Direct estimation of quantile functions using the maximum entropy principle

The paper presents a distribution free method for estimating the quantile f unction of a non-negative random variable using the principle of maximum en tropy (MaxEnt) subject to constraints specified in terms of the probability -weighted moments estimated from observed data. Traditionally, MaxEnt is us ed for estimating the probability density function under specified moment c onstraints. The density function is then integrated to obtain the cumulativ e distribution function, which needs to be inverted to obtain a quantile co rresponding to some specified probability. For correct modelling of the dis tribution tail, higher order moments must be considered in the analysis. Ho wever, the higher order (> 2) moment estimates from a small sample of data tend to be highly biased and uncertain. The difficulty in obtaining accurat e moment estimates from small samples has been the main impediment to the a pplication of the MaxEnt Principle in extreme quantile estimation. The pres ent paper is an attempt to overcome this problem by the use of probability- weighted moments (PWMs), which are essentially the expectations of order st atistics. In contrast with ordinary statistical moments, higher order PWMs can be accurately estimated from small samples. By interpreting the PWM as the moment of quantile function, the paper derives an analytical form of qu antile function using MaxEnt principle. Monte Carlo simulations are perform ed to assess the accuracy of MaxEnt quantile estimates computed from small samples. (C) 2000 Elsevier Science Ltd. All rights reserved.