Generalized Estimating Equation for the Student-t Distributions

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In KumarS15J2, it was shown that a generalized maximum likelihood estimation problem on a (canonical) α-power-law 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 Iα-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 KumarS15J2 to the general M⁽α⁾-family. Finally we apply this result to the Student-t distribution and find generalized estimators for its parameters.