2012 Volume 41 Issue 1 Pages 1-15
A polynomial that is nonnegative over a given domain T is called a positive polynomial. We consider the likelihood ratio test for the hypothesis of positivity that the estimand quadratic polynomial regression curve is a positive polynomial when T is the union of intervals. We define hierarchical hypotheses including the hypothesis of positivity, and derive their null distributions as mixtures of chi-square distributions. According to the volume-oftube method, the mixing probabilities are obtained through the evaluation of the volumes of boundaries of the closed convex cone K consisting of quadratic positive polynomials and its dual K*. We introduce the parameterizations of the boundaries of K and K*, and then provide expressions for the mixing probabilities. We demonstrate that the symmetric cone programming is useful for obtaining numerically the test statistics.
