The estimation of heavy-tailed probability density functions, their mixtures and quantiles

This paper is devoted to the estimation of heavy-tailed probability density functions (p.d.f.s), their mixtures and high quantiles. We discuss the relevance of this issue in teletraffic engineering and propose a new combined estimation technique for such p.d.f.s. The "tail" of the p.d.f. is estimated by a parametric tail model and its "body" by a nonparametric method in terms of a finite linear combination of trigonometric functions. To provide the minimum of the mean-squared error of the estimation, the parameters of the parametric and non-parametric parts are estimated by means of the bootstrap method and the structural risk minimization method. The latter parameters are determined by the number of extreme-valued data that are used in Hill's estimate of the tail index and the number of terms and coefficients of the linear combination. The new method is illustrated using some relevant mixtures of heavy-tailed p.d.f.s and applied to construct a high-quantile estimate. Furthermore, its effectiveness is shown by an application to real data arising from Web-traffic characteristics. (C) 2002 Elsevier Science B.V. All rights reserved.