Likelihood ratio tests for positivity in polynomial regressions

Details Likelihood ratio tests for positivity in polynomial regressions Likelihood ratio tests for positivity in polynomial regressions

2011-08-04

arxiv.org/abs/1108.1033v2

Mathematics

Statistics Theory

23 pages, 1 figure

Scientific paper

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 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-tube method. The mixing probabilities are obtained by utilizing the parameterizations for the cone $K$ and its dual provided in the framework of the Tchebycheff systems when the degree of polynomials is up to 4. Moreover, we propose the associated simultaneous confidence bound for polynomial regression curves. Regarding computation, we demonstrate that symmetric cone programming is useful to obtain the test statistics.

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