Degenerate U-Statistics Based on Non-Independent Observations

Abstract

Many important statistics are actually degenerate U-statistics; examples include the χ2 goodness-of-fit statistic, the generalized Cramer-von Mises goodness-of-fit statistics, Hoeffding's nonparametric measure of bivariate dependence, the sample variance, ahd the cross-product statistic. Although these statistics were originally proposed for iid data, they remain intuitively reasonable and useful even when the underlying data contain serial dependence. The presence of such dependence alters the limiting distributions for these statistics, and this in turn should be reflected in any concomitant confidence intervals or critical regions. This paper presents straightforward asymptotic distribution theory for degenerate U-statistics computed from dependent observations : the results are applied to the examples mentioned above. The dependence in the data is characterized using standard model-free mixing conditions. There has not been much other work on degenerate U-statistics in the non-independent case, and our geueral formulation is the first to permit a unified treatment of all the examples discussed above. In fact, the asymptotic distributions of the Cramér-von Mises and Hoeffding statistics have not previously been derived in the case of non-independent data.