On the validity of the formal Edgeworth expansion

Bhattacharya, R. N. ; Ghosh, J. K. (1978) On the validity of the formal Edgeworth expansion Annals of Statistics, 6 (2). pp. 434-451. ISSN 0090-5364

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Let {Yn}n≥ 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 = n½[ H(Z)-- H(µ)] are obtained for functions H on Rk having continuous derivatives of order s in a neighborhood of µ = 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.

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