A robust multivariate Birnbaum–Saunders distribution: EM estimation

ABSTRACT

We propose here a robust multivariate extension of the bivariate Birnbaum–Saunders (BS) distribution derived by Kundu et al. [Bivariate Birnbaum–Saunders distribution and associated inference. J Multivariate Anal. 2010;101:113–125], based on scale mixtures of normal (SMN) distributions that are used for modelling symmetric data. This resulting multivariate BS-type distribution is an absolutely continuous distribution whose marginal and conditional distributions are of BS-type distribution of Balakrishnan et al. [Estimation in the Birnbaum–Saunders distribution based on scalemixture of normals and the EM algorithm. Stat Oper Res Trans. 2009;33:171–192]. Due to the complexity of the likelihood function, parameter estimation by direct maximization is very difficult to achieve. For this reason, we exploit the nice hierarchical representation of the proposed distribution to propose a fast and accurate EM algorithm for computing the maximum likelihood (ML) estimates of the model parameters. We then evaluate the finite-sample performance of the developed EM algorithm and the asymptotic properties of the ML estimates through empirical experiments. Finally, we illustrate the obtained results with a real data and display the robustness feature of the estimation procedure developed here.

KEYWORDS:

• Multivariate Birnbaum–Saunders distribution
• EM algorithm
• kurtosis
• maximum likelihood method
• robust estimation
• scale mixtures of normal distributions

Acknowledgments

The authors thank the Editor-in-Chief, and two referees for their constructive comments on an earlier version of this manuscript, which resulted in this improved version.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors also acknowledge the partial financial support received from, FAPESP (2013/25935-2) and CNPq (309086/2009-4), Brazil.