Generalized Estimating Equation for the Student-t Distributions

In [12], it was shown that a generalized maximum likelihood estimation problem on a (canonical) α-power-law model (M-family) can be solved by solving a system of linear equations. This was due to an orthogonality relationship between the M-family and a linear family with respect to the relative α-entropy (or the Iα-divergence). Relative α-entropy is a generalization of the usual relative entropy (or the KullbackLeibler divergence). M-family is a generalization of the usual exponential family. In this paper, we first generalize the Mfamily 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 M-family. Finally we apply this result to the Student-t distribution and find generalized estimators for its parameters.