Uniformly More Powerful Tests in a One-Sided Multivariate Problem
针对正态均值向量在零假设下为零、备择假设下在锥内的检验问题，提出比似然比检验和Hotelling T2更优且更简单的检验方法，并证明其无偏性，给出临界值。
Abstract We consider a testing problem in which a normal mean vector is at the origin under the null hypothesis and in a cone under the alternative and the covariance matrix is unknown. Perlman studied the likelihood ratio (LR) test and found that it is biased and hence does not compare favorably with Hotelling's T2. We point out that certain tests of Perlman are uniformly more powerful as well as simpler than the LR test. These are the corresponding LR tests when the alternative is replaced with a half space that contains the cone. We prove that these tests are unbiased by developing some basic results on stochastic ordering. Critical values for these tests are given. Simulation results suggest that the new tests are also uniformly more powerful than Hotelling's T2. For the special case when the cone is the positive orthant, we discuss an issue about invariant test. Finally, we discuss the assumption of unknown covariance matrix, and conclude that the new tests are more useful for small samples than for large samples.