On the Asymptotic Normality of Statistics with Estimated Parameters

Abstract

Often a statistic of interest would take the form of a member of a common family, except that some vital parameter is unknown and must be estimated. This paper describes methods for showing the asymptotic normality of such statistics with estimated parameters. Whether or not the limiting distribution is affected by the estimator is primarily a question of whether or not the limiting mean (derived by replacing the estimator by a mathematical variable) has a nonzero derivative with respect to that variable. Section 2 contains conditions yielding the asymptotic normality of $U$-statistics with estimated parameters. These results generalize previous theorems by Sukhatme (1958). As an example, we show the limiting normality of a resubstitution estimator of a correct classification probability when using Fisher's linear discriminant function. The results for $U$-statistics are extended to cover a broad class of families of statistics through the differential. Specifically, conditions are given which yield the asymptotic normality of adaptive $L$-statistics and an example due to de Wet and van Wyk (1979) is examined.