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adjusted likelihood ancillary statistic approximation assume asymptotic normality Bayes estimate Bayes risk Bayesian Bhattacharya and Ghosh bias Borel sets calculations Chapter coefficient compact sets conditional likelihood continuously differentiable cumulants curvature curved exponential defined delta method estimate Tn exponential family finite Fisher consistent estimates Fisher information Fisher information matrix FOE estimates frequentist frequentist Bartlett correction Ghosh and Mukerjee Ghosh and Subramanyam Haar measure Hence higher order asymptotics information matrix integral interval Jeffreys prior Lemma likelihood equation likelihood ratio test linear lower bound matching minimal minimaxity noninformative priors Note nuisance parameter o(n l orthogonal parameter of interest perturbation polynomial posterior density posterior probability profile likelihood proof reference prior regularity conditions result sample satisfies Section Sinha and Joshi solution Subramanyam 1974 sufficient statistic term Theorem 2.1 third order efficiency thrice continuously differentiable unbiased estimate uniformly on compact valid Edgeworth expansion variance zero
Page 28 - ... the sum of the first three terms on the right side of Eq.
Page 15 - So if H is (s — 1) times continuously differentiable in a neighborhood of //-(0), the estimate Tn = H(Z) has a valid Edgeworth expansion correct up to o(n-(s~2)/2).