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([theta]i, [sigma]2). Let [theta]' = ([theta]1, ..., [theta]k) and let A1 be a (k - m) - k matrix of rank (k - m), 0 <= m <= k - 1. The problem is to test H: A1[theta] = 0 vs K - H where K: A1[theta] >= 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 [sigma]2 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 [theta]'s lie on a line against the alternative that they are convex.