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March, 1978 On the Validity of the Formal Edgeworth Expansion
R. N. Bhattacharya, J. K. Ghosh
Ann. Statist. 6(2): 434-451 (March, 1978). DOI: 10.1214/aos/1176344134


Let $\{Y_n\}_{n\geqq 1}$ be a sequence of i.i.d. $m$-dimensional random vectors, and let $f_1,\cdots, f_k$ be real-valued Borel measurable functions on $R^m$. Assume that $Z_n = (f_1(Y_n),\cdots, f_k(Y_n))$ has finite moments of order $s \geqq 3$. Rates of convergence to normality and asymptotic expansions of distributions of statistics of the form $W_n = n^{\frac{1}{2}}\lbrack H(\bar{Z}) - H(\mu)\rbrack$ are obtained for functions $H$ on $R^k$ having continuous derivatives of order $s$ in a neighborhood of $\mu = EZ_1$. This asymptotic expansion is shown to be identical with a formal Edgeworth expansion of the distribution function of $W_n$. 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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R. N. Bhattacharya. J. K. Ghosh. "On the Validity of the Formal Edgeworth Expansion." Ann. Statist. 6 (2) 434 - 451, March, 1978.


Published: March, 1978
First available in Project Euclid: 12 April 2007

zbMATH: 0396.62010
MathSciNet: MR471142
Digital Object Identifier: 10.1214/aos/1176344134

Primary: 62E20
Secondary: 62G05 , 62G10 , 62G20

Keywords: asymptotic expansion , Cramer's condition , Delta method , minimum contrast estimators

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 2 • March, 1978
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