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absolutely continuous distribution binomial distribution Chapter characteristic function class of probability conditional distribution conditional probability confidence region consider continuous distribution function convex function corresponding decision function defined definition density function distribution with mean easily seen equivalent Example Fe(x fundamental lemma given hence Hoeffding hypothesis testing problem induced distribution integers invariant test Lebesgue measure Let Xv limiting distribution limiting normal distribution location parameter loss function Math nonparametric normal with mean obtain order statistic outcome permutations power function powerful test probability density probability distribution probability measure Proof prove Radon-Nikodym theorem random variable rank test real line real-valued restriction risk function satisfies Section sequence statistic t(x statistician subset sufficient statistic symmetric test for Hypothesis test function Theorem 3.1 theory tolerance region transformations two-sample problem unbiased estimator unbiased test uniformly most powerful Xn be independent zero