## The Annals of Statistics

### On the Effect of Substituting Parameter Estimators in Limiting $\chi^2 U$ and $V$ Statistics

#### Abstract

Consider statistics $T_n(\lambda)$ that take the form of limiting chi-square (degenerate) $U$ or $V$ statistics. Here the phrase "limiting chi-square" means they have the same asymptotic distribution as a weighted sum of (possibly infinitely many) independent $\chi^2_1$ random variables. This paper examines the limiting distribution of $T_n(\hat{\lambda})$ and compares it to that of $T_n(\lambda)$, where $\hat{\lambda}$ denotes a consistent estimator of $\lambda$ based on the same data. Whether or not $T_n(\hat{\lambda})$ and $T_n(\lambda)$ have the same limiting distribution is primarily a question of whether or not a certain mean function has a zero derivative. Some statistics that are appropriate for testing hypotheses are used to illustrate the theory.

#### Article information

Source
Ann. Statist., Volume 15, Number 1 (1987), 398-412.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176350274

Digital Object Identifier
doi:10.1214/aos/1176350274

Mathematical Reviews number (MathSciNet)
MR885745

Zentralblatt MATH identifier
0617.62017

JSTOR
de Wet, Tertius; Randles, Ronald H. On the Effect of Substituting Parameter Estimators in Limiting $\chi^2 U$ and $V$ Statistics. Ann. Statist. 15 (1987), no. 1, 398--412. doi:10.1214/aos/1176350274. https://projecteuclid.org/euclid.aos/1176350274