A New Graphical Device and Related Tests for the Shape of Non-parametric Regression Function

TL;DR

This paper introduces a new graphical tool for assessing the shape of non-parametric regression functions by examining their derivatives, which helps identify properties like monotonicity and convexity, supported by theoretical and empirical validation.

Contribution

The paper presents a novel graphical device for shape analysis of regression functions and develops related tests with asymptotic distributions, enhancing shape inference methods.

Findings

The graphical device effectively identifies the shape of regression functions.

The tests based on the device perform well in simulations and real data.

The method includes checks for monotonicity and convexity as special cases.

Abstract

We consider a non-parametric regression model $y = m(x) + \epsilon$ and propose a novel graphical device to check whether the $r$-th ($r \geqslant 1$) derivative of the regression function $m(x)$ is positive or otherwise. Since the shape of the regression function can be completely characterized by its derivatives, the graphical device can correctly identify the shape of the regression function. The proposed device includes the check for monotonicity and convexity of the function as special cases. We also present an example to elucidate the practical utility of the graphical device. In addition, we employ the graphical device to formulate a class of test statistics and derive its asymptotic distribution. The tests are exhibited in various simulated and real data examples.