Generalized Estimating Equation for the Student-t Distributions

Abstract

In [12], it was shown that a generalized maximum likelihood estimation problem on a (canonical) \alpha -power-law model (\mathbb{M}^{(\alpha)} -family) can be solved by solving a system of linear equations. This was due to an orthogonality relationship between the \mathbb{M}^{(\alpha)} -family and a linear family with respect to the relative \alpha -entropy (or the \mathscr{I}-{\alpha} -divergence). Relative \alpha -entropy is a generalization of the usual relative entropy (or the Kullback-Leibler divergence). \mathbb{M}^{(\alpha)} -family is a generalization of the usual exponential family. In this paper, we first generalize the \mathbb{M}^{(\alpha)}- family including the multivariate, continuous case and show that the Student-t distributions fall in this family. We then extend the above stated result of [12] to the general \mathbb{M}^{(\alpha)} -family. Finally we apply this result to the Student-t distribution and find generalized estimators for its parameters. © 2018 IEEE.