Asymptotic Learning Theory for Conditional U–Statistics Based on Delta Sequences Under Missing at Random Mechanisms

This article develops a unified asymptotic theory for conditional U-statistics based on delta-sequence smoothing, thereby extending, in a substantial and conceptually coherent manner, the classical kernel-based framework for localized nonlinear conditional functionals. The proposed methodology is formulated in a highly general nonparametric setting and includes, as particular cases, the estimator of Stute, histogram-type procedures, orthogonal series methods, and a broad family of approximation schemes generated by positive delta sequences. In contrast with the existing literature, the present work explicitly incorporates response missingness under a Missing-at-Random mechanism, a setting of considerable methodological importance in modern statistical inference. Within this incomplete-data framework, we introduce a complete-case conditional U-statistic estimator and establish its asymptotic properties under general smoothness, integrability, and positivity conditions. Our first main contribution is the derivation of non-asymptotic exponential concentration inequalities for the proposed estimator, both in the bounded-kernel case and in the more delicate unbounded regime, with the latter being handled through a conditional Bernstein-type moment assumption. These inequalities provide a sharp probabilistic control of the stochastic fluctuations and constitute a fundamental technical device for the subsequent asymptotic analysis. Our second contribution is the establishment of strong consistency with explicit convergence rates, together with asymptotic normality of the localized estimator. In particular, the analysis makes precise the manner in which smoothing, dimensionality, interaction order, and missingness jointly determine the asymptotic bias and variance structure. The missing-data mechanism enters the limiting theory in a nontrivial yet fully quantifiable way through the observation probabilities, thereby yielding a refined description of the effective loss of information induced by incomplete responses. The scope of the theory is sufficiently broad to cover a wide class of nonlinear statistical functionals arising in discrimination, metric learning, multipartite ranking, conditional dependence analysis, generalized multi-sample U-statistics, and set-indexed conditional inference. To complement the theoretical developments, we conduct an extensive simulation study under several data-generating schemes, smoothing configurations, and missingness intensities. The numerical results corroborate the asymptotic theory, illustrate the finite-sample bias–variance trade-off inherent in delta-sequence localization, and demonstrate the stability and practical accuracy of the proposed estimator over a wide range of relevant regimes. Taken together, these results show that delta-sequence conditional U-statistics provide a flexible, mathematically rigorous, and broadly applicable framework for higher-order nonparametric inference with incomplete data.