1. Let X 1 , X 2 , ... , X n be a random sample of size n from the distribution with probability density function
f ( x ; λ ) =λ 2 2 λ e- x π , x > 0, λ > 0.
a) Obtain the maximum likelihood estimator of λ, λˆ .
d) Suppose n = 4, and x 1 = 0.2, x 2 = 0.6, x 3 = 1.1, x 4 = 1.7.
Find the maximum likelihood estimate of λ.
c) Obtain the method of moments estimator of λ, λ
d) Suppose n = 4, and x 1 = 0.2, x 2 = 0.6, x 3 = 1.1, x 4 = 1.7.
Find a method of moments estimate of λ.
2. Let X 1 , X 2 , ... , X n be a random sample from the distribution with probability
density function
( ) ( ) ( ) ( )
X X θ 2 1 1
f x = f x; = θ + θ x θ - - x , 0 < x < 1, θ > 0.
a) Obtain the method of moments estimator of θ, θ ~
b) Is θ ~
an unbiased estimator of θ ? Justify your answer.
c) Suppose n = 6, and x 1 = 0.3, x 2 = 0.5, x 3 = 0.6, x 4 = 0.65, x 5 = 0.75, x 6 = 0.8.
Find a method of moments estimate of θ.
3. Let θ > 0 and let X 1 , X 2 , ... , X n be a random sample from a Uniform distribution
on interval ( 0, θ ).
a) Obtain the method of moments estimator of θ, θ ~
.
b) Is θ ~
an unbiased estimator of θ ? Justify your answer.
c) Find Var ( θ ~ ). d) Find MSE ( θ ~ ).
4. Let X 1 , X 2 , ... , X n be a random sample from the distribution with probability
density function
( ) θ
;θ θ
12 8
3
+
+
=
f x x , 0 < x < 4, θ >
4
3
- .
a) Find the method of moments estimator of θ, θ ~
.b) Suppose n = 5, and x 1 = 1.2, x 2 = 1.8, x 3 = 2.6, x 4 = 3.1, x 5 = 3.8.
Find the method of moments estimate of θ.
c) Suppose n = 4, and x 1 = 1.3, x 2 = 2.2, x 3 = 3.1, x 4 = 3.8.
Find the method of moments estimate of θ.