Some results for beta Fr\'echet distribution

TL;DR

This paper explores the properties of the beta Fréchet distribution, including moments, order statistics, and maximum likelihood estimation, providing new mathematical insights and applications to real data.

Contribution

It derives explicit moments, order statistics, and the information matrix for the beta Fréchet distribution, extending prior work by Nadarajah and Gupta.

Findings

Explicit formulas for moments and order statistics.

Maximum likelihood estimation and information matrix calculation.

Successful application to real data sets.

Abstract

Nadarajah and Gupta (2004) introduced the beta Fr\'echet (BF) distribution, which is a generalization of the exponentiated Fr\'echet (EF) and Fr\'echet distributions, and obtained the probability density and cumulative distribution functions. However, they do not investigated its moments and the order statistics. In this paper the BF density function and the density function of the order statistics are expressed as linear combinations of Fr\'echet density functions. This is important to obtain some mathematical properties of the BF distribution in terms of the corresponding properties of the Fr\'echet 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 known. The information matrix is easily numerically determined. Two applications to real data sets are given to illustrate the potentiality of this distribution.