On the unbiasedness of the likelihood ratio test for the multivariate one-sided problem

Abstract

Let $X_1,\dots ,X_n$ be a random sample from a multivariate normal distribution with nonnegative mean $\mu$ and unknown covariance matrix $\Sigma$. The likelihood ratio test of $H_0:\mu =0$ conditional on $V=\sum X_i{X}_i^{\prime }$ is proven to be unbiased. Some related topics are also discussed.