Some Results for Beta Fréchet Distribution

• Published in: Communications in statistics. Theory and methods Vol. 40; no. 5; pp. 798 - 811
• Main Authors: Barreto-Souza, Wagner, Cordeiro, Gauss M., Simas, Alexandre B.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Taylor & Francis Group 01-01-2011; Taylor & Francis; Taylor & Francis Ltd
• Subjects: Beta; Beta distribution; Density; Distribution functions; Distribution theory; Exact sciences and technology; Exponentiated Fréchet; Fréchet distribution; General topics; Information matrix; Mathematical analysis; Mathematical models; Mathematics; Maximum likelihood estimation; Nonparametric inference; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Statistics; Statistical distribution; Fréchet differentiability; Statistical theory; Non parametric estimation; Exponentiated Frechet; Information measure; Linear combination; 62E20; Probability density function; Distribution function; Approximation theory; Mathematical expansion; Expansion; 60E05; Maximum likelihood estimation; Beta distribution; Linear function; Statistical moment; Order statistic; Statistical estimation; Information matrix; Frechet distribution; Statistical method; Density function; Maximum likelihood; Information theory
• ISSN: 0361-0926; 1532-415X
• DOI: 10.1080/03610920903366149

Nadarajah and Gupta (
2004
) introduced the beta Fréchet (BF) distribution, which is a generalization of the exponentiated Fréchet (EF) and Fréchet distributions, and obtained the probability density and cumulative distribution functions. However, they did not investigate the moments and the order statistics. In this article, the BF density function and the density function of the order statistics are expressed as linear combinations of Fréchet density functions. This is important to obtain some mathematical properties of the BF distribution in terms of the corresponding properties of the Fréchet distribution. We derive explicit expansions for the ordinary moments and L-moments and obtain the order statistics and their moments. We also discuss maximum likelihood estimation and calculate the information matrix which was not given in the literature. The information matrix is numerically determined. The usefulness of the BF distribution is illustrated through two applications to real data sets.