1. Consider a sample X1,...,Xn from the Unif (0,θ) distribution. The MLE of θ is given by
.
a. Find the cdf of θˆ, and use it to find the pdf of θˆ. [Hint: use the fact that maxi xi ≤ t iff xi ≤ t for every i.]
b. Derive an expression for the bias of θˆ.
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 θˆ. 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
The MLEs of the unknown parameters, which you have derived in the previous homework, are
.
a. Find the (joint) sampling distribution of , and σc2.
b. Find the bias of the three estimators. Which one is unbiased? Which one is biased?
