Shape-constrained Estimation in Functional Regression with Bernstein Polynomials

TL;DR

This paper introduces a new estimation method for functional regression models that incorporates shape constraints like monotonicity and convexity using Bernstein polynomials, with theoretical guarantees and practical applications.

Contribution

The paper extends nonparametric regression techniques to functional data with shape constraints, providing asymptotic theory, confidence intervals, and bootstrap tests for the estimators.

Findings

Improved efficiency of estimators under shape constraints in simulations.

Successful application to drug effect modeling and physical activity quantile functions.

Provision of R software implementation for the proposed methods.

Abstract

Shape restrictions on functional regression coefficients such as non-negativity, monotonicity, convexity or concavity are often available in the form of a prior knowledge or required to maintain a structural consistency in functional regression models. A new estimation method is developed in shape-constrained functional regression models using Bernstein polynomials. Specifically, estimation approaches from nonparametric regression are extended to functional data, properly accounting for shape-constraints in a large class of functional regression models such as scalar-on-function regression (SOFR), function-on-scalar regression (FOSR), and function-on-function regression (FOFR). Theoretical results establish the asymptotic consistency of the constrained estimators under standard regularity conditions. A projection based approach provides point-wise asymptotic confidence intervals for the constrained estimators. A bootstrap test is developed facilitating testing of the shape constraints. Numerical analysis using simulations illustrate improvement in efficiency of the estimators from the use of the proposed method under shape constraints. Two applications include i) modeling a drug effect in a mental health study via shape-restricted FOSR and ii) modeling subject-specific quantile functions of accelerometry-estimated physical activity in the Baltimore Longitudinal Study of Aging (BLSA) as outcomes via shape-restricted quantile-function on scalar regression (QFOSR). R software implementation and illustration of the proposed estimation method and the test is provided.