This paper develops a boundary-sensitive asymptotic theory for nonparametric conditional U-statistics smoothed by support-adapted asymmetric kernels when the response variable is subject to Missing-at-Random observation. The problem lies at the intersection of three well-established but traditionally separate lines of research: conditional U-statistics, asymmetric smoothing on constrained supports, and incomplete-data inference under MAR sampling. The contribution of the paper is not a novelty claim concerning any of these components in isolation. Rather, it consists in deriving a kernel-specific and MAR-aware limit theory for their simultaneous occurrence, where the estimators are nonlinear complete-case ratios of localized U-statistics and the localization devices are point-dependent approximate identities adapted to the geometry of the covariate support. The analysis covers three principal classes of support-respecting smoothers: Dirichlet kernels on the simplex, Bernstein polynomial smoothers, and multivariate beta kernels on hypercubes, with an additional extension to mixed continuous–categorical regressors. These smoothing schemes are not translation-invariant, and their local moments, effective support, normalizing constants and L2-masses vary with the evaluation point, especially near the boundary. Consequently, their incorporation into conditional U-statistics requires more than a direct transfer of ordinary asymmetric-kernel regression theory. The numerator and denominator of the estimators are localized U-statistics whose stochastic expansions are governed by Hoeffding projections, including canonical components that must be controlled uniformly over the conditioning domain. Under regularity, smoothness and positivity assumptions adapted to the MAR setting, we establish uniform consistency, weak and strong uniform convergence rates, stochastic expansions and asymptotic normality. The results are obtained both on fixed compact subsets and on interior regions approaching the boundary, thereby identifying how support geometry enters the bias and stochastic normalizations. A central feature of the theory is the separation between the deterministic effect of complete-case sampling and its stochastic effect. For the complete-case estimator, the natural deterministic equivalent is obtained by replacing the design density f with the effective complete-case density pf, where p is the propensity score. Thus, the MAR mechanism may enter higher-order deterministic bias constants through the local design tilt, whereas the leading stochastic dispersion reflects the loss of effective information through propensity score factors. The precise variance constants and normalizing rates remain kernel-specific, depending on the local L2-structure of the Dirichlet, Bernstein or beta smoothing device. The paper should therefore be viewed as a MAR extension and refinement of the complete-data asymmetric-kernel conditional U-statistic theory. It provides a common probabilistic architecture for several boundary-adapted smoothing schemes while retaining the kernel-dependent bias operators, variance constants, boundary regimes and Hoeffding-projection structures required for sharp asymptotic interpretation. Numerical experiments illustrate the finite-sample behavior predicted by the theory and highlight the interaction between support-adapted smoothing, boundary effects and incomplete response observation.
Salim Bouzebda (Fri,) studied this question.