STATS 306A: HW 2 SOLUTIONS
1. Question 2.1
1.1. Part A. The natural parameter space can be dened as
A=
ey g0 (y) d (y) < .
:
Y
Our goal is to show that A is convex. To do so, let 1 and 2 be in A; we need to show that
= 1 + (1 )2 A for any 0 < < 1. Now, l
1
Stat 306a Homework 3
(due March 7, 2012; solutions posted March 20, 2012)
4.1 (a) We must verify the formulas for l , , i , and . We begin with
=
=
Continuing on with l we see
l (y) = y
So taking a derivative, and applying chain rule we get that
l
STATS 306A: HW 3 SOLUTIONS
1. Question 4.5A
Recall that given observations Y , a statistic T (Y ) is sucient for if the distribution of
Y only depends on through T :
P [Y, T ] = P [T ] P Y T .
Thus, in terms of likelihood, suciency implies that
(; Y ) = (
Stats 306A Hw1
Question 1.1:
Let A denote the natural parameter space for an one parameter exponential
family. That is,
A = cfw_ :
ey g0 (y) m(dy) <
Prove that if 1 , 2 A, then so is any 1 2 (i.e. A is a possibly innite
interval).
Answer:
For any y, ey i