[Paper Review] Testing nonparametric shape restrictions

Tatiana Komarova, Javier Hidalgo|arXiv (Cornell University)|Sep 4, 2019

Statistical Methods and Inference64 references4 citations

TL;DR

This paper proposes a nonparametric test for a broad class of shape restrictions—such as monotonicity, convexity, U-shape, and log-convexity—using partial sums empirical processes. After a transformation, the test’s asymptotic distribution is a functional of standard Brownian motion, enabling critical values; a valid bootstrap algorithm is also provided to improve finite-sample performance.

ABSTRACT

We describe and examine a test for a general class of shape constraints, such as constraints on the signs of derivatives, U-(S-)shape, symmetry, quasi-convexity, log-convexity, $r$-convexity, among others, in a nonparametric framework using partial sums empirical processes. We show that, after a suitable transformation, its asymptotic distribution is a functional of the standard Brownian motion, so that critical values are available. However, due to the possible poor approximation of the asymptotic critical values to the finite sample ones, we also describe a valid bootstrap algorithm.

Motivation & Objective

• To develop a general test for nonparametric shape restrictions in regression models without assuming a parametric form for the alternative.
• To address the challenge of testing qualitative features like monotonicity, convexity, or S-shaped relationships in nonparametric regression.
• To provide a test whose asymptotic distribution is distribution-free under the null via Khmaladze’s transformation.
• To improve finite-sample performance by introducing a valid bootstrap procedure due to potential inaccuracies in asymptotic critical values.
• To ensure the test is implementable without bandwidth selection, leveraging B-spline approximations and empirical process theory.

Proposed method

• Uses partial sums empirical processes to construct a test statistic sensitive to deviations from shape constraints.
• Employs B-splines of degree $ q $ with $ L $ knots to approximate the regression function, mapping shape constraints into constraints on spline coefficients $ \beta_\ell $.
• Applies Khmaladze’s martingale transformation to convert the empirical process into a functional of standard Brownian motion under the null.
• Imposes shape constraints via a non-stochastic set $ S_{q,L} \subseteq \mathbb{R}^L $, ensuring consistency as $ L \to \infty $.
• Derives asymptotic critical values from the distribution of the transformed process, enabling size control.
• Proposes a bootstrap algorithm to correct for finite-sample bias in critical values, ensuring valid inference.

Experimental results

Research questions

• RQ1 Can a general test be developed for nonparametric shape restrictions such as monotonicity, convexity, and U-shape without assuming a parametric model?
• RQ2 How can the asymptotic distribution of the test statistic be derived and made distribution-free under the null hypothesis?
• RQ3 What is the impact of finite-sample approximation errors in asymptotic critical values, and how can they be corrected?
• RQ4 Can a bootstrap procedure be constructed that is valid and computationally feasible for this class of nonparametric shape tests?
• RQ5 How do B-spline approximations with increasing knot density ensure consistency in capturing shape constraints?

Key findings

• The test statistic, after transformation, converges in distribution to a functional of standard Brownian motion, enabling analytical critical values.
• The Hausdorff distance between the true shape class $ \mathcal{M}_0 $ and the B-spline approximation class $ \mathcal{M}_{S_{q,L}} $ converges to zero as $ L \to \infty $, ensuring consistency.
• The asymptotic distribution is free of nuisance parameters, making the test distribution-free under the null under regularity conditions.
• Finite-sample critical values are poorly approximated by the asymptotic distribution, necessitating a bootstrap correction.
• The proposed bootstrap algorithm is valid and improves size accuracy in small to moderate samples.
• The method avoids bandwidth selection, offering a robust alternative to existing nonparametric tests based on kernel smoothing.