We develop a reproducing kernel Hilbert space (RKHS) framework for nonparametric mean-variance optimization and inference on shape constraints of the optimal rule. We derive statistical properties of the sample estimator and provide rigorous theoretical guarantees, such as asymptotic consistency, a functional central limit theorem, and a finite-sample deviation bound that matches the Monte Carlo rate up to regularization. Building on these findings, we introduce a joint Wald-type statistic to test for shape constraints over finite grids. The approach comes with an efficient computational procedure based on a pivoted Cholesky factorization, facilitating scalability to large datasets. Empirical tests suggest favorably of the proposed methodology.

翻译：我们开发了一种再生核希尔伯特空间（RKHS）框架，用于非参数均值-方差优化及对最优规则形状约束的推断。我们推导了样本估计量的统计性质，并提供了严格的理论保证，例如渐近一致性、泛函中心极限定理，以及与蒙特卡洛率匹配（直至正则化项）的有限样本偏差界。基于这些发现，我们引入了一种联合Wald型统计量，用于在有限网格上检验形状约束。该方法配备了一种基于主元Cholesky分解的高效计算流程，有助于扩展到大规模数据集。实证检验表明所提方法具有良好性能。