A Generalization of the Lehmann-Scheff\'e Theorem
Agustín García Nogales · arXiv (Cornell University) · 2013
Markov kernels play a decisive role in probability and mathematical statistics theories, conditional distributions being the main example; as it is noted below, every Markov kernel is a conditional distribution of some random variable given another. In statistical decision theory, randomized procedures are Markov kernels; it is well known that, in some situations, the optimum procedure is randomized. In Bayesian inference, sampling probabilities and posterior distributions are Markov kernels. A Markov kernel is also an extension of the concepts of $\sigma$-field and statistic, and well known concepts of probability theory or mathematical statistics, such as independence, completeness, ancillarity or conditional distribution have been extended to Markov kernels in Nogales (2012a) and Nogales (2012b), in a similar way that Heyer (1982) extend the concept of sufficiency. In this paper we extend to Markov kernels the theorems of Rao-Blackwel and Lehmann-Scheff\'e. In fact both results are generalized for randomized estimators when a sufficient and complete Markov kernel is known. We introduce in passing the concept of conditional expectation of a Markov kernel given another. Finally, a generalization of a result about the completeness of the family of nonrandomized estimators is given.