Likelihood ratio tests for positivity in polynomial regressions

TL;DR

This paper develops likelihood ratio tests for assessing whether polynomial regression curves are positive over an interval, deriving their null distributions, and applying the methods to growth data analysis.

Contribution

It introduces nested likelihood ratio tests for positivity in polynomial regression, deriving their null distributions using volume-of-tubes and cone parameterizations, with applications to growth curves.

Findings

Null distributions are mixtures of chi-square distributions.

Symmetric cone programming effectively computes test statistics.

Applied to growth data, the tests assess differences in growth rates.

Abstract

A polynomial that is nonnegative over a given interval is called a positive polynomial. The set of such positive polynomials forms a closed convex cone $K$. In this paper, we consider the likelihood ratio test for the hypothesis of positivity that the estimand polynomial regression curve is a positive polynomial. By considering hierarchical hypotheses including the hypothesis of positivity, we define nested likelihood ratio tests, and derive their null distributions as mixtures of chi-square distributions by using the volume-of-tubes method. The mixing probabilities are obtained by utilizing the parameterizations for the cone $K$ and its dual provided in the framework of Tchebycheff systems for polynomials of degree at most 4. For polynomials of degree greater than 4, the upper and lower bounds for the null distributions are provided. Moreover, we propose associated simultaneous confidence bounds for polynomial regression curves. Regarding computation, we demonstrate that symmetric cone programming is useful to obtain the test statistics. As an illustrative example, we conduct data analysis on growth curves of two groups. We examine the hypothesis that the growth rate (the derivative of growth curve) of one group is always higher than the other.