In this research, inferential theory for hypothesis testing under general
convex cone alternatives for correlated data is developed. While there exists
extensive theory for hypothesis testing under smooth cone alternatives with
independent observations, extension to correlated data under general convex
cone alternatives remains an open problem. This long-pending problem is
addressed by (1) establishing that a "generalized quasi-score" statistic is
asymptotically equivalent to the squared length of the projection of the
standard Gaussian vector onto the convex cone and (2) showing that the
asymptotic null distribution of the test statistic is a weighted chi-squared
distribution, where the weights are "mixed volumes" of the convex cone and its
polar cone. Explicit expressions for these weights are derived using the
volume-of-tube formula around a convex manifold in the unit sphere.
Furthermore, an asymptotic lower bound is constructed for the power of the
generalized quasi-score test under a sequence of local alternatives in the
convex cone. Applications to testing under order restricted alternatives for
correlated data are illustrated.Comment: 31 page