Relations, Bounds and Approximations for Order Statistics

1. Front Matter

   Pages I-IX

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2. The Distribution of Order Statistics

   • Barry C. Arnold, Narayanaswamy Balakrishnan

   Pages 1-4

3. Recurrence Relations and Identities for Order Statistics

   • Barry C. Arnold, Narayanaswamy Balakrishnan

   Pages 5-37

4. Bounds on Expectations of Order Statistics

   • Barry C. Arnold, Narayanaswamy Balakrishnan

   Pages 38-72

5. Approximations to Moments of Order Statistics

   • Barry C. Arnold, Narayanaswamy Balakrishnan

   Pages 73-107

6. Order Statistics from a Sample Containing a Single Outlier

   • Barry C. Arnold, Narayanaswamy Balakrishnan

   Pages 108-140

7. Record Values

   • Barry C. Arnold, Narayanaswamy Balakrishnan

   Pages 141-151

8. Back Matter

   Pages 152-176

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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.

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

• Department of Statistics, University of California, Riverside, USA

  Barry C. Arnold

• Department of Mathematics and Statistics, McMaster University, Hamilton, Canada

  Narayanaswamy Balakrishnan

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