# Relations, Bounds and Approximations for Order Statistics

• Barry C. Arnold
• Narayanaswamy Balakrishnan
Book

Part of the Lecture Notes in Statistics book series (LNS, volume 53)

1. Front Matter
Pages I-IX
2. Barry C. Arnold, Narayanaswamy Balakrishnan
Pages 1-4
3. Barry C. Arnold, Narayanaswamy Balakrishnan
Pages 5-37
4. Barry C. Arnold, Narayanaswamy Balakrishnan
Pages 38-72
5. Barry C. Arnold, Narayanaswamy Balakrishnan
Pages 73-107
6. Barry C. Arnold, Narayanaswamy Balakrishnan
Pages 108-140
7. Barry C. Arnold, Narayanaswamy Balakrishnan
Pages 141-151
8. Back Matter
Pages 152-176

### Introduction

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.

### Keywords

Area Maxima Variance approximation behavior computation control development distribution form function functional identity maximum statistics

#### Authors and affiliations

• Barry C. Arnold
• 1
• Narayanaswamy Balakrishnan
• 2
1. 1.Department of StatisticsUniversity of CaliforniaRiversideUSA
2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-3644-3
• Copyright Information Springer-Verlag New York 1989
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-0-387-96975-6
• Online ISBN 978-1-4612-3644-3
• Series Print ISSN 0930-0325
• Buy this book on publisher's site
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