Some results for beta Frechet distribution

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.

💡 Research Summary

The paper revisits the beta‑Frechet (BF) distribution originally introduced by Nadarajah and Gupta (2004). While the original work provided the probability density function (pdf) and cumulative distribution function (CDF), it left the moments, L‑moments, order‑statistics, and the Fisher information matrix unexplored. The authors fill these gaps by expressing the BF pdf as an infinite linear combination of standard Frechet pdfs. This representation is achieved by writing the BF CDF in terms of the beta function and then differentiating, which yields a series

 f_BF(x) = Σ_{j=0}^{∞} c_j f_Frechet(x;α,λ_j)

where the coefficients c_j are explicit functions of the beta parameters (a, b) and gamma functions, and λ_j are scaled versions of the original Frechet scale parameter. Because each term in the series is a known Frechet density, any statistical property of the BF distribution can be derived from the corresponding Frechet property by linearity.

Using this expansion, the authors derive closed‑form series for the ordinary moments E