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.
Projection theorems of divergence functionals reduce certain estimation problems under specific families of probability distributions to linear problems. In this paper, we study projection theorems concerning Kullback-Leibler, Rényi, density power, and logarithmic-density power divergences which are popular in robust inference. We first extend these projection theorems to the continuous case by directly solving the associated estimating equations. We then apply these ideas to solve certain estimation problems concerning Student and Cauchy distributions. Finally, we explore the projection theorems by a generalized notion of principle of sufficiency. In particular, we show that the statistics of the data that influence the projection theorems are also a minimal sufficient statistics with respect to this generalized notion.
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