Projection Theorems, Estimating Equations, and Power-Law Distributions

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, R\'enyi, 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.