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a)-level absolutely continuous assume asymptotically balls Bayes estimator binomial choose common PDF compute conditional distribution confidence interval consider constant continuous type converges Corollary defined Definition DF F discrete equivariant equivariant estimator Example exists exponential family family of distributions fe(x finite following result given iid RVs independent RVs inequality integer invariant joint PDF Lemma Let X Let Xi linear loss function marginal method minimax nonnegative normal distribution normal RVs Note observations order statistics otherwise parameter PDF PMF probability space problem proof random sample random variable real numbers reject Remark ri ri ri RV with PDF RVs with common sample space Section sequence of RVs Show subsets sufficient statistic Suppose symmetric Theorem UMP test UMVUE unbiased estimator unknown values var(X variance a2 X2 Xn