Exponential families. The exponential family (3.19) with T (x) = x and Q() = is STP,

Exponential families. The exponential family (3.19) with T (x) = x and Q(θ) = θ is STP∞, with the natural parameter space and X = (−∞,∞).

[That the determinant |eθi x j |,i, j = 1,..., n, is positive can be proved by induction. Divide the ith column by eθ1 xi,i = 1,..., n; subtract in the resulting determinant the (n − 1)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 n is seen to have the same sign as

n = |eηi x j − eηi x j−1

|, i, j = 2,..., n, where ηi = θi − θ1. If this determinant is expanded by the first column, one obtains a sum of the form a2(eη2 x2 − eη2 x1 ) +···+ an(eηn x2 − eηn x1 ) = h(x2) − h(x1)

= (x2 − x1)h

(y2), where x1 < y2 < x2. Rewriting h

(y2) as a determinant of which all columns but the first coincide with those of n and proceeding in the same manner with the columns, one reduces the determinant to |eηi yj |, i, j = 2,..., n, which is positive by the induction hypothesis.]