Contiguity — Hellinger Transforms

Abstract

We saw, in Chapter 2, that when Θ is finite, convergence of experiments in the sense of our distance is equivalent to convergence in distribution of likelihood ratios. Here we shall describe, in Section 3.1, some consequences of a condition, called contiguity that simplifies many arguments in passages to the limit. Contiguity is simply an asymptotic form of absolute continuity. Theorem 1 establishes several equivalent forms of the conditions for contiguity. One of the most useful consequences of contiguity is its application to the joint limiting distribution of statistics and likelihood ratios, described in Proposition 1. We also introduce, in Section 3.2, a technical tool, the Hellinger transform, that is often convenient in studies involving independent observations. In that case it provides the same flexibility as that given by characteristic functions in the study of sums of independent random variables.KeywordsLikelihood RatioProbability MeasureIndependent ObservationCluster PointAbsolute ContinuityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.