Testing Functional Inequalities

TL;DR

This paper introduces new nonparametric regression inequality tests based on kernel estimators, which are asymptotically distribution free, consistent, and have local power, supported by Monte Carlo simulations.

Contribution

It develops a novel testing methodology for inequality constraints in nonparametric regression using $L_p$-type functionals and establishes their asymptotic properties.

Findings

Tests are asymptotically distribution free and normally approximated.

Tests are consistent against fixed alternatives.

Monte Carlo simulations support the theoretical results.

Abstract

This paper develops tests for inequality constraints of nonparametric regression functions. The test statistics involve a one-sided version of $L_p$-type functionals of kernel estimators $(1 \leq p < \infty)$. Drawing on the approach of Poissonization, this paper establishes that the tests are asymptotically distribution free, admitting asymptotic normal approximation. In particular, the tests using the standard normal critical values have asymptotically correct size and are consistent against general fixed alternatives. Furthermore, we establish conditions under which the tests have nontrivial local power against Pitman local alternatives. Some results from Monte Carlo simulations are presented.