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

• Published in: Probabilistic engineering mechanics Vol. 16; no. 1; pp. 31 - 42
• Main Author: Pandey, M.D.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 2001; Elsevier Science
• Subjects: Distribution theory and monte carlo studies; Entropy; Exact sciences and technology; Extreme value analysis; Information theory; Mathematical methods in physics; Minimum cross-entropy principle; Order statistics; Pareto distribution; Physics; Probability; Probability theory, stochastic processes, and statistics; Probability weighted moment; Quantile function; Extreme value analysis; Minimum cross-entropy principle; Pareto distribution; Probability weighted moment; Probability; Entropy; Order statistics; Quantile function; Information theory; Uncertain systems; Statistical analysis; Statistical moment; Value engineering; Order statistic; Computerized simulation; Pulse modulation; Monte Carlo methods; Extreme value; Probability density function; Minimum principle; Distribution tail; Modelling; Random variable; Biased estimation
• ISSN: 0266-8920; 1878-4275
• DOI: 10.1016/S0266-8920(00)00004-7

The paper presents a general approach to the estimation of the quantile function of a non-negative random variable using the principle of minimum cross-entropy (CrossEnt) subject to constraints specified in terms of expectations of order statistics estimated from observed data. Traditionally CrossEnt is used for estimating the probability density function under specified moment constraints. In such analyses, consideration of higher order moments is important for accurate modelling of the distribution 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 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 a PWM as the moment of quantile function, the paper derives an analytical form of quantile function using the CrossEnt principle. Monte Carlo simulations are performed to assess the accuracy of CrossEnt quantile estimates obtained from small samples.