Robust tests for model selection

Abstract

It was shown almost 40 years ago by Lucien Le Cam that the existence of suitable tests between Hellinger balls in the parameter set led to the construction of some sort of universal estimators for parametric statistical problems with i.i.d. observations. This idea of deriving estimators from families of robust tests was developed and substantially generalized in some of my previous work and more recently extended to Model Selection based estimation. Since the key ingredient for the design of such estimators for a given statistical framework is the construction of the relevant tests for this particular framework, it is essential to explain how to build them for as many different frameworks as possible. The purpose of this paper is to provide improved results about the existence of such tests for the problems of estimation based on independent (not necessarily i.i.d.) observations, estimation of conditional densities and of Markov transitions.