Locally minimax test of independence in elliptically symmetrical distributions with additional observations
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LetX=(X ij )=(X 1, ...,X n )’,X’ i =(X i1, ...,X ip )’,i=1,2, ...,n be a matrix having a multivariate elliptical distribution depending on a convex functionq with parameters, 0,σ. Let ϱ2=ϱ 2 -2 be the squared multiple correlation coefficient between the first and the remainingp 2+p 3=p−1 components of eachX i . We have considered here the problem of testingH 0:ϱ2=0 against the alternativesH 1:ϱ 1 -2 =0, ϱ 2 -2 >0 on the basis ofX andn 1 additional observationsY 1 (n 1×1) on the first component,n 2 observationsY 2(n 2×p 2) on the followingp 2 components andn 3 additional observationsY 3(n 3×p 3) on the lastp 3 components and we have derived here the locally minimax test ofH 0 againstH 1 when ϱ 2 -2 →0 for a givenq. This test, in general, depends on the choice ofq of the familyQ of elliptically symmetrical distributions and it is not optimality robust forQ.
Key wordsElliptically symmetric distributions invariance locally best invariant test locally minimax test optimality robust.
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