Nonparametric estimation of long-tailed density functions and its application to the analysis of World Wide Web traffic

The study of WWW-traffic measurements has shown that different traffic characteristics can be modeled by long-tail distributed random variables (r.v.s). In this paper we discuss the nonparametric estimation of the probability density function of long-tailed distributions. Two nonparametric estimates, a Parzen–Rosenblatt kernel estimate and a histogram with variable bin width called polygram, are considered. The consistency of these estimates for heavy-tailed densities is discussed. To provide the consistency of the estimates in the metric space L1, the transformation of the initial r.v. to a new r.v. distributed on the interval [0,1] is proposed. Then the proposed estimates are applied to analyze real data of WWW-sessions. The latter are characterized by the sizes of the responses and inter-response intervals as well as the sizes and durations of sub-sessions. By these means the effectiveness of the nonparametric procedures in comparison to parametric models of the WWW-traffic characteristics is demonstrated.