The Annals of Statistics

On the Validity of the Formal Edgeworth Expansion

R. N. Bhattacharya and J. K. Ghosh

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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.

Article information

Ann. Statist., Volume 6, Number 2 (1978), 434-451.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62E20: Asymptotic distribution theory
Secondary: 62G05: Estimation 62G10: Hypothesis testing 62G20: Asymptotic properties

Asymptotic expansion delta method Cramer's condition minimum contrast estimators


Bhattacharya, R. N.; Ghosh, J. K. On the Validity of the Formal Edgeworth Expansion. Ann. Statist. 6 (1978), no. 2, 434--451. doi:10.1214/aos/1176344134.

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  • See Correction: R. N. Bhattacharya, J. K. Ghosh. Correction to "On the Validity of the Formal Edgeworth Expansion". Ann. Statist., Volume 8, Number 6 (1980), 1399--1399.