Generalized multivariate Birnbaum–Saunders distributions and related inferential issues

Birnbaum and Saunders introduced in 1969 a two-parameter lifetime distribution which has been used quite successfully to model a wide variety of univariate positively skewed data. Diaz-Garcia and Leiva-Sanchez [8] proposed a generalized Birnbaum–Saunders distribution by using an elliptically symmetric distribution in place of the normal distribution. Recently, Kundu et al. [13] introduced a bivariate Birnbaum–Saunders distribution, based on a transformation of a bivariate normal distribution, and discussed its properties and associated inferential issues. In this paper, we construct a generalized multivariate Birnbaum–Saunders distribution, by using the multivariate elliptically symmetric distribution as a base kernel for the transformation instead of the multivariate normal distribution. Different properties of this distribution are obtained in the general case. Special emphasis is placed on statistical inference for two particular cases: (i) multivariate normal kernel and (ii) multivariate-t kernels. We use the maximized log-likelihood values for selecting the best kernel function. Finally, a data analysis is presented for illustrative purposes.