Download Problem Set 04

Middle East Technical University
Faculty of Economics and Administrative Sciences
Department of Economics
Econ 206 – Fall, 2014
November 21
PROBLEM SET 4
1. Let Y1, Y2, . . . , Yn denote a random sample from a population with mean μ and
variance σ2. Consider the following three estimators for μ:
a. Show that each of the three estimators is unbiased.
b. Find the efficiency of ̂
μ3 relative to ̂
μ2 and ̂
μ1 respectively.
Solution:
2. Suppose that Y1, Y2, . . . , Yn is a random sample from a normal distribution with
mean μ and variance σ2. Two unbiased estimators of σ2 are
̂1 2 relative to σ
̂2 2
Find the efficiency of σ
Solution:
1
3. Let Y1 , Y2 , …….,Yn denote a random sample from the probability density function
θ-1
f(y)= { θy ,0<y<1
0 ,elsewhere
θ
where θ>0. Show that ̅
Y is a consistent estimator for
θ+1
Solution:
2
4. Suppose that populations are normally distributed with σ21 = σ22 = σ2 . Show that
2
̅ ) + ∑n (Yi -Y
̅)
∑ni=1 (Xi -X
i=1
2n-2
2
is a consistent estimator of σ2 .
Solution:
5. If X1 , X2 , …….,Xn constitute a random sample of size n from an exponential
population, show that ̅
X is a sufficient estimator of the parameter θ.
Solution:
3
6. Let Y1, Y2, . . . , Yn denote independent and identically distributed random
variables from
a. Power family distribution with parameters α and θ (α, θ > 0)
If θ is known, show that ∏ni=1 Yi is sufficient for α.
b. Pareto distribution with parameters α and β (α, β > 0)
If β is known, show that ∏ni=1 Yi is sufficient for α.
Solution:
4