Inference Under Convex Cone Alternatives for Correlated Data

This paper develops inferential theory for hypothesis testing under general convex cone alternatives for correlated data. Often, interest lies in detecting order among treatment effects, while simultaneously modeling relationships with regression parameters. Incorporating shape or order restrictions in the modeling framework improves the efficiency of statistical methods. 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 problem is addressed by 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. It is shown that the asymptotic null distribution of this 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 via the Hotelling-Weyl-Naiman volume-of-tube formula around a convex manifold in the unit sphere. Furthermore, an asymptotic lower bound for the power of the generalized quasi-score test under a sequence of local alternatives in the convex cone is established. Applications to testing for order restricted alternatives in correlated data and testing for a monotone regression function are discussed.