| Abstract: | The paper presents a distribution free method for estimating the quantile function of a non-negative random variable using the principle of maximum entropy (MaxEnt) subject to constraints specified in terms of the probability-weighted moments estimated from observed data. Traditionally, MaxEnt is used for estimating the probability density function under specified moment constraints. The density function is then integrated to obtain the cumulative distribution function, which needs to be inverted to obtain a quantile corresponding to some specified probability. For correct modelling of the distribution tail, higher order moments must be considered in the analysis. However, the higher order (>2) moment estimates from a small sample of data tend to be highly biased and uncertain. The difficulty in obtaining accurate moment estimates from small samples has been the main impediment to the application of the MaxEnt Principle in extreme quantile estimation. The present paper is an attempt to overcome this problem by the use of probability-weighted moments (PWMs), which are essentially the expectations of order statistics. 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 quantile function using MaxEnt principle. Monte Carlo simulations are performed to assess the accuracy of MaxEnt quantile estimates computed from small samples. |
