Inference using shape-restricted regression splines

TL;DR

This paper introduces algorithms for shape-restricted regression splines, including monotone and convex constraints, which improve robustness, accuracy, and computational efficiency in nonparametric regression analysis.

Contribution

The paper develops a cubic monotone spline algorithm, extends it to convex and other shape constraints, and demonstrates improved inference and testing power over traditional methods.

Findings

Restricted splines have lower squared error loss than unrestricted ones.

Shape-restricted splines are less sensitive to knot placement, enhancing practical inference.

Tests using shape-restricted splines outperform standard methods in power.

Abstract

Regression splines are smooth, flexible, and parsimonious nonparametric function estimators. They are known to be sensitive to knot number and placement, but if assumptions such as monotonicity or convexity may be imposed on the regression function, the shape-restricted regression splines are robust to knot choices. Monotone regression splines were introduced by Ramsay [Statist. Sci. 3 (1998) 425--461], but were limited to quadratic and lower order. In this paper an algorithm for the cubic monotone case is proposed, and the method is extended to convex constraints and variants such as increasing-concave. The restricted versions have smaller squared error loss than the unrestricted splines, although they have the same convergence rates. The relatively small degrees of freedom of the model and the insensitivity of the fits to the knot choices allow for practical inference methods; the computational efficiency allows for back-fitting of additive models. Tests of constant versus increasing and linear versus convex regression function, when implemented with shape-restricted regression splines, have higher power than the standard version using ordinary shape-restricted regression.