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{1176350274,
author = {de Wet, Tertius and Randles, Ronald H.},
title = {On the Effect of Substituting Parameter Estimators in Limiting $\chi^2 U$ and $V$ Statistics},
journal = {Ann. Statist.},
volume = {15},
number = {1},
year = {1987},
pages = { 398-412},
language = {en},
url = {http://dml.mathdoc.fr/item/1176350274}
}
de Wet, Tertius; Randles, Ronald H. On the Effect of Substituting Parameter Estimators in Limiting $\chi^2 U$ and $V$ Statistics. Ann. Statist., Tome 15 (1987) no. 1, pp. 398-412. http://gdmltest.u-ga.fr/item/1176350274/