28. Exponential families . The exponential family (12) with T(x) = x and Q(8) = 8 is STPoo' with 0 the natural parameter space and !l' = (- 00, 00). [That the determinant leUjxJI, i, j = 1, . . . , n, is positive can be proved by induction. Divide the ith column by eU'x" i = 1, . . . , n; subtract in the resulting determinant the (n - l)st column from the nth, the (n - 2)nd from the (n - 1)st, . . . , the 1st from the 2nd; and expand the determinant obtained in this way by the first row. Then 6.n is seen to have the same sign as ~n = IelJ;Xj - eTJ ,Xj-11, i , j = 2, . . . , n, where 11; = 8j - 8i - If this determinant is expanded by the first column one obtains a sum of the form a 2(e'l2X2 - e'l2 X,) + ... +an(e'ln X2 - e'lnX,) = h(X2) - h(x.) = (X2 - xl)h'(Y2)' where XI < Y2 < x2' Rewriting h'(Y2) as a determinant of which all columns but the first coincide with those of s; and proceeding in the same manner with the other columns, one reduces the determinant to le'lI))I, i, j = 2, .. . , n, which is positive by the induction hypothesis.]