Maximum Entropy Principle and Partial Probability Weighted Moments

Maximum entropy principle (MaxEnt) is usually 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. In such analysis, consideration of higher order moments is important for accurate modelling of the distribution tail. There are three drawbacks for this conventional methodology: (1) Estimates of higher order (>2) moments from a small sample of data tend to be highly biased; (2) It can merely cope with problems with complete or non-censored samples; (3) Only probability weighted moments of integer orders have been utilized. These difficulties inevitably induce bias and inaccuracy of the resultant quantile estimates and therefore have been the main impediments to the application of the MaxEnt Principle in extreme quantile estimation. This paper attempts to overcome these problems and presents a distribution free method for estimating the quantile function of a non-negative random variable using the principle of maximum partial entropy subject to constraints of the partial probability weighted moments estimated from censored sample. The main contributions include: (1) New concepts, i.e., partial entropy, fractional partial probability weighted moments, and partial Kullback-Leibler measure are elegantly defined; (2) Maximum entropy principle is re-formulated to be constrained by fractional partial probability weighted moments; (3) New distribution free quantile functions are derived. Numerical analyses are performed to assess the accuracy of extreme value estimates computed from censored samples.