Stat610: Intermediate Theory of Statistics
Homework 0 (8/29/06)
Instructor: Jianhua Huang Editor: Student A
TA: Seokho Lee
Send your solution (latex file) of the problem to the assigned editor of the week. After combining the files
and editing, the assigned editor should send the file to the TA (Seokho Lee, [email protected]u.edu).
Thanks!
Jianhua
Problem 1.
A random variable Xhas piecewise exponential distribution with parameters λ1,...,λmand partition
t1,...,tmif its density is
f(x) = kjλ−1
je−(x−tj−1)/λj, x ∈(tj−1, tj], j = 1,...,m
where t0= 0, tm=∞,k1= 1 and
kj= exp−
j−1
X
l=1
(tl−tl−1)/λl, j = 2,...,m.
a) Show that f(x) is indeed a probability density function, that is,
R∞
0f(x)dx = 1.
b) Show that P(X > tj−1) = kjand
E[(Xi−tj−1)I(tj−1≤Xi≤tj)] = λj(tj−tj+1)−(tj−tj+1)kj+1 .
c) Suppose X1,...,Xnis a random sample from above piecewise exponential distribution and λ1,...,λmare
unknown parameters that need to be estimated. Give the closed-from expression of the MLE of λ1,...,λm.
d) Show that the MLE is consistent.
Solution (by Student B).
1
