Ebook: Relations, Bounds and Approximations for Order Statistics

Bounds on moments of order statistics have been of interest since Sir Francis Galton (1902) flrst addressed the problem of fairly dividing flrst and second prize money in a competition. The present compendium of results represents our effort to sort the plethora of results into some semblance of order. We have tried to assign priority for results appropriately. We will cheerfully accept corrections. Omissions of interesting results have inevitably occurred. On this too we await (cheerful) corrections. We are grateful to Peggy Franklin (University of California), Janet Leach, Domenica Calabria and Patsy Chan (McMaster University) who shared the responsibility of typing the manuscript. The flnal form of the manuscript owes much to their skill and patience. Barry C. Arnold Riverside, California U. S. A. N. Balakrishnan Hamilton, Ontario Canada November, 1988 Table of Contents Chapter 1: TIlE DISTRIBUTION OF ORDER STATISTICS Exercises 4 Chapter 2: RECURRENCE RELATIONS AND IDENTITIES FOR ORDER STATISTICS 2. 0. Introduction 5 2. 1. Relations for single moments 6 2. 2. Relations for product moments 9 2. 3. Relations for covariances 13 15 2. 4. Results for symmetric populations 2. 5. Results for normal population 17 20 2. 6. Results for two related populations 2. 7. Results for exchangeable variates 23 25 Exercises Chapter 3: BOUNDS ON EXPECTATIONS OF ORDER STATISTICS 3. 0. Introduction 38 3. 1. Universal bounds in the Li. d. case 38 3. 2. Variations on the Samuelson-Scott theme 43 3. 3.

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This book describes in great length some relations satisfied by moments of order statistics and some methods of deriving bounds and approximations for these moments. The main purpose of the book is to present various old, as well as recent, developments in the above-mentioned three topics in order statistics and also to illustrate some of their uses. Statisticians working in the areas of order statistics, approximation theory, robust inference, goodness-of-fit, outliners, etc., will find this book quite useful. Various new results, particularly involving order statistics from outliner models, have been presented and their uses in robustness studies have been demonstrated. Some inter-relationships between various results are pointed out; some cautionary notes are given regarding their use. These will be of interest to those who are working on theoretical as well as computational, problems in order statistics and related areas. Some generalizations of well-known results are presented and these will be of interest to researchers working in the area of order statistics and also to those who are applying the theory of order statistics to other fields, including quality control, reliability, control theory, etc.

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This book describes in great length some relations satisfied by moments of order statistics and some methods of deriving bounds and approximations for these moments. The main purpose of the book is to present various old, as well as recent, developments in the above-mentioned three topics in order statistics and also to illustrate some of their uses. Statisticians working in the areas of order statistics, approximation theory, robust inference, goodness-of-fit, outliners, etc., will find this book quite useful. Various new results, particularly involving order statistics from outliner models, have been presented and their uses in robustness studies have been demonstrated. Some inter-relationships between various results are pointed out; some cautionary notes are given regarding their use. These will be of interest to those who are working on theoretical as well as computational, problems in order statistics and related areas. Some generalizations of well-known results are presented and these will be of interest to researchers working in the area of order statistics and also to those who are applying the theory of order statistics to other fields, including quality control, reliability, control theory, etc.
Content:
Front Matter....Pages I-IX
The Distribution of Order Statistics....Pages 1-4
Recurrence Relations and Identities for Order Statistics....Pages 5-37
Bounds on Expectations of Order Statistics....Pages 38-72
Approximations to Moments of Order Statistics....Pages 73-107
Order Statistics from a Sample Containing a Single Outlier....Pages 108-140
Record Values....Pages 141-151
Back Matter....Pages 152-176

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This book describes in great length some relations satisfied by moments of order statistics and some methods of deriving bounds and approximations for these moments. The main purpose of the book is to present various old, as well as recent, developments in the above-mentioned three topics in order statistics and also to illustrate some of their uses. Statisticians working in the areas of order statistics, approximation theory, robust inference, goodness-of-fit, outliners, etc., will find this book quite useful. Various new results, particularly involving order statistics from outliner models, have been presented and their uses in robustness studies have been demonstrated. Some inter-relationships between various results are pointed out; some cautionary notes are given regarding their use. These will be of interest to those who are working on theoretical as well as computational, problems in order statistics and related areas. Some generalizations of well-known results are presented and these will be of interest to researchers working in the area of order statistics and also to those who are applying the theory of order statistics to other fields, including quality control, reliability, control theory, etc.
Content:
Front Matter....Pages I-IX
The Distribution of Order Statistics....Pages 1-4
Recurrence Relations and Identities for Order Statistics....Pages 5-37
Bounds on Expectations of Order Statistics....Pages 38-72
Approximations to Moments of Order Statistics....Pages 73-107
Order Statistics from a Sample Containing a Single Outlier....Pages 108-140
Record Values....Pages 141-151
Back Matter....Pages 152-176
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