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.

Autorentext

Albert W. Marshall is Professor Emeritus of Statistics at the University of British Columbia. His fundamental contributions to reliability theory have had a profound effect in furthering its development. Ingram Olkin is Professor Emeritus of Statistics at Stanford University. He has made fundamental contributions in multivariate analysis, and in the development of statistical methods in meta-analysis, which have resulted in its use in many applications. Barry C. Arnold is Distinguished Professor of Statistics at the University of California, Riverside. His previous books deal with Pareto Distributions, Order Statistics, Record Values, Conditionally Specified Distributions, and the Lorenz Order.

Klappentext

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. TOC:Contents: The Distribution of Order Statistics.- Recurrence Relations and Identities for Order Statistics.- Bounds on Expectations of Order Statistics.- Approximations to Moments of Order Statistics.- Order Statistics From a Sample Containing a Single Outliner.- Record Values.- References.- Author Index.- Subject Index.

Inhalt
1: The Distribution of Order Statistics.- Exercises.- 2: Recurrence Relations and Identities for Order Statistics.- 2.0. Introduction.- 2.1. Relations for single moments.- 2.2. Relations for product moments.- 2.3. Relations for covariances.- 2.4. Results for symmetric populations.- 2.5. Results for normal population.- 2.6. Results for two related populations.- 2.7. Results for exchangeable variates.- Exercises.- 3: Bounds on Expectations of Order Statistics.- 3.0. Introduction.- 3.1. Universal bounds in the i.i.d. case.- 3.2. Variations on the Samuelson-Scott theme.- 3.3. Bounds via maximal dependence.- 3.4. Restricted families of parent distributions.- Exercises.- 4: Approximations to Moments of Order Statistics.- 4.0. Introduction.- 4.1. Uniform order statistics and moments.- 4.2. David and Johnson's approximation.- 4.3. Clark and Williams' approximation.- 4.4. Plackett's approximation.- 4.5. Saw's error analysis.- 4.6. Sugiura's orthogonal inverse expansion.- 4.7. Joshi's modified bounds and approximations.- 4.8. Joshi and Balakrishnan's improved bounds for extremes.- Exercises.- 5: Order Statistics From a Sample Containing a Single Outlier.- 5.0. Introduction.- 5.1. Distributions of order statistics.- 5.2. Relations for single moments.- 5.3. Relations for product moments.- 5.4. Relations for covariances.- 5.5. Results for symmetric outlier model.- 5.6 Results for two related outlier models.- 5.7. Functional behaviour of order statistics.- 5.8. Applications in robustness studies.- Exercises.- 6: Record Values.- 6.0. Introduction.- 6.1. Record values.- 6.2. Bounds on mean record values.- 6.3. Record values in dependent sequences.- Exercises.- References.- Author Index.