Limit theorems for nonparametric conditional U-statistics smoothed by asymmetric kernels

$U$-statistics represent a fundamental class of statistics used to model quantities derived from responses of multiple subjects. These statistics extend the concept of the empirical mean of a $d$-variate random variable $X$ by considering sums over all distinct $m$-tuples of observations of $X$. Within this realm, W. Stute ^[134] introduced conditional $U$-statistics, a generalization of the Nadaraya-Watson estimators for regression functions, and demonstrated their strong point-wise consistency. This paper presented a first theoretical examination of the Dirichlet kernel estimator for conditional $U$-statistics on the $dm$-dimensional simplex. This estimator, being an extension of the univariate beta kernel estimator, effectively addressed boundary biases. Our analysis established its asymptotic normality and uniform strong consistency. Additionally, we introduced a beta kernel estimator specifically tailored for conditional $U$-statistics, demonstrating both weak and strong uniform convergence. Our investigation considered the expansion of compact sets and various sequences of smoothing parameters. For the first time, we examined conditional $U$-statistics based on mixed categorical and continuous regressors. We presented new findings on conditional $U$-statistics smoothed by multivariate Bernstein kernels, previously unexplored in the literature. These results are derived under sufficiently broad conditions on the underlying distributions. The main ingredients used in our proof were truncation methods and sharp exponential inequalities tailored to the $U$-statistics in connection with the empirical processes theory. Our theoretical advancements significantly contributed to the field of asymmetric kernel estimation, with potential applications in areas such as discrimination problems, $\ell$-sample conditional $U$-statistics, and the Kendall rank correlation coefficient. Finally, we conducted some simulations to demonstrate the small sample performances of the estimators.