More on Stochastic Inference

Abstract

Most of the work in Chapters IV and V on inference has focussed on a simple hypothesis versus a simple alternative. When either (or both) of the latter is composite, several new problems arise. In this chapter, we consider some of these questions in detail. The results again depend on likelihood ratios, and an extension of the Neyman-Pearson-Grenander theorem is once more of importance. Parameter estimation plays a key role, often involving uncountable sets of measures, and some of the work of Pitcher and his associates is presented. This utilizes certain abstract methods, and moreover includes some results appearing here for the first time. The Gaussian dichotomy in an alternative form (without separability restrictions) is considered. A general Girsanov theorem is also established because these assertions are needed in many important applications, including the Wiener-Itô chaos, white noise and the Wick products as complements. With further detailed analysis on the more tractable Gaussian processes, the following work substantially advances that of the earlier chapters.