LRT conjecture for bivariate normal

摘要

In this paper, we consider testing H0:(θ1,θ2)=(0,0) vs H1:(θ1,θ2)∈Θ−{(0,0)}, where Θ is a closed square with center at the origin in the case of a bivariate normal distribution with mean vector (θ1,θ2) and identity covariance matrix. We show by numerical calculations that the conjecture of Marden [1982], which says that if more restrictions are put on the alternative space, then the power of the LRT increases, is not true in this case. Moreover, some properties of the LRT are given.