The effects on convergence of substituting parameter estimates into U-statistics and other families of statistics

Summary Substituting an estimator in a statistic will often affect its limiting distribution. Sukhatme (1958), Randles (1982), and Pierce (1982) all consider the changes, if any, in the statistic's limiting normal distribution. This paper gives conditions for a law of the iterated logarithm for U-statistics which have a kernel with an estimator substituted into it. It also gives conditions for both strong and weak convergence. Applications of the theory are illustrated by constructing a sequential test for scale differences with power one. The theory also produces convergence results for adaptive M-estimators and for cross-validation assessment statistics. In addition, it is shown how to extend LIL results to a broad class of statistics with estimators substituted into them by use of the differential. In particular, a law of the iterated logarithm is described for adaptive L-statistics and is illustrated by an example of de Wet and van Wyk (1979).