Limit Theorems for Wavelet Conditional $$\boldsymbol{U}$$-Statistics for Time Series Models

What the paper says

U-statistics constitute a foundational class of statistical methodologies employed to model quantities derived from responses across multiple subjects. They extend the classical concept of the empirical mean of a random variable $$X$$ by aggregating over all possible 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–Watson estimators tailored for regression functions, as detailed in [104]. Stute established the robust pointwise consistency of these statistics 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 nonparametric statistical curve estimation. It introduces an innovative class of wavelet-based estimators, marking their inaugural appearance in the existing literature, and establishes uniform rates of almost sure convergence over compact subsets of $$\mathbb{R}^{dm}$$ for strongly mixing processes. Moreover, the proposed estimators are demonstrated to exhibit asymptotic normality. These theoretical advancements provide a robust foundation for further developments in regression estimation and suggest potential applications in areas such as classification problems and Kendall’s tau. Additionally, within the same framework, we establish the uniform consistency of nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under conditions of random censorship or left-truncated and right-censored data, which holds significant independent interest.