## The Annals of Statistics

- Ann. Statist.
- Volume 10, Number 2 (1982), 475-478.

### The Asymptotic Effect of Substituting Estimators for Parameters in Certain Types of Statistics

#### Abstract

In a variety of statistical problems, one is interested in the limiting distribution of statistics $\hat{T}_n = T_n(y_1, y_2, \cdots, y_n; \hat{\lambda}_n)$, where $\hat{\lambda}_n$ is an estimator of a parameter in the distribution of the $y_i$ and where the limiting distribution of $T_n = T_n(y_1, y_2, \cdots, y_n; \lambda)$ is relatively easy to find. For cases in which the limiting distribution of $T_n$ is normal with mean independent of $\lambda$, a useful method is given for finding the limiting distribution of $\hat{T}_n$. A simple application to testing normality in regression models is given.

#### Article information

**Source**

Ann. Statist., Volume 10, Number 2 (1982), 475-478.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176345788

**Digital Object Identifier**

doi:10.1214/aos/1176345788

**Mathematical Reviews number (MathSciNet)**

MR653522

**Zentralblatt MATH identifier**

0488.62012

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 62F05: Asymptotic properties of tests

**Keywords**

Asymptotic distributions goodness-of-fit tests nuisance parameters residuals

#### Citation

Pierce, Donald A. The Asymptotic Effect of Substituting Estimators for Parameters in Certain Types of Statistics. Ann. Statist. 10 (1982), no. 2, 475--478. doi:10.1214/aos/1176345788. https://projecteuclid.org/euclid.aos/1176345788