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A monotonicity property of the power function of multivariate tests. (English) Zbl 0970.62038
Summary: Let S= k=1 n X k X k ' , where the X k are independent observations from a 2-dimensional normal N(μ k ,Σ) distribution, and let Λ= k=1 n μ k μ k ' Σ -1 be a diagonal matrix of the form λI, where λ0 and I is the identity matrix. It is shown that the density φ of the vector ˜=( 1 , 2 ) of characteristic roots of S can be written as G(λ, 1 , 2 )φ 0 ( ˜), where G satisfies the FKG condition on + 3 . This implies that the power function of tests with monotone acceptance region in 1 and 2 , i.e. a region of the form {g( 1 , 2 )c}, where g is nondecreasing in each argument, is nondecreasing in λ. It is also shown that the density φ of ( 1 , 2 ) does not allow a decomposition φ( 1 , 2 )=G(λ, 1 , 2 )φ 0 ( ˜), with G satisfying the FKG condition, if Λ=diag(λ,0) and λ>0, implying that this approach to proving monotonicity of the power function fails in general.
MSC:
62H15Multivariate hypothesis testing
62H10Multivariate distributions of statistics