Stat610: Intermediate Theory of Statistics
Homework 8, Additional Problems (10/20/06)
Instructor: Jianhua Huang Editor: Krista Rister
TA: Seokho Lee
Problem A1.
When we derived the saddlepoint approximation for the distribution of the sample mean, we considered the
exponential family
pθ(x) = exp{θx −K(θ) + h(x)}
and used the fact that the MLE
b
θis the solution of the “saddlepoint equation” K′(
b
θ) = ¯x.
Show that (i) K(θ) is a convex function; (ii) the MLE
b
θis uniquely defined, so there is an 1-to-1 correspon-
dence between
b
θand ¯x.
1