Nonparametric Analysis of Univariate

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.

Natalia Markovich – Institute of Control Sciences, Russian Academy of Sciences, Moscow

Having been the Leading Scientist at the Institute of Control Sciences for the last eleven years, Dr Markovich has had much experience in this area. An extremely active member of the statistical community, she has presented many seminars and invited talks, as well as being involved in numerous international research projects. She has published over 50 articles and has written chapters in two books, for Springer-Verlag and Elsevier.

Preface	p. xi
Definitions and rough detection of tail heaviness	p. 1
Definitions and basic properties of classes of heavy-tailed distributions	p. 1
Tail index estimation	p. 6
Estimators of a positive-valued tail index	p. 6
The choice of k in Hill's estimator	p. 8
Estimators of a real-valued tail index	p. 13
On-line estimation of the tail index	p. 17
Detection of tail heaviness and dependence	p. 27
Rough tests of tail heaviness	p. 27
Analysis of Web traffic and TCP flow data	p. 30
Dependence detection from univariate data	p. 42
Dependence detection from bivariate data	p. 49
Bivariate analysis of TCP flow data	p. 51
Notes and comments	p. 56
Exercises	p. 57
Classical methods of probability density estimation	p. 61
Principles of density estimation	p. 61
Methods of density estimation	p. 70
Kernel estimators	p. 70
Projection estimators	p. 74
Spline estimators	p. 76
Smoothing methods	p. 76
Illustrative examples	p. 83
Kernel estimation from dependent data	p. 85
Statement of the problem	p. 86
Numerical calculation of the bandwidth	p. 89
Data-driven selection of the bandwidth	p. 91
Applications	p. 91
Finance: evaluation of market risk	p. 91
Telecommunications	p. 93
Population analysis	p. 94
Exercises	p. 95
Heavy-tailed density estimation	p. 99
Problems of the estimation of heavy-tailed densities	p. 100
Combined parametric-nonparametric method	p. 101
Nonparametric estimation of the density by structural risk minimization	p. 103
Illustrative examples	p. 107
Web data analysis by a combined parametric-nonparametric method	p. 109
Barron's estimator and [chi][superscript 2]-optimality	p. 111
Kernel estimators with variable bandwidth	p. 113
Retransformed nonparametric estimators	p. 117
Exercises	p. 119
Transformations and heavy-tailed density estimation	p. 123
Problems of data transformations	p. 123
Estimates based on a fixed transformation	p. 124
Estimates based on an adaptive transformation	p. 128
Estimation algorithm	p. 128
Analysis of the algorithm	p. 129
Further remarks	p. 133
Estimating the accuracy of retransformed estimates	p. 135
Boundary kernels	p. 136
Accuracy of a nonvariable bandwidth kernel estimator	p. 139
The D method for a nonvariable bandwidth kernel estimator	p. 141
The D method for a variable bandwidth kernel estimator	p. 142
Method and results	p. 142
Application to Web traffic characteristics	p. 144
The [omega][superscript 2] method for the projection estimator	p. 147
Exercises	p. 149
Classification and retransformed density estimates	p. 151
Classification and quality of density estimation	p. 151
Convergence of the estimated probability of misclassification	p. 154
Simulation study	p. 155
Application of the classification technique to Web data analysis	p. 160
Intelligent browser	p. 160
Web data analysis by traffic classification	p. 161
Web prefetching	p. 161
Exercises	p. 161
Estimation of high quantiles	p. 163
Introduction	p. 163
Estimators of high quantiles	p. 164
Distribution of high quantile estimates	p. 167
Simulation study	p. 169
Comparison of high quantile estimates in terms of relative bias and mean squared error	p. 169
Comparison of high quantile estimates in terms of confidence intervals	p. 170
Application to Web traffic data	p. 175
Exercises	p. 176
Nonparametric estimation of the hazard rate function	p. 179
Definition of the hazard rate function	p. 180
Statistical regularization method	p. 182
Numerical solution of ill-posed problems	p. 185
Estimation of the hazard rate function of heavy-tailed distributions	p. 187
Hazard rate estimation for compactly supported distributions	p. 188
Estimation of the hazard rate from the simplest equations	p. 188
Estimation of the hazard rate from a special kernel equation	p. 193
Estimation of the ratio of hazard rates	p. 197
Failure time detection	p. 199
Hormesis detection	p. 200
Hazard rate estimation in teletraffic theory	p. 207
Teletraffic processes at the packet level	p. 207
Estimation of the intensity of a nonhomogeneous Poisson process	p. 208
Semi-Markov modeling in teletraffic engineering	p. 210
The Gilbert-Elliott model	p. 210
Estimation of a retrial process	p. 212
Exercises	p. 217
Nonparametric estimation of the renewal function	p. 219
Traffic modeling by recurrent marked point processes	p. 220
Introduction to renewal function estimation	p. 221
Histogram-type estimator of the renewal function	p. 224
Convergence of the histogram-type estimator	p. 225
Selection of k by a bootstrap method	p. 228
Selection of k by a plot	p. 232
Simulation study	p. 234
Application to the inter-arrival times of TCP connections	p. 245
Conclusions and discussion	p. 247
Exercises	p. 248
Appendices
Proofs of Chapter 2	p. 251
Proofs of Chapter 4	p. 253
Proofs of Chapter 5	p. 267
Proofs of Chapter 6	p. 271
Proofs of Chapter 7	p. 275
Proofs of Chapter 8	p. 285
List of Main Symbols and Abbreviations	p. 291
References	p. 295
Index	p. 307
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Nonparametric Analysis of Univariate Heavy-Tailed Data Research and Practice

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