Comment: Fuzzy and Randomized Confidence Intervals and P-Values

Abstract

Geyer and Meeden are to be congratulated for a ma jor idea on how to express uncertainty in classical fre quentist statistics. Their development of the trinity of fuzzy test functions, confidence intervals and P -values brings a new coherence to the relationship among these statistical entities when the test is randomized. In Geyer and Meeden, the uncertainty or randomization is due to the discreteness of data random variables, but another form of uncertainty or randomness arises in la tent variable problems and may be treated in an analo gous fashion. In this discussion, I will focus on fuzzy P -values in that context. Following the notation of Geyer and Meeden, let X denote the observed data random variables, but sup pose there are latent variables W of scientific interest, so that the ideal test statistic is t(X, W), a function of both X and W. Many examples of this situation arise in the analysis of genetic data, where W may be un observable DNA types or paths of descent of DNA to the observed individuals of a pedigree or population. Indeed, in this case the hypothesis of interest often concerns the probability distribution of W and hence the ideal test statistic is a function of W alone. The data random variable X is only of interest for the in formation it provides about W through some proba bility model Pr(X|W). For convenience, we consider here the case where the latent test statistic t(W) is a function only of W and assume the random variable W to be continuous. The discrete case is considered by Thompson and Geyer (2005). In the area of statistical genetic methodology, a stan dard procedure has been to average over the latent variable uncertainty and form test statistics E(t (W)|X) (Whittemore and Halpern, 1994; Kruglyak, Daly, Reeve-Daly and Lander, 1996). However, there appear to be neither theoretical justification nor optimality properties for such a proceeding. Moreover, the dis tribution of such a test statistic is not only hard or im possible to obtain, but depends on the distribution of X given W, which may itself be subject to consider able uncertainty. The second key point made by Geyer and Meeden is that the distribution function of the ab