All other items including (and especially) CELL PHONES must be left at the front of the room.

Transcription

1 TEST #2 / STA 5327 (Inference) / Spring 2017 (April 24, 2017) Name: Directions This exam is closed book and closed notes. You will be supplied with scratch paper, and a copy of the Table of Common Distributions from the back of our textbook. During the exam, you may use ONLY what you need to write with (pens, pencils, erasers, etc). No calculator is needed and calculators are NOT allowed. All other items including (and especially) CELL PHONES must be left at the front of the room. For all problems, you must show and explain all your work. Do not quote homework results. If you wish to use a result from homework in a solution, you must prove this result. You may use results from lecture (unless you are being asked to prove that very result), but you should clearly state the result you are using. In multi-part problems, if you cannot solve one part, in some cases it is still possible to go on and do some (or all) of the later parts. Partial credit is available. (If you know part of a solution write it down. If you know an approach to a problem, but cannot carry it out write down this approach. If you know facts, theorems, or definitions which are relevant to a problem, write them down. ) The exam has 5 problems and 11 pages. The points total

2 Problem 1. Let X 1, X 2,..., X n be iid Uniform(0, θ). This is similar to exercise 7.49 but uses the Uniform(0, θ) distribution. The calculations in exercises 7.9, 7.37, 7.46 are similar to some extent. (a) (8%) Find an unbiased estimator of θ based only on Y = min{x 1, X 2,..., X n }. Assume the unbiased estimator has the form cy. Evaluate EY to determine the appropriate value of c. Calculations show that EY = 1 θ so the desired unbiased estimator is (n + 1)Y. n + 1 2

3 (b) (8%) Find the best unbiased estimator of θ. (You may use without proof the fact that Z = max{x 1, X 2,..., X n } is a complete sufficient statistic for θ.) Assume the best unbiased estimator has the form cz. Evaluate EZ to find the value of c. Calculations show that EZ = n n+1 θ so that Z is unbiased for θ. Since it is a function of the CSS Z, n+1 n it is automatically the best unbiased estimator by a result stated in lecture. 3

4 (c) (8%) Find E(Y Z). The important result we need was stated in the lecture notes: If S is an unbiased estimator of τ(θ) with finite variance, and T is a CSS, then E(S T ) is the best unbiased estimator of τ(θ) (the UMVUE). See page 14 of notes11.pdf. If students state this result clearly, that is worth 4 points (even if they don t do anything else). Applying this result to the results of parts (a) and (b) we learn that E ((n + 1)Y Z) = n + 1 n Z which implies E(Y Z) = Z/n 4

5 Problem 2. (14%) Let X 1, X 2,..., X n be iid Gamma(α, β) with α known. Find the best unbiased estimator of 1/β 2. Similar to exercise See the solution given on page 7 of ch7 extra handwritten solutions.pdf. From this it is clear that c/t 2 will be unbiased for 1/β 2 where c = Γ(nα)/Γ(nα 2). The family of Gamma distributions with α known and β unknown forms a 1pef with CSS T = n i=1 X i. Since c/t 2 is unbiased for 1/β 2 and also a function of the CSS T, we know it is best unbiased. 5

6 Problem 3. (10%) Suppose that X 1, X 2,..., X n are iid with density f(x θ) = ψ(x θ), θ R, where 1 ψ(x) =, < x <. 2(1 + x ) 2 Show that the MLE of θ exists (that is, there exists a value of θ at which the likelihood attains its sup). Clearly state the result from lecture you are using. This is exercise X1.5. It uses the result stated in notes8.pdf, page 5. See the solution in mordor. 6

7 Problem 4. Suppose that the random variables Y 1, Y 2,..., Y n satisfy Y i = βx i + ε i, i = 1,..., n where x 1,..., x n are fixed known constants, and ε 1,..., ε n are iid N(0, σ 2 ), and β and σ 2 are unknown. This problem consists of parts of exercises 7.19 and See the solution manual posted in mordor. Another solution for the comparison of variances is given on page 1 of ch7 misc solutions.pdf. (a) (12%) Find the MLE of β and show it is an unbiased estimator of β. This is exercise 7.19(b). 7

8 (b) (8%) Show that Y i / x i is an unbiased estimator of β. This is exercise 7.20(a). 8

9 (c) (8%) Calculate the variance of Y i / x i and compare it to the variance of the MLE. This is exercise 7.20(b). 9

10 (d) (10%) Suppose we alter the model stated earlier by now assuming that σ 2 has a known value σ0. 2 Find I Y (β), the Fisher information in the data Y 1, Y 2,..., Y n. The easiest way to compute the Fisher information is using the Alternate Formula (Fact 5 on page 5 of notes12.pdf). The solution manual solution of exercise 7.19(b) gives the second partial of the log-likelihood with respect to β (but with a sign error!) which easily leads to the Fisher information being equal to 1 x 2 σ 2 i. i Note: Since Y 1, Y 2,..., Y n are independent but NOT identically distributed, one cannot simply calculate the Fisher information for one observation and multiply by n. That is a serious error. (e) of β. (4%) Find the Cramér-Rao lower bound (CRLB) for the variance of an unbiased estimator The answer is 1/I Y (β) = σ2 i x2 i If students state the general result CRLB = (τ (θ)) 2 /I X (θ) and get no further, give them 2 points. If they know that this reduces to CRLB = 1/I Y (β) in this case and get no further, give them 3 points. If they actually evaluated the Fisher information in part (d) and get the correct CRLB given above, give them all 4 points. 10

11 Problem 5. (10%) Let X 1, X 2,..., X n be iid Bernoulli(θ). Define T = n i=1 X i. An estimator of θ is given by ( ) T ˆθ = (1 p)a + p n where a and p are known constants. Find the MSE (mean squared error) of the estimator ˆθ. This is done on page 4 of notes11.pdf. The answer is given at the bottom of that page. 11