Distributions with maximum entropy subject to constraints on their L-moments or expected order statistics

• Published in: Journal of statistical planning and inference Vol. 137; no. 9; pp. 2870 - 2891
• Main Author: Hosking, J.R.M.
• Format: Journal Article
• Language: English
• Published: Lausanne Elsevier B.V 01.09.2007; New York,NY Elsevier Science; Amsterdam
• Subjects: Density estimation; Density-quantile function; Distribution theory; Exact sciences and technology; General topics; Logistic distribution; Mathematics; Nonparametric; Nonparametric inference; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Statistics; Density estimation; Density-quantile function; Nonparametric; Logistic distribution; Polynomial; Conditional distribution; Exponential distribution; Sample size; Optimization method; L statistics; Non parametric method; Non parametric estimation; Entropy; Linear constraint; Uniform distribution; Linear combination; Distribution function; Quantile; Integration; Fitting; Statistical moment; Order statistic; Method of maximum entropy; Statistical decision; Constrained optimization; Polynomial function; Statistical method; Function derivative; Density function
• ISSN: 0378-3758; 1873-1171
• DOI: 10.1016/j.jspi.2006.10.010

We find the distribution that has maximum entropy conditional on having specified values of its first r L -moments. This condition is equivalent to specifying the expected values of the order statistics of a sample of size r. The maximum-entropy distribution has a density-quantile function, the reciprocal of the derivative of the quantile function, that is a polynomial of degree r; the quantile function of the distribution can then be found by integration. This class of maximum-entropy distributions includes the uniform, exponential and logistic, and two new generalizations of the logistic distribution. It provides a new method of nonparametric fitting of a distribution to a data sample. We also derive maximum-entropy distributions subject to constraints on expected values of linear combinations of order statistics.