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

Exponential families. The exponential family (3.19) with T(x) =

x and Q(θ) = θ is STP∞, with Ω the natural parameter space and X = (−∞, ∞).

[That the determinant |eθixj |, i, j = 1,...,n, is positive can be proved by induction. Divide the ith column by eθ1xi , 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 = |e

ηixj − e

ηixj−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(e

η2x2 − e

η2x1 ) + ··· + an(e

ηnx2 − e

ηnx1 ) = 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 |eηiyj |, i, j = 2,...,n, which is positive by the induction hypothesis.]