Asymptotics for conditional U-statistics with applications

In 1948 Wassily Hoeffding introduced the class of U-statistics as unbiased estimators of statistical functionals. As U-statistics generalize statistics based on sums of independent random variables, so does Winfried Stute generalize a basic tool of nonparametric regression, the Nadaraya-Watson estimator. Rather than taking a weighted average of the values of a response variable within a localized range of a predictor variable, Stute suggests taking a weighted average of values of some kernel function of k variables over all possible permutations of its arguments, essentially a ratio of U-statistics. Stute calls the resulting estimator a conditional U-statistic, as it estimates the conditional expectation of the U-statistic kernel function given a particular value of the random predictor(s). Stute proves results about conditional U-statistics which mirror Hoeffding's, such as normality and consistency.;While the design of averaging over all possible permutations of data values is similar to classical U-statistics, however, the ratio structure of conditional U-statistics leads to an important difference: conditional U-statistics are not unbiased. We use functional expansions to estimate the asymptotic bias and variance of conditional U-statistics, specifically to determine the order of bias and variance, and thus to find the order of an optimal bandwidth for this estimator in terms of minimizing asymptotic mean squared error.;Further, we suggest uses for conditional U-statistics in testing distributional assumptions of classical multivariate analysis. Assuming multinormality for a set of random variables implies linearity of regression, homoscedasticity, and constant partial correlations among those variables. Conditional U-statistics provide a nonparametric means of estimating a smooth curve not just for regression functions, but also for higher order functions like conditional variances and partial correlations. By estimating functional values along the range of the conditioning variables we can formally test the assumption that such variances and correlations are constant.