This volume is based on a course of lectures delivered at the St. Petersburg State University (Russia) and at Ohio State University (Columbus). It is intended as a textbook for graduate students and postdocs. The book presents the theory of records and some information on order statistics. Also included are exercises illustrating the examples and developing the ideas. The past 20 years has seen tremendous progress in the topic, giving forth a large number of new models that reflect the dynamics of records in a wide range of areas. This volume presents systematic main results with a special emphasis on non-classical record schemes. The material is presented in a comprehensive style succinctly outlining the current state of the theory. The work is geared toward statisticians, actuarians, engineers, hydrologists, meteorologists, and sports and market analysts.
Author(s): Valery B. Nevzorov
Year: 2000
Language: English
Pages: 164
Cover......Page 1
Title page......Page 4
Copyright......Page 5
Dedication......Page 6
Contents ......Page 8
Preface ......Page 10
Introduction. Lecture 1 ......Page 12
Lecture 2. Order Statistics and Their Distributions ......Page 16
Lecture 3. Three Classical Representations for Order Statistics ......Page 21
Lecture 4. Markov Property and Conditional Independence of Order Statistics ......Page 26
Lecture 5. Order Statistics for Nonstationary Sequences ......Page 29
Lecture 6. Moments of Order Statistics ......Page 32
Lecture 7. Moments of Order Statistics (continued) ......Page 37
Lecture 8. Asymptotic Distributions of the Middle Order Statistics ......Page 43
Lecture 9. Asymptotic Distributions of the Middle Order Statistics (continued) ......Page 49
Lecture 10. Asymptotic Distributions of Maxima ......Page 53
Lecture 11. Domains of Attraction of the Limiting Distributions of Extremes ......Page 58
Lecture 12. Some Topics Related to the Distribution of Extremes ......Page 61
Lecture 13. Maxima and Records ......Page 66
Lecture 14. Distributions of Record Times ......Page 70
Lecture 15. Distributions of Record Values ......Page 76
Lecture 16. Shorrock's Representation for Discrete Records ......Page 80
Lecture 17. Joint Distributions of Record Times and Record Values ......Page 87
Lecture 18. Asymptotic Distributions of Record Values ......Page 90
Lecture 19. The kth Records ......Page 93
Lecture 20. Generating Function of the kth Record Times ......Page 97
Lecture 21. Moment Characteristics of the kth Record Times ......Page 100
Lecture 22. Tata's Representation and Its Generalizations ......Page 103
Lecture 23. Correlation Coefficients for Records ......Page 106
Lecture 24. Records for Nonstationary Sequences of Random Variables ......Page 112
Lecture 25. Record Times in the Fe-Scheme ......Page 118
Lecture 26. Independence of Record Indicators and Maxima ......Page 125
Lecture 27. Asymptotic Distribution of Record Values in the Fa-Scheme ......Page 127
Lecture 28. Records in Sequences of Dependent Random Variables ......Page 130
Lecture 29. Records and the Secretary Problem ......Page 136
Lecture 30. Statistical Procedures Related to Records ......Page 140
Appendix 1. Theory of Records: Historical Review ......Page 144
Appendix 2. Hints, Solutions, and Answers ......Page 148
Bibliography ......Page 164
