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Let \Yₙ\₍ ₁ be a sequence of i. i. d. m-dimensional random vectors, and let f₁, , fₖ be real-valued Borel measurable functions on Rᵐ. Assume that Zₙ = (f₁ (Yₙ), , fₖ (Yₙ) ) has finite moments of order s 3. Rates of convergence to normality and asymptotic expansions of distributions of statistics of the form Wₙ = n^1{2} H (Z) - H () are obtained for functions H on Rᵏ having continuous derivatives of order s in a neighborhood of = EZ₁. This asymptotic expansion is shown to be identical with a formal Edgeworth expansion of the distribution function of Wₙ. This settles a conjecture of Wallace (1958). The class of statistics considered includes all appropriately smooth functions of sample moments. An application yields asymptotic expansions of distributions of maximum likelihood estimators and, more generally, minimum contrast estimators of vector parameters under readily verifiable distributional assumptions.
Bhattacharya et al. (Wed,) studied this question.