivotal roles in the study of records because of their wide applicability in the modeling and analysis of lifetime data. In this handbook, various properties of record values of univariate exponential and Rayleigh distributions, including most of the recent works, are reviewed. Some new results on mean residual life based on records and the minimum variance linear unbiased estimators of past records from univariate exponential distribution are presented"-- Provided by publisher.

Contents:

Intro

Contents

Preface

Acknowledgments

Chapter 1

Introduction

1.1. Record Values

1.2. The Exact Distribution of Record Values

1.3. Inter Record Times

1.4. Numbers of Records in a Sequence of Observations

1.5. Moments of Record Values

1.6. Representations of Records

Chapter 2

Univariate Exponential Distribution

1. Introduction

2. Records of the Univariate Exponential Distribution

2.1. Probability Density Functions (pdf's) of Single, Joint and Conditional Record Values

2.2. Moments

2.3. Limiting Distribution of Record Values

2.4. Mean Residual Life of Records

2.5. Minimum Variance Linear Unbiased Estimator (MVLUE)

2.5.1. MVLUE of Records

2.5.2. Best Linear Invariant Estimateur (BLIE)

2.5.3. MVLUE of Past Records

Conclusion

Chapter 3

Rayleigh Distribution

2. Record Values of Rayleigh Distribution

2.1. Distribution of Record Values: Definitions and Notations

2.2. Distribution of Record Values for Rayleigh Distribution

3. Moments and Recurrence Relations

3.1. Moments

3.2. Odd and Even Moments of the Upper Records

3.3. Recurrence Relations between Moments of the Upper Records

4. Survival, Hazard and Entropy Functions

4.1. Survival and Hazard Functions

4.2. Entropy

5. Estimation of Parameters

5.1. Maximum Likelihood Estimates

5.2. Minimum Variance Linear Unbiased Estimates

6. Prediction of Record Values

6.1. Prediction of the sth Upper Record Value

6.2. A Numerical Example

Chapter 4

Characterization of Exponential Distribution Based on Record Values

2. Characterization of Exponential Distribution Based on Record Values

Chapter 5

Characterizations of Rayleigh Distribution Based on Record Values

1. Introduction.

2. Characterizations Based on Conditional Expectations

3. Characterizations Based on Record Values

3.1. We First Prove the Following Theorem Based on Conditional Expectation

4. Characterizations Based on Failure Rate

Chapter 6

Ratio of Two Exponential Random Variables

2. Ratio of Exponential Random Variables

3. Distribution of Record Values of the Ratio of Exponentiel Distributions

4. Distributional Properties of Record Values of the Ratio of Exponential Distributions

4.1. Moments

4.2. Representation of Records and Existence of Moments

4.2.1. Representation of Records

4.3. Survival and Hazard Functions

4.4. Entropy

5. Percentiles

Chapter 7

Ratio of Two Rayleigh Random Variables

2. Ratio of Rayleigh Random Variables

3. Record Values of the Ratio of Two Rayleigh Random Variables

4. Distributional Properties of Record Values of the Ratio of Rayleigh Random Variables

4.2. Representation of Records

Appendix

Formulas and Results

A.1. Gamma Function

A.2. Digamma Function

A.3. Exponential Integral

A.4. Error Function

A.5. Properties of Gamma Function

A.6. Special Functions

A.7. Additional Formulas

A.8. Shannon's Entropy of Commonly Used Continuous Probability Density Functions

References

Index

Authors' Contact Information

Blank Page.

Notes:

Description based on print version record.

Includes bibliographical references and index.

Other Format:

Print version: Ahsanullah, Mohammad Record Values and Their Applications for Exponential and Rayleigh Distributions - a Handbook

ISBN:

9798886975086