Bound on FWER for correlated normal distribution
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In this paper,our main focus is to obtain an asymptotic bound on the family wise error rate (FWER) for Bonferroni-type procedure in the simultaneous hypotheses testing problem when the observations corresponding to individual hypothesis are correlated. In particular, we have considered the sequence of null hypotheses H_0i : X_i follows N(0,1) , (i=1,2,....,n) and equicorrelated structure of the sequence (X_1,....,X_n). Distribution free bound on FWER under equicorrelated setup can be found in Tong(2014). But the upper bound provided in Tong(2014) is not a bounded quantity as the no. of hypotheses(n) gets larger and larger and as a result,FWER is highly overestimated for the choice of a particular distribution (e.g.- normal). In the equicorrelated normal setup, we have shown that FWER asymptotically is a convex function (as a function of correlation (rho)) and hence an upper bound on the FWER of Bonferroni-(alpha) procedure is alpha(1-ρ).This implies,Bonferroni's method actually controls the FWER at a much smaller level than the desired level of significance under the positively correlated case and necessitates a correlation correction.
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