The Annals of Statistics

Projected Tests for Order Restricted Alternatives

Arthur Cohen, J. H. B. Kemperman, and H. B. Sackrowitz

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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.

Article information

Ann. Statist., Volume 22, Number 3 (1994), 1539-1546.

First available in Project Euclid: 11 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F03: Hypothesis testing
Secondary: 62C07: Complete class results

Unbiased tests complete class admissible tests cone alternatives simple order tree order umbrella order Schur convexity majorization


Cohen, Arthur; Kemperman, J. H. B.; Sackrowitz, H. B. Projected Tests for Order Restricted Alternatives. Ann. Statist. 22 (1994), no. 3, 1539--1546. doi:10.1214/aos/1176325641.

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