Likelihood Ratio Test for the Nonnegative Polynomial Cone Hypotheses and Associated Confidence Bands
Naohiro Kato1, Satoshi Kuriki2
1Department of Statistical Science, The Graduate University for Advanced Studies, Tachikawa, Tokyo, Japan; 2The Institute of Statistical Mathematics, Tachikawa, Tokyo, Japan

A polynomial that is nonnegative over a given interval [a,b], say, is called a nonnegative polynomial. The nonnegative polynomial cone K of order n is defined as the closed convex cone consisting of all coefficients of nonnegative polynomials of n-th order (at most). Based on the n+1 dimensional Gaussian observation x with mean vector c and a given (known) covariance matrix, we consider statistical tests for the null hypothesis H0: c=0 against H1 that c belongs to the cone K, and for the null hypothesis H1 against H2 that c is arbitrary. The likelihood ratio statistics for these two problems are given by the lengths of the orthogonal projections of x onto the cones K and its dual, respectively. According to the volume-of-tube method, we see that these null distributions are mixtures of chi-square distributions, and their mixing probabilities are obtained as the coefficients of the (spherical) volume-of-tube formula about the intersection of the cone K and the unit sphere. Using the parameterizations for the interior points and the boundaries of the nonnegative polynomial cone K and its dual (Karlin and Studden, 1966), as well as the Gauss-Bonnet theorem, we provide explicit formulas for the mixing probabilities up to n=4. We apply our formulas to the test for the equivalence of two polynomial regression curves against the alternative that one of the regression curve uniformly dominates the other curve over a given covariate region. The associated simultaneous confidence bound for polynomial regression curves is also constructed.

Keywords: Cone of nonnegative polynomials; Volume-of-tube formula; Confidence bands for polynomial regression

Biography: The speaker is a professor of the Institute of Statistical Mathematics of Japan. His research interests are Geometry of Random Fields, Multiple comparisons, and Genetic Data Analysis. Since last year, he was the former Editor of the Annals of the Institute of Statistical Mathematics.