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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.
Sommario
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.
Info autore
N. BALAKRISHNAN, PhD, is Professor of Mathematics and Statistics at McMaster University in Hamilton, Ontario, Canada.V. B. NEVZOROV, PhD, DS, is Professor of Probability and Statistics at St. Petersburg State University in St. Petersburg, Russia.
