Generalized Estimating Equation for the Student-t Distributions

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