Limit Theorems in the Nonparametric Conditional Single-Index U -Processes for Locally Stationary Functional Random Fields under Stochastic Sampling Design

In his work published in (Ann. Probab. 19, No. 2 (1991), 812–825), W. Stute introduced the notion of conditional U -statistics, expanding upon the Nadaraya–Watson estimates used for regression functions. Stute illustrated the pointwise consistency and asymptotic normality of these statistics. Our research extends these concepts to a broader scope, establishing, for the first time, an asymptotic framework for single-index conditional U -statistics applicable to locally stationary random fields { X s , A n : s i n R n } observed at irregularly spaced locations in R n , a subset of R d . We introduce an estimator for the single-index conditional U -statistics operator that accommodates the nonstationary nature of the data-generating process. Our method employs a stochastic sampling approach that allows for the flexible creation of irregularly spaced sampling sites, covering both pure and mixed increasing domain frameworks. We establish the uniform convergence rate and weak convergence of the single conditional U -processes. Specifically, we examine weak convergence under bounded or unbounded function classes that satisfy specific moment conditions. These findings are established under general structural conditions on the function classes and underlying models. The theoretical advancements outlined in this paper form essential foundations for potential breakthroughs in functional data analysis, laying the groundwork for future research in this field. Moreover, in the same context, we show the uniform consistency for the nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship, which is of its own interest. Potential applications of our findings encompass, among many others, the set-indexed conditional U -statistics, the Kendall rank correlation coefficient, and the discrimination problems.