[SOLVED] STAT5113 Homework2

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

1. Find the cdf of θ^ˆ, and use it to find the pdf of θ^ˆ. [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 θ^ˆ.
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 θ and of the bias of the estimator.

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

X_i ∼N(µ₁,σ²),              i = 1,…,n₁,                   Y_j ∼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

.

2. Find the (joint) sampling distribution of, and σ_c².
3. Find the bias of the three estimators. Which one is unbiased? Which one is biased?