Generalized Estimating Equation for the Student-t Distributions

Abstract

In [12], 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 [12] to the general M (α) -family. Finally we apply this result to the Student-t distribution and find generalized estimators for its parameters.

Generalized Fisher-Darmois-Koopman-Pitman Theorem and Rao-Blackwell Type Estimators for Power-Law Distributions

“…(ii) In [34], this family was derived as projections of Jones et al divergence on a set of probability distributions determined by linear constraints. More about M (α) -family can be found in [2], [24], [26].…”

“…However, in the following, we address this problem by deriving a Rao-Blackwell type theorem for Basu et al likelihood function (8). Note that, for this model, Basu et al estimator and Jones et al estimator are the same [24], [26]. as in (8).…”