Projection Theorems, Estimating Equations, and Power-Law Distributions
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Projection theorems of divergence functionals reduce certain estimation problems on some specific families of probability distributions to a linear problem. Most of these divergences are also popular in the context of robust statistics. In this paper we first extend these projection theorems to a more general setup including the continuous case by directly solving the associated estimating equations. We then apply these ideas to solve the estimation problems concerning the Student and Cauchy distributions. Finally we explore the projection theorems by a generalized principle of sufficiency. In particular, we show that the statistics of the data that influence the projection theorems are a minimal sufficient statistics with respect to this generalized notion of sufficiency.
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