Characterizations of Exponential Distribution by Ordered Random Variables
Published on 15. May 2019
192 pages
978-1-5361-5403-0 (ISBN)
Description
Exponential distribution is one of the most-used distributions in the theory and practice of statistics. It has several important properties like being memoryless and having a constant hazard rate. The field of characterization is developed in different branches of statistics and applied probability. Ordered random variables are common in various applications in practice. In this book, characterizations of exponential distribution using ordered random variables are presented. Most of the known results as well as many new results are given in this book. The aim of the book is to present various characterizations of exponential distribution based on ordered random variables. The book is written on a lower technical level and requires basic knowledge of mathematics and statistics. Chapter 1 gives some basic properties of exponential distribution. Chapters 2, 3, and 4 give the characterization of exponential distribution based on order statistics, record values, and generalized order statistics.
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Mohammad Ahsanullah
Characterizations of Exponential Distribution by Ordered Random Variables
Book
06/2019
Nova Science Publishers Inc
€126.26
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Content
- Intro
- Contents
- Preface
- Chapter 1
- Basic Properties
- 1.1. Introduction
- Theorem 1.1.1.
- Proof
- Theorem 1.1.2.
- Theorem 1.1.3.
- Proof of Theorem 1.1.3.
- Theorem 1.1.4.
- Proof of Theorem 1.1.4.
- Theorem 1.1.5.
- Proof
- Theorem 1.1.6.
- Proof
- Theorem 1.1.7.
- Proof
- Proof
- 1.2. Order Statistics
- Theorem 1.2.1.
- Proof
- Theorem 1.2.2.
- Theorem 1.2.3.
- Theorem 1.2.4.
- Theorem 1.2.5.
- Theorem 1.2.6.
- 1.3. Record Values
- 1.3.1. Definition of Record Values and Record Times
- 1.3.2. The Exact Distribution of Record Values for Continuous Random Variables
- 1.3.3. Conditional Distribution
- Theorem 1.3.1.
- Proof
- Theorem 1.3.2.
- 1.3.4. Representations of Records
- 1.3.4.1. Representations of Upper Records
- Theorem 1.3.1. The Following Equality Holds
- Proof
- Proof
- 1.3.4.2. Representation of Lower Records
- 1.4. Generalized Order Statistics
- Definitions
- 1.4.1. Dual Generalized Order Statistics (dgos)
- Chapter 2
- Characterizations by Order Statistics
- 2.1. Characterizations by Conditional Expectations
- Theorem 2.1.1.
- Theorem 2.1.2.
- Theorem 2.1.3.
- Theorem 2.1.4.
- Proof
- Theorem 2.1.5.
- Proof
- Theorem 2.1.6.
- Theorem 2.1.7.
- Proof
- Theorem 2.1.8.
- Theorem 2.1.9.
- 2.2. Characterization by Equality of Distributions
- Theorem 2.2.1.
- Theorem 2.2.2.
- Proof
- Theorem 2.2.3.
- Proof
- Theorem 2.2.4.
- Theorem 2.2.5.
- Proof
- Proof
- Theorem 2.2.6.
- Proof
- Result 2.2.1.
- Result 2.2.2.
- Theorem 2.2.7.
- Theorem 2.2.8.
- Theorem 2.2.9.
- Theorem 2.2.10.
- Theorem 2.2.11.
- 2.3. Characterization by Independence Properties
- Theorem 2.3.1.
- Proof
- Theorem 2.3.2.
- Theorem 2.3.3.
- Chapter 3
- Characterizations by Record Values
- 3.1. Characterizations by Independence Properties
- Theorem 3.1.1.
- Proof
- Theorem 3.1.2.
- Proof
- Theorem 3.1.3.
- Proof
- Lemma 3.1.1.
- Proof
- 3.2. Characterizations by Equality in Distributions
- Proof
- Theorem 3.2.3.
- Proof
- Theorem 3.2.4.
- Proof
- Theorem 3.2.5.
- Proof
- Remark 3.2.1.
- Theorem 3.2.6.
- Proof
- 3.3. Characterizations by Moments and Conditional Moments
- Theorem 3.3.1.
- Proof
- Theorem 3.3.2.
- Theorem 3.3.3.
- Proof
- Theorem 3.3.4.
- Proof
- 3.4. Characterizations by Equality in Hazard Rates
- Theorem 3.4.1.
- Proof
- 3.5. Characterizations by Lower Records
- Theorem 3.5.1.
- Lemma 3.5.1.
- Proof
- Proof of the Theorem 3.3.4.
- Define
- Theorem 3.5.2.
- Chapter 4
- Characterizations by Generalized Order Statistics
- 4.1. Characterizations by Conditional Expectations
- Theorem 4.1.1.
- Proof
- Theorem 4.1.2.
- Proof
- Theorem A
- Theorem 4.1.3.
- Remark 4.1.1.
- Theorem 4.1.4.
- Theorem 4.1.5.
- Proof
- 4.2. Characterizations by Independence Properties
- Theorem 4.2.1.
- Proof
- Remark 4.2.1.
- 4.3. Characterizations by Identical Distributions
- Theorem 4.3.1.
- Proof
- Remark 4.3.1.
- Remark 4.3.2.
- Theorem 4.3.2.
- Theorem 4.3.3.
- Appendix
- Assumptions A
- Lemma 2.1.
- Lemma 2.2.
- References
- About the Author
- Index
- Blank Page
