Testing Under a General Convex Cone Alternative for Correlated Data

• Activity Number: 221
• Type: Topic Contributed
• Date/Time: Tuesday, August 9, 2005 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Ramani S. Pilla*+
• Companies: Case Western Reserve University
• Keywords: Asymptotic distribution ; Correlated data ; General cone alternative ; Mixed volumes ; Regression function ; Volume-of-tube formula

Abstract

We consider a testing problem for order-restricted parameters in correlated data. For example, in biomedical studies, the problem is to detect ordering of treatment effects while simultaneously modeling relationships with other covariates. Taking order restrictions into account improves the efficiency of a statistical analysis. While there exists extensive theory for testing for ordered alternatives with independent observations, extensions to correlated data is not straightforward. We propose a test statistic based on the quadratic inference function and derive its asymptotic distribution. First, it is established that the proposed test statistic is asymptotically equivalent to the squared length of the projection of the standard normal vector onto a convex cone. Next, it is shown that the asymptotic 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." Furthermore, an asymptotic lower bound for the power of the test under a sequence of local alternatives is derived. The proposed theory can be generalized to other testing problems with convex cones, such as monotone regression.