Extreme quantile estimation using order statistics with minimum cross-entropy principle

The paper presents a general approach to the estimation of the quantile fun ction of a non-negative random variable using the principle of minimum cros s-entropy (CrossEnt) subject to constraints specified in terms of expectati ons of order statistics estimated from observed data. Traditionally CrossEnt is used for estimating the probability density funct ion under specified moment constraints. In such analyses, consideration of higher order moments is important for accurate modelling of the distributio n tail. Since the higher order (>2) moment estimates from a small sample of data tend to be highly biased and uncertain, the use of CrossEnt quantile estimates in extreme value analysis is fairly limited. The present paper is an attempt to overcome this problem via the use of pro bability weighted moments (PWMs), which are essentially the expectations of order statistics. In contrast with ordinary statistical moments, higher or der PWMs can be accurately estimated from small samples. By interpreting a PWM as the moment of quantile function, the paper derives an analytical for m of quantile function using the CrossEnt principle. Monte Carlo simulation s are performed to assess the accuracy of CrossEnt quantile estimates obtai ned from small samples. (C) 2000 Elsevier Science Ltd. All rights reserved.