On the validity of the formal Edgeworth expansion. (English) Zbl 0396.62010

Bhattacharya, R. N.; Ghosh, J. K.

Ann. Stat. 6, 434-451 (1978).

Authors’ summary: Let \(\{Y_n\}_{n\geq 1}\) be a sequence of i.i.d. \(m\)-dimensional random vectors, and let \(f_1,\ldots,f_k\) be real-valued Borel measurable functions on \(\mathbb R^m\). Assume that \(Z_n=(f_1(Y_n),\ldots,f_k(Y_n))\) has finite moments of order \(s\geq 3\). Rates of convergence to normality and asymptotic expansions of distributions of statistics of the form \(W_n=n^{1/2}[H(\bar z)-H(\mu)]\) are obtained for functions \(H\) on \(\mathbb R^k\) 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 D. L. Wallace [Ann. Math. Stat. 29, 635–654 (1958; Zbl 0086.34004)]. 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.
For corrections see Zbl 0465.62025.

Reviewer: B. Penkov

MSC:

62E20	Asymptotic distribution theory in statistics
62G05	Nonparametric estimation
62G10	Nonparametric hypothesis testing
62G20	Asymptotic properties of nonparametric inference

Keywords:

formal Edgeworth expansion; maximum likelihood estimators; minimum contrast estimators of vector parameters; asymptotic expansions of distributions

Citations:

Zbl 0086.34004; Zbl 0465.62025