The Annals of Statistics

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

Tertius de Wet and Ronald H. Randles

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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

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

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62E20: Asymptotic distribution theory
Secondary: 62G10: Hypothesis testing

Limiting $\chi^2$ distributions $U$ statistics $V$ statistics asymptotic distribution


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.

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