Unbiased Tests for Normal Order Restricted Hypotheses

✍ Scribed by A. Cohen; J.H.B. Kemperman; H.B. Sackrowitz

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 516 KB
• Volume: 46
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1993.1053

✦ Synopsis

Consider the model where (X_{i j}, i=1, \ldots, k ; j=1,2, \ldots, n_{i}) are observed. Here (X_{i j}) are independent (N\left(\theta_{i}, \sigma^{2}\right)). Let (\boldsymbol{\theta}^{\prime}=\left(\theta_{1}, \ldots, \theta_{k}\right)) and let (A_{1}) be a ((k-m) \times k) matrix of rank ((k-m), 0 \leqslant m \leqslant k-1). The problem is to test (H: A_{1} \boldsymbol{\theta}=\mathbf{0}) vs (K-H) where (K: A, \boldsymbol{\theta} \geqslant \mathbf{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}=\left(A_{1} A_{1}^{\prime}\right)^{-1} \geqslant 0) we do the following: