Unbiased Tests for Normal Order Restricted Hypotheses

Consider the model where Xij, i = 1, ..., k; j = 1, 2, ..., ni are observed. Here Xij are independent N(θi, σ2). Let θ′ = (θ1, ..., θk) and let A1 be a (k − m) × k matrix of rank (k − m), 0 ≤ m ≤ k − 1. The problem is to test H: A1θ = 0 vs K − H where K: A1θ ≥ 0. A wide variety of order restricted alternative problems are included in this formulation. Robertson, Wright, and Dykstra (1988) list many such problems. We offer sufficient conditions for a test to be unbiased. For problems where G−1 = (A1A′1)−1 ≥ 0 we do the following: (1) give an additional easily verifiable condition for unbiased tests in terms of variables used to describe a complete class; (2) show that the likelihood ratio test is unbiased; (3) for σ2 known, we identify a class of unbiased tests that contain all admissible unbiased tests. Considerable effort is devoted to determining which particular problems are such that G−1 ≥ 0. Four important examples are offered. These include testing homogeneity vs simple order and testing whether the θ′s lie on a line against the alternative that they are convex.