Records

The first and only comprehensive guide to modern record theory and its applications Although it is often thought of as a special topic in order statistics, records form a unique area, independent of the study of sample extremes. Interest in records has increased steadily over the years since Chandler formulated the theory of records in 1952. Numerous applications of them have been developed in such far-flung fields as meteorology, sports analysis, hydrology, and stock market analysis, to name just a few. And the literature on the subject currently comprises papers and journal articles numbering in the hundreds. Which is why it is so nice to have this book devoted exclusively to this lively area of statistics. Written by an exceptionally well-qualified author team, Records presents a comprehensive treatment of record theory and its applications in a variety of disciplines. With the help of a multitude of fascinating examples, Professors Arnold, Balakrishnan, and Nagaraja help readers quickly master basic and advanced record value concepts and procedures, from the classical record value model to random and multivariate record models. The book follows a rational textbook format, featuring witty and insightful chapter introductions that help smooth transitions from one topic to another and challenging chapter-end exercises, which expand on the material covered. An extensive bibliography and numerous references throughout the text specify sources for further readings on relevant topics. Records is a valuable professional resource for probabilists and statisticians, in addition to applied statisticians, meteorologists, hydrologists, market analysts, and sports analysts. It also makes an excellent primary text for courses in record theory and a supplement to order statistics courses. Cover......Page 1 Records......Page 5 Contents......Page 9 Notations and Abbreviations......Page 15 Preface......Page 19 1.1 Who Cares?......Page 23 1.2 A Road Map......Page 24 2.2 Standard Record Values Processes......Page 29 2.3 Record Values From the Classical Model......Page 31 2.4.1 Weibull Records......Page 42 2.4.3 Pareto Records......Page 43 2.5 Record Times and Related Statistics......Page 44 2.6 Markov Chains......Page 50 2.7 Moments of Record Values......Page 51 2.7.1 Weibull Distribution......Page 52 2.7.3 Pareto Distribution......Page 53 2.7.5 Normal Distribution......Page 54 2.7.6 Covariance and Correlations......Page 55 2.8 A Discrete Interlude......Page 56 2.9 Geometric Results......Page 60 2.10.1 A Point Process View......Page 63 2.10.2 k-Record Statistics......Page 64 Exercises......Page 66 3.1 Introduction......Page 73 3.2 Exponential Distribution......Page 74 3.3 Weibull Distribution......Page 76 3.4 Gumbel Distribution......Page 78 3.5 Lomax Distribution......Page 83 3.6 Normal Distribution......Page 85 3.7 Logistic Distribution......Page 87 3.8 Bounds and Approximations......Page 90 3.9 Results for k-Records......Page 103 Exercises......Page 111 4.1 Introduction......Page 115 4.2 Characterizing Properties of Record Values......Page 116 4.2.1 The Moment Sequence......Page 117 4.2.2 Regression of Adjacent Record Values......Page 118 4.3.1 Families Defined by Reliability Properties......Page 119 4.3.2 Linear Regressions of Adjacent Record Values......Page 121 4.4.1 The Incomplete Catalog......Page 123 4.4.2 Integrated Cauchy Functional Equation......Page 125 4.4.3 Characterizations Based on the ICFE and Other Functional Equations......Page 127 4.4.4 Lower Record Statistics......Page 129 4.5 Other Continuous Distributions......Page 130 4.6 Geometric-Tail Distributions......Page 133 4.7 Dependence Structures of Record Values and Order Statistics......Page 136 Exercises......Page 138 5.1 Introduction......Page 143 5.2 Maximum Likelihood Estimation......Page 144 5.3 Best Linear Unbiased Estimation......Page 149 5.4 Best Linear Invariant Estimation......Page 165 5.5 Interval Estimation and Tests of Hypotheses......Page 167 5.6.1 Best Linear Unbiased Prediction......Page 172 5.6.2 Best Linear Invariant Prediction......Page 175 5.6.3 Asymptotic Linear Prediction......Page 176 5.7.1 Prediction Intervals Based on BLUE's......Page 178 5.7.2 Conditional Prediction Intervals......Page 181 5.7.3 Tolerance Region Prediction......Page 183 5.7.4 Bayesian Prediction Intervals......Page 184 5.8 Illustrative Examples......Page 185 5.9 Inference with Records and Inter-Record Times......Page 191 5.10 Distribution-Free Tests in Time-Series Using Records......Page 192 Exercises......Page 197 6.1 Introduction......Page 205 6.2 Geometrically Increasing Populations......Page 206 6.3 The Fa Record Model......Page 209 6.3.1 Finite-sample Properties......Page 210 6.3.2 Asymptotic Properties......Page 212 6.4 Linear Drift Record Model......Page 216 6.5 The Pfeifer Model......Page 220 6.6.1 Fa and Linear Drift Record Models......Page 224 6.6.2 The Pfeifer Model......Page 229 6.7 Records From Dependent Sequences......Page 230 6.7.1 Markov Sequences......Page 231 6.7.2 Exchangeable Observations......Page 233 6.7.3 Dependent Models Based on Archimedean Copula......Page 235 6.7.4 A Random Power Record Model......Page 236 Exercises......Page 237 7.1 Introduction......Page 245 7.2.1 Joint Distribution of Record Values......Page 246 7.2.2 Dependence Structure of Record Values and Record Counts......Page 251 7.2.3 Number of Records......Page 253 7.3 Basic Point Process Model and Applications......Page 255 7.4.1 Homogeneous Poisson Pacing Process......Page 258 7.4.2 Comparison with the Classical Model......Page 261 7.4.3 Nonhomogeneous Poisson Pacing Process......Page 263 7.5 Records Over a Renewal Process......Page 265 7.6.1 General Birth Process......Page 268 7.6.2 Yule Process......Page 270 7.6.3 Poisson Record Arrival Process......Page 271 7.7 Records Over Other Pacing Processes......Page 274 7.8 The Secretary Problem......Page 276 Exercises......Page 280 8.1 How Should We Define Multivariate Records?......Page 287 8.2 Bivariate Records With Independent Coordinates......Page 289 8.3 Concomitants of Records......Page 293 8.4 Lower and Upper Records and the Record Range......Page 296 Exercises......Page 299 Bibliography......Page 301 Author Index......Page 325 Subject Index......Page 331

