Description
A. The following problem might be graded.
1. Consider a sample X1, . . . , Xn from the Unif (0, θ) distribution. The MLE of θ is given by
ˆθ = max
1≤i≤n
Xi
.
a. Find the cdf of ˆθ, and use it to find the pdf of ˆθ. [Hint: use the fact that maxi xi ≤ t iff xi ≤ t
for every i.]
b. Derive an expression for the bias of ˆθ.
c. Suppose the sample consisted of the following numbers:
6.83 8.85 1.46 7.81 5.89 7.20 6.60 11.98 10.55 8.12 7.59 4.50
10.51 0.18 8.62 9.58 6.89 2.30 7.55 4.12 10.67 1.08 0.53 9.47
Provide an estimate of θ and of the bias of the estimator.
d. Using the data provided above, give an estimate of the MSE of ˆθ.
B. The following problem will be graded.
2. As in problem A.3 of Homework 1, consider independent samples
Xi ∼ N (µ1, σ2
), i = 1, . . . , n1, Yj ∼ N (µ2, σ2
), j = 1, . . . , n2.
Define the one-sample MLEs
X¯ =
1
n1
Xn1
i=1
Xi Y¯ =
1
n2
Xn2
j=1
Yj
S
2
1 =
1
n1
Xn1
i=1
(Xi − X¯)
2 S
2
2 =
1
n2
Xn2
j=1
(Yj − Y¯ )
2
The MLEs of the unknown parameters, which you have derived in the previous homework, are
µc1 = X¯
µc2 = Y¯
σc2 =
n1S
2
1 + n2S
2
2
n1 + n2
.
a. Find the (joint) sampling distribution of µc1, µc2, and σc2.
b. Find the bias of the three estimators. Which one is unbiased? Which one is biased?
