Abstract
U-statistics constitute a foundational class of statisticalmethodologies employed to model quantities derived from responsesacross multiple subjects. They extend the classical concept of theempirical mean of a random variable \(X\) by aggregating over allpossible distinct \(m\) -tuples of \(X\) observations. In 1991, W.Stute introduced a novel subclass of conditional U-statistics,which can be interpreted as extensions of Nadaraya–Watsonestimators tailored for regression functions, as detailed in[104]. Stute established the robust pointwise consistency of thesestatistics with respect to the function:
\(r^{(m)}(\varphi,\mathbf{t})=\mathbb{E}\left[\varphi(Y_{1},\ldots,Y_{m})|(\mathbf{X}_{1},\ldots,\mathbf{X}_{m})=\mathbf{t}\right],\)
where \(\mathbf{t}\in\mathbb{R}^{dm}\) and \(\varphi(\cdot):\mathbb{R}^{mq}\rightarrow\mathbb{R}\) is a measurable function.This paper presents a comprehensive framework for nonparametricstatistical curve estimation. It introduces an innovative class ofwavelet-based estimators, marking their inaugural appearance inthe existing literature, and establishes uniform rates of almostsure convergence over compact subsets of \(\mathbb{R}^{dm}\) forstrongly mixing processes. Moreover, the proposed estimators aredemonstrated to exhibit asymptotic normality. These theoreticaladvancements provide a robust foundation for further developmentsin regression estimation and suggest potential applications inareas such as classification problems and Kendall’s tau.Additionally, within the same framework, we establish the uniformconsistency of nonparametric inverse probability of censoringweighted (I.P.C.W.) estimators of the regression function underconditions of random censorship or left-truncated andright-censored data, which holds significant independent interest.
