Nonparametric Analysis of Univariate Heavy-Tailed Data: Research and Practice

Natalia Markovich0470510870, 9780470510872

Heavy-tailed distributions are typical for phenomena in complex multi-component systems such as biometry, economics, ecological systems, sociology, web access statistics, internet traffic, biblio-metrics, finance and business. The analysis of such distributions requires special methods of estimation due to their specific features. These are not only the slow decay to zero of the tail, but also the violation of Cramer’s condition, possible non-existence of some moments, and sparse observations in the tail of the distribution.The book focuses on the methods of statistical analysis of heavy-tailed independent identically distributed random variables by empirical samples of moderate sizes. It provides a detailed survey of classical results and recent developments in the theory of nonparametric estimation of the probability density function, the tail index, the hazard rate and the renewal function.Both asymptotical results, for example convergence rates of the estimates, and results for the samples of moderate sizes supported by Monte-Carlo investigation, are considered. The text is illustrated by the application of the considered methodologies to real data of web traffic measurements.

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Table of contents :
Nonparametric Analysis of Univariate Heavy-Tailed Data……Page 1
Frontmatter……Page 2
1 Definitions and Rough Detection of Tail Heaviness……Page 22
2 Classical Methods of Probability Density Estimation……Page 81
3 Heavy-Tailed Density Estimation……Page 118
4 Transformations and Heavy-Tailed Density Estimation……Page 141
5 Classification and Retransformed Density Estimates……Page 169
6 Estimation of High Quantiles……Page 181
7 Nonparametric Estimation of the Hazard Rate Function……Page 196
8 Nonparametric Estimation of the Renewal Function……Page 236
Appendix A: Proofs of Chapter 2……Page 267
Appendix B: Proofs of Chapter 4……Page 268
Appendix C: Proofs of Chapter 5……Page 282
Appendix D: Proofs of Chapter 6……Page 286
Appendix E: Proofs of Chapter 7……Page 290
Appendix F: Proofs of Chapter 8……Page 299
List of Main Symbols and Abbreviations……Page 304
References……Page 307
Index……Page 318
Wiley Series in Probability and Statistics……Page 322