AbstractLet X1X2… XNbe independent normal p‐vectors with common mean vector $$ = ($$) and common nonsingular covariance matrix $$ = Diag ($sGi) [(1–p) I + pE] Diag ($sGi), $sGi>0, i = 1… p, 1>p>=1/p–1. Write rij= sample correlation between the i th and the j th variable i j = 1,… p. It has been proved that for testing the hypothesis H0: p = 0 against the alternative H1: p>0 where $$ and $sG1,…, $sGpare unknown, the test which rejects H0for large value of $$ rijis locally best invariant for every $aL: 0>$aL>1 and locally minimax as p $$ 0 in the sense of Giri and Kiefer, 1964, for every $aL: 0>$aL $$ $aL0>1 wher