Kernel-Based Nonparametric Tests For Shape Constraints
This work addresses statistical inference for shape constraints in optimization, which is incremental but provides computational improvements for large datasets.
The authors tackled the problem of testing shape constraints in nonparametric mean-variance optimization by developing an RKHS framework, resulting in theoretical guarantees like asymptotic consistency and a finite-sample deviation bound, along with an efficient computational procedure that scales to large datasets.
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.