Unbiased tests for normal order restricted hypotheses

Consider the model where X_ij, i = 1, …, k; j = 1, 2, …, n_i are observed. Here X_ij are independent N(θ_i, σ²). Let θ′ = (θ₁, …, θ_k) 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.