Projected Tests for Order Restricted Alternatives

Abstract

Consider the model where $X_{ij}, i = 1, 2, \ldots, k, j = 1, 2, \ldots, n,$ are independent random variables distributed according to a one-parameter exponential family, with natural parameter $\theta_j$. We test $H_0: \theta_1 = \ldots = \theta_k$ versus $H_1: \theta \in \mathscr{C} - \{\theta: \theta \in H_0\},$ where $\theta = (\theta_1, \ldots, \theta_k)'$ and $\mathscr{C}$ is a cone determined by $A\theta \geq 0,$ where the rows of $A$ are contrasts with two nonzero elements. We offer a method of generating "good" tests for $H_0$ versus $H_1$. The method is to take a "good" test for $H_0$ versus $H_2:$ not $H_0,$ and apply the test to projected sample points, where the projection is onto $\mathscr{C}$. "Good" tests for $H_0$ versus $H_2$ are tests that are Schur convex. "Good" tests for $H_0$ versus $H_1$ are tests which are monotone with respect to a cone order. We demonstrate that if the test function for $H_0$ versus $H_2$ is a constant size Schur convex test, then the resulting projected test is monotone.