## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 42, Number 4 (1971), 1412-1424.

### On Obtaining Large-Sample Tests from Asymptotically Normal Estimators

#### Abstract

This is an extension of Wald's asymptotic test procedure based on unrestricted maximum-likelihood estimators. Wald showed that under certain regularity conditions the test statistic has a limiting central chi-square distribution under the hypothesis and a limiting noncentral chi-square distribution under a sequence of local alternatives. We extend this procedure, allowing it to be based on a broader class of estimators and to obey simpler and less restrictive conditions. Sufficient conditions for validity of the limiting distributions are local twice-differentiability of the left side of the hypothesis and, under a sequence of local alternatives, asymptotic normality of the estimator of the parameter defining the distribution and stochastic convergence (to the appropriate asymptotic value) of the estimator of the covariance matrix. The required asymptotic behavior is verified for the case of independent sampling from two normal distributions and formulas are presented which aid in computing the test statistic.

#### Article information

**Source**

Ann. Math. Statist., Volume 42, Number 4 (1971), 1412-1424.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177693252

**Digital Object Identifier**

doi:10.1214/aoms/1177693252

**Mathematical Reviews number (MathSciNet)**

MR290491

**Zentralblatt MATH identifier**

0223.62030

**JSTOR**

links.jstor.org

#### Citation

Stroud, T. W. F. On Obtaining Large-Sample Tests from Asymptotically Normal Estimators. Ann. Math. Statist. 42 (1971), no. 4, 1412--1424. doi:10.1214/aoms/1177693252. https://projecteuclid.org/euclid.aoms/1177693252

#### Corrections

- See Correction: T. W. F. Stroud. Correction Note: Correction to "On Obtaining Large-Sample Tests from Asymptotically Normal Estimators". Ann. Math. Statist., Volume 43, Number 2 (1972), 695--695.Project Euclid: euclid.aoms/1177692658