tailieunhanh - On estimation of high quantiles for certain classes of distributions

Estimation of large quantiles of an unknown distribution function is a statistical problem of great practical importance. Let us mention estimation of the Value-at-Risk parameter for a given financial portfolio as an important problem that directly involves high quantile estimators. | Yugoslav Journal of Operations Research 25 (2015), Number 2, 299–312 DOI: ON ESTIMATION OF HIGH QUANTILES FOR CERTAIN CLASSES OF DISTRIBUTIONS ´ Jelena STANOJEVIC Faculty of Economics, University of Belgrade Serbia jelenas@ Received: June 2013 / Accepted: April 2014 Abstract: We investigate the rate of convergence of the direct-simulation estimator xˆp (n) of a large quantile xp of the Pareto and Gamma distributions. The upper bound of the probability P{|xˆp (n) − xp | > ε} is determined. Keywords: High Quantile Estimation, Negative Dependence, the Pareto Distribution, Gamma Distribution. MSC: 62F12, 62G32. 1. INTRODUCTION Estimation of large quantiles of an unknown distribution function is a statistical problem of great practical importance. Let us mention estimation of the Value-at-Risk parameter for a given financial portfolio as an important problem that directly involves high quantile estimators. Different estimators of high quantiles based on the upper order statistics of a sample were proposed and many important properties were proved. See, for example, Feldman and Tucker [7], Dekkers and de Haan [5], Embrechts et al. [6], Matthus and Beirlant [13] and references therein. In this paper we consider the rate of convergence of the directsimulation estimator of large quantiles and the aim of this paper is to calculate the rate of convergence of the Pareto and Gamma distributions. Applications of that distributions in theory as in empirical analyzes are well known. For example, it is well established that the burst and idle times for on/off traffic are modeled by the Pareto and Gamma distributions, respectively. Also, the inter arrival times between on/off-traffic is the convolution of the Pareto and Gamma random variables. For details see Nadarajah and Kotz [14]. The Pareto distribution is widely applied in different fields such as finance, insurance, physics, hydrology, geology, 300 Jelena Stanojevi´c / On Estimation of