4. Let the distribution of X depend on the parameters «(J, ii) = «(JI' . . . , (Jr' iiI' ... , iis)' A test of H : (J = (J0 is locally strictly unbiased if for each ii,

(a) fJ",«(Jo, ii) =

a,

(b) there exists a (J-neighborhood of (J0 in which fJ",«(J, ii) > a for (J ", (J0. (i) Suppose that the first and second derivatives a I a 2 fJ~( {;) = a(J fJ",( (J, ii) and fJ:;( ii) = a(J a(J fJ",( (J, ii)I I 00 I J 00 exist for all critical functions

among all similar (and hence all locally unbiased) tests where S = {('IIll" " 'IIr) :E~-I'II7 =la 2 } . Letting p tend to zero and utilizing the conditions f3~{ 1'i) = 0, [ 'IIi'llj dA =0 for i 4: i. ['117 dA = k{pa), one finds that CPo maximizes E~_If3~i('II, a2 ) among all locally unbiased tests. Since for any positive definite matrix, J(f3~)1 nf3~, it follows that for any locally strictly unbiased test cp, 1(f3~j)1 s ne s [E~~ir s [E~~~ r= [f3;~r =1(f3~)I·