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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.

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This paper generalizes the notion of sufficiency for estimation problems beyond maximum likelihood. In particular, we consider estimation problems based on Jones et al. and Basu et al. likelihood functions that are popular among distance-based robust inference methods. We first characterize the probability distributions that always have a fixed number of sufficient statistics with respect to these likelihood functions. These distributions are power-law extensions of the usual exponential family and contain Student distributions as special case. We then extend the notion of minimal sufficient statistics and compute it for these power-law families. Finally, we establish a Rao-Blackwell type theorem for finding best estimators for a power-law family. This enables us to derive certain generalized Cramér-Rao lower bounds for power-law families.

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