The first and only comprehensive guide to modern record theory and its applications

Although it is often thought of as a special topic in order statistics, records form a unique area, independent of the study of sample extremes. Interest in records has increased steadily over the years since Chandler formulated the theory of records in 1952. Numerous applications of them have been developed in such far-flung fields as meteorology, sports analysis, hydrology, and stock market analysis, to name just a few. And the literature on the subject currently comprises papers and journal articles numbering in the hundreds. Which is why it is so nice to have this book devoted exclusively to this lively area of statistics.

Written by an exceptionally well-qualified author team, Records presents a comprehensive treatment of record theory and its applications in a variety of disciplines. With the help of a multitude of fascinating examples, Professors Arnold, Balakrishnan, and Nagaraja help readers quickly master basic and advanced record value concepts and procedures, from the classical record value model to random and multivariate record models. The book follows a rational textbook format, featuring witty and insightful chapter introductions that help smooth transitions from one topic to another and challenging chapter-end exercises, which expand on the material covered. An extensive bibliography and numerous references throughout the text specify sources for further readings on relevant topics. Records is a valuable professional resource for probabilists and statisticians, in addition to applied statisticians, meteorologists, hydrologists, market analysts, and sports analysts.It also makes an excellent primary text for courses in record theory and a supplement to order statistics courses.

Booknews

This textbook provides a resource for a one-term course on record statistics. In their preface the authors apologize to readers who might be seeking the latest information on the tallest resident of Lithuania or the most recent record of the renowned pole vaulter Sergei Bubka; the material covered here focuses on the theory behind sequences of record values and timestheory that the authors contend is both elegant and elementary. Prerequisites include a one- year course in introductory mathematical statistics, and additional basic knowledge of stochastic processes. Annotation c. by Book News, Inc., Portland, Or.

Records in areas such as weather, sports, hydrology, and the stock market have always interested both the specialist and the general public. Despite several books in this field, no single source for record value results has been available until now. This book covers both theory and application on sequences of record value.