## The Annals of Statistics

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

### On the Validity of the Formal Edgeworth Expansion

R. N. Bhattacharya and J. K. Ghosh

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176344134

**Digital Object Identifier**

doi:10.1214/aos/1176344134

**Mathematical Reviews number (MathSciNet)**

MR471142

**Zentralblatt MATH identifier**

0396.62010

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 62G05: Estimation 62G10: Hypothesis testing 62G20: Asymptotic properties

**Keywords**

Asymptotic expansion delta method Cramer's condition minimum contrast estimators

#### Citation

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. https://projecteuclid.org/euclid.aos/1176344134

#### Corrections

- 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.Project Euclid: euclid.aos/1176345213