5. (i) Let X = (Xl' . . . , Xn) have probability density (1/0 n)f[(XI - ~)/O" ",(xn - ~)/O] where -00 < < 00 , 0<0 are unknown, and where I is even. The problem of testing I = 10 against I = II remains invariant under the transformations xf = ax, + b (i = 1, ... , n), a ", 0, - 00 < b < 00 , and the most powerful invariant test is given by the rejection region f oo 100 2 v n- Il(VXl + u"",VXn + u) dodu -00 0 f oo 100 2 > C vn- lo(vxl + u" " ,vXn + u) dodu, -00 0 (ii) Let X = (Xl" ' " Xn) have probability density I (Xl - Ej_lWlA, .. . , x.; - Ej-lW"j,8), where k < n, the w's are given constants, the matrix (Wi) is of rank k, the ,8's are unknown, and we wish to test 1=10 against 1=11 ' The problem remains invariant under the transformations xf = Xi + E;_lWi/Yj' - 00 < Yl" '" Yk < 00, and the most powerful invariant test is given by the rejection region j . .. j/l(Xl - I:WIA"" ,xn - LwnA} d,8l, .. ·, d,8k ---------------->

c. j .. . jlo(xi - LwlA" ",xn - LwnA} d,8l ,· .. ,d,8k [A maximal invariant is given by y = ( n n n) Xl - L alrx" X2 - L a2r x" " " Xn- k - L «:«.», r-n-k+l r-n-k+l r-n-k+l for suitably chosen constants air']