The Beta Generalized Half-Normal Distribution: New Properties

We study some mathematical properties of the beta generalized half-normal distribution recently proposed by Pescim et al. (2010). This model is quite flexible for analyzing positive real data since it contains as special models the half-normal, exponentiated half-normal, and generalized half-normal distributions. We provide a useful power series for the quantile function. Some new explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability, and entropy. We demonstrate that the density function of the beta generalized half-normal order statistics can be expressed as a mixture of generalized half-normal densities. We obtain two closed-form expressions for their moments and other statistical measures. The method of maximum likelihood is used to estimate the model parameters censored data. The beta generalized half-normal model is modified to cope with long-term survivors may be present in the data. The usefulness of this distribution is illustrated in the analysis of four real data sets. 1. Introduction Cooray and Ananda [1] pioneered the generalized half-normal (GHN) distribution with shape parameter and scale parameter defined by the cumulative distribution function (cdf) where the standard normal cdf and the error function are given by Following an idea due to Eugene et al. [2], Pescim et al. [3] proposed the beta generalized half-normal (BGHN) distribution, which seems to be superior over the GHN model for some applications. The justification for the practicability of this model is based on the fatigue crack growth under variable stress or cyclic load. In this paper, we study several mathematical properties of the BGHN model with the hope that it will attract wider applications in reliability, engineering and in other areas of research. The four-parameter BGHN cdf is defined from (1) by (for ), where is the beta function, is the incomplete beta function ratio, and and are two additional shape parameters. The probability density function (pdf) and the hazard rate function (hrf) corresponding to (3) are respectively. Hereafter, a random variable with pdf (4) is denoted by . Pescim et al. [3] demonstrated that the cdf and pdf of can be expressed as infinite power series of the GHN cumulative distribution. Here, all expansions in power series are around the point zero. If is a real noninteger, we can expand the binomial term in (3) to obtain where . The pdf corresponding to (6) can be expressed as If is an integer, (7) provides the BGHN density function as an infinite power series of the GHN cumulative distribution.