We propose a kernel-based nonparametric framework for mean-variance optimization that enables inference on economically motivated shape constraints in finance, including positivity, monotonicity, and convexity. Many central hypotheses in financial econometrics are naturally expressed as shape relations on latent functions (e.g., term premia, CAPM relations, and the pricing kernel), yet enforcing such constraints during estimation can mask economically meaningful violations; our approach therefore separates learning from validation by first estimating an unconstrained solution and then testing shape properties. We establish statistical properties of the regularized sample estimator and derive rigorous guarantees, including asymptotic consistency, a functional central limit theorem, and a finite-sample deviation bound achieving the Monte Carlo rate up to a regularization term. Building on these results, we construct a joint Wald-type statistic to test shape constraints on finite grids. An efficient algorithm based on a pivoted Cholesky factorization yields scalability to large datasets. Numerical studies, including an options-based asset-pricing application, illustrate the usefulness of the proposed method for evaluating monotonicity and convexity restrictions.

翻译：我们提出了一种基于核的非参数均值-方差优化框架，该框架能够对金融领域中具有经济动机的形状约束（包括正性、单调性和凸性）进行统计推断。金融计量经济学中的许多核心假设自然地表现为潜在函数（如期限溢价、CAPM关系与定价核）的形状关系，但在估计过程中强制施加此类约束可能掩盖具有经济意义的违背现象；因此，我们的方法通过先估计无约束解再检验形状性质，实现了学习与验证的分离。我们建立了正则化样本估计量的统计性质，并推导出严格的理论保证，包括渐近一致性、泛函中心极限定理以及达到蒙特卡洛速率（仅相差一个正则项）的有限样本偏差界。基于这些结果，我们构建了联合Wald型统计量以检验有限网格上的形状约束。采用基于主元Cholesky分解的高效算法，实现了对大尺度数据集的扩展计算。数值研究（包括一项基于期权的资产定价应用）验证了所提方法在评估单调性与凸性限制方面的有效性。