[Solved] STAT5113 Homework2

1. The following problem might be graded.
2. Consider a sample X₁,,X_n from the Unif (0,) distribution. The MLE of is given by

1. Find the cdf of , and use it to find the pdf of . [Hint: use the fact that max_i x_i t iff x_i t for every i.]
2. Derive an expression for the bias of .
3. 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 and of the bias of the estimator.

1. Using the data provided above, give an estimate of the MSE of . B. The following problem will be graded.
2. As in problem A.3 of Homework 1, consider independent samples

X_i N(₁,²), i = 1,,n₁, Y_j N(₂,²), j = 1,,n₂.

Define the one-sample MLEs

The MLEs of the unknown parameters, which you have derived in the previous homework, are

.

2. Find the (joint) sampling distribution of, and _c².
3. Find the bias of the three estimators. Which one is unbiased? Which one is biased?