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Kernel-Based Nonparametric Tests For Shape Constraints

Rohan Sen

TL;DR

This work develops a kernel-based, nonparametric RKHS framework to perform mean-variance optimization and inference for shape constraints on an unknown function and its derivatives. By embedding a general derivative-inclusive target functional into an RKHS and applying a representer theorem, the authors derive population and empirical optimizers, establishing consistency, a functional CLT, and finite-sample deviation bounds. They then construct a joint Wald-type test for shape constraints over a finite grid, with a plug-in covariance estimator leading to a chi-bar-squared limit, and provide an efficient computation pipeline using pivoted Cholesky decomposition to scale to large datasets. Numerical experiments demonstrate accurate size control and increasing power with sample size across diverse covariance structures, underscoring the method’s practical potential for economics, finance, and risk analysis where positivity, monotonicity, and convexity constraints are crucial. The framework offers a principled, scalable approach to nonparametric shape testing in settings where derivatives of the underlying function drive decision rules and risk-sensitive objectives, bridging RKHS learning with shape-constrained inference and hypothesis testing.

Abstract

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.

Kernel-Based Nonparametric Tests For Shape Constraints

TL;DR

This work develops a kernel-based, nonparametric RKHS framework to perform mean-variance optimization and inference for shape constraints on an unknown function and its derivatives. By embedding a general derivative-inclusive target functional into an RKHS and applying a representer theorem, the authors derive population and empirical optimizers, establishing consistency, a functional CLT, and finite-sample deviation bounds. They then construct a joint Wald-type test for shape constraints over a finite grid, with a plug-in covariance estimator leading to a chi-bar-squared limit, and provide an efficient computation pipeline using pivoted Cholesky decomposition to scale to large datasets. Numerical experiments demonstrate accurate size control and increasing power with sample size across diverse covariance structures, underscoring the method’s practical potential for economics, finance, and risk analysis where positivity, monotonicity, and convexity constraints are crucial. The framework offers a principled, scalable approach to nonparametric shape testing in settings where derivatives of the underlying function drive decision rules and risk-sensitive objectives, bridging RKHS learning with shape-constrained inference and hypothesis testing.

Abstract

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.
Paper Structure (29 sections, 28 theorems, 195 equations, 1 figure)

This paper contains 29 sections, 28 theorems, 195 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Outline
  5. Preliminaries
  6. Notation and setting
  7. Reproducing kernel Hilbert spaces
  8. Mean-variance optimization in RKHS
  9. Population and empirical problem
  10. Formulation in RKHS
  11. Representer theorem
  12. Statistical properties of sample estimator
  13. Assumptions and setting
  14. Asymptotic properties
  15. Finite-sample properties
  16. ...and 14 more sections

Key Result

Theorem 2.1

Let $s\in \mathbb{N}$ and $\mathcal{K}\colon \mathcal{X}\times \mathcal{X}\to \mathbb{R}$ is a reproducing kernel such that $\mathcal{K} \in \mathscr{C}^{2s}( \mathcal{X} \times \mathcal{X})$. Then, it holds,

Figures (1)

  • Figure 1: Performance of test statistic: size and power vs. $N$. Columns: covariance designs ($\boldsymbol \Omega=\boldsymbol I_n$, SVD-decay, SVD-spike); Rows: grid sizes $n\in\{10,50,100\}$; Curves show empirical size ($H_0$) and power under mild/moderate/strong violations with equal-$\ell_2$ scaling; dashed line marks the nominal size $\alpha=0.05$.

Theorems & Definitions (60)

  • Theorem 2.1: zhou2008derivative
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: Representer theorem
  • proof : Proof of Theorem \ref{['thm:representer_thm']}
  • Remark 2.5
  • Proposition 3.1: Moment and operator properties
  • Proposition 3.2: Optimal solution
  • Proposition 3.3: Asymptotic gaussianity
  • proof
  • ...and 50 more