Generalized Estimating Equation for the Student-t Distributions

In \cite{KumarS15J2}, it was shown that a generalized maximum likelihood estimation problem on a (canonical) -power-law model (-family) can be solved by solving a system of linear equations. This was due to an orthogonality relationship between the -family and a linear family with respect to the relative -entropy (or the -divergence). Relative -entropy is a generalization of the usual relative entropy (or the Kullback-Leibler divergence). -family is a generalization of the usual exponential family. In this paper, we first generalize the -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 \cite{KumarS15J2} to the general -family. Finally we apply this result to the Student-t distribution and find generalized estimators for its parameters.
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