Year: 1978
Type: article
Abstract: 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.
Source: The Annals of Statistics
Authors: R. N. Bhattacharya, J. K. Ghosh
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2106730112