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Nonparametric Analysis of Univariate Heavy-Tailed Data: Research and Practice (Wiley Series in Probability and Statistics #753)
By: Natalia MarkovicheBook Publisher: John Wiley & Sons
Imprint: Wiley-Interscience
Format: Adobe Encrypted (DRM)
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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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| Title of eBook: Nonparametric Analysis of Univariate Heavy-Tailed Data: Research and Practice (Wiley Series in Probability and Statistics #753) | |
| Release Date: 05-12-2008 | |
| Publisher: Wiley-Interscience |
This eBook download is available in the following formats:
| Parent title | Nonparametric Analysis of Univariate... |
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Nonparametric Analysis of Univariate Heavy-Tailed Data: Research and Practice (Wiley Series in Probability and Statistics #753)
Chapter One
Definitions and rough detection of tail heaviness
In this chapter, the basic definitions and properties of heavy-tailed distributions are presented. Tail index estimation and methods for selecting the number of largest order statistics in the Hill estimator are discussed. Rough methods for the detection of heavy tails, the number of finite moments, dependence and long-range dependence are described. Elements of bivariate analysis are presented: estimation of the Pickands function and bivariate quantiles. The latter methods are applied to the analysis of telecommunication data.
1.1 Definitions and basic properties of classes of heavy-tailed distributions
We start with the common definitions.
Definition 1 The set ([OMEGA], A, P) is called the probability space, where [OMEGA] is the space of elementary events, A is a [sigma]-algebra of subsets of [OMEGA], and P is a probability measure on A.
Let ([OMEGA, A) be some measurable space, (R, B (R)) be the real line with the [sigma]-algebra B(R) of Borelian sets on R.
Definition 2 The real-valued function X = X([omega]) defined on ([OMEGA], A), is called a random variable (r.v.), if for any B [[subset].bar] B(R) {[omega]: X ([omega] [member of]} B} [[subset].bar] A holds.
Definition 3 The function [F.sub.X](x) = P{[omega] : X([omega])[less than or equal to] x}, x [member of] R, is called the distribution function (DF) of the r.v. X.
Definition 4 Let a nonnegative real-valued function f (t), t [member of] R, exist such that for all x [member of] R,
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The function f(t), t [me
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