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GROUND RULES:
- This exam contains 7 questions. Point totals are in []. The maximum number of points on this exam is 100.
- Print your name at the top of this page in the upper right hand corner.
- This is a closed-book and closed-notes exam. You may not use a calculator.
- Show all of your work. Explain all of your reasoning.
- Any discussion or otherwise inappropriate communication between examinees, as well as the appearance of any unnecessary material, will be dealt with severely.
- You have 3 hours to complete this exam. GOOD LUCK!
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After you have finished the exam, please read the following statement and sign your name below it.
I promise that I did not discuss any aspect of this exam with anyone other than the instructor, that I neither gave nor received any unauthorized assistance on this exam, and that the work presented herein is entirely my own.
- [9] Suppose that z ∼ N (μ, 1) and v ∼ χ^2 k. If z and v are independent, then
t =
z √ v/k
has a noncentral t distribution with k degrees of freedom and noncentrality parameter μ. For k > 1, show that
E(t) =
μ
kΓ{(k − 1)/ 2 } √ 2Γ(k/2)
where Γ(·) is the usual gamma function. Note: For k ≤ 1, E(t) is not finite.
- [15] Consider the one-way, fixed-effects ANOVA model
yij = μ + αi + eij ,
for i = 1, 2 , ..., a and j = 1, 2 , ..., n. Suppose that eij are iid N (0, σ^2 ) random variables. The parameter σ^2 is unknown.
(a) Show that γ =
∑a i=1 ciαi^ is estimable if and only if^
∑a i=1 ci^ = 0.
(b) Show that the MLE of γ is γ̂ =
∑a i=1 ciyi+, where^ yi+^ is the^ ith sample treatment mean.
(c) Derive a 100(1 − α) percent confidence interval for γ. Explain as many details as possible.
- [9] Let M 1 and M 2 be perpendicular projection matrices on Rn. Prove that if C(M 1 )⊥C(M 2 ), then M 1 + M 2 is a perpendicular projection matrix onto C(M 1 , M 2 ).
- [20] Consider the model
yij = μ + τi + γi(xij − xi+) + eij ,
for i = 1, 2 and j = 1, 2 , ..., n. Assume that eij are iid N (0, σ^2 ) random variables, where σ^2 > 0. As a frame of reference, suppose that we are studying the effects of two drugs on weight loss (y) in the situation wherein a covariable x (e.g., initial weight, etc.) is available for each subject. Assume that
∑n j=1(xij^ −^ xi+) (^2) > 0 for i = 1, 2 and that
x 1 j 6 = x 2 j for at least one j.
(a) Write the model in matrix form y = Xb + e, identifying all vectors and matrices.
(b) Find a maximal set of linearly independent estimable functions. Verify your answer.
(c) The researcher believes that τ 1 = τ 2 and that γ 1 = γ 2. If the researcher is correct, sketch a picture of what you would expect the observed data to look like. Use different plotting symbols for the two drugs.
(d) Suppose that we were interested in testing the hypothesis
H 0 : τ 1 = τ 2 , γ 1 = γ 2 versus H 1 : not H 0.
Write a reduced model, say y = Wc + e, that would be appropriate when H 0 is true. Identify all vectors and matrices in your reduced model.
(e) Describe a procedure, in full detail, how you would test the hypothesis in (d). Be very precise, rigorous, and thorough in your explanation. Your description should end with how the test statistic is formed (you can leave it in terms of matrices and vectors), what its distribution is under H 0 , and when we reject H 0.
- [7] Consider the linear model y = Xb+e, where X is N ×p with rank r ≤ p. Assume that E(e) = 0. Let PX denote the perpendicular projection matrix onto C(X).
(a) True or False. If δ can be written as a linear combination of the basis vectors for N (X′), then PXδ = 0.
(b) True or False. If b̂ 1 and ̂b 2 both solve the normal equations X′Xb = X′y, then λ′^ b̂ 1 = λ′^ b̂ 2.
(c) True or False. Either the vector y ∈ C(X) or y ∈ C(X)⊥.
(d) True or False. r(X) = r(X′X) = r(PX).
(e) True or False. 1 ′(I − PX)y = 0 ; i.e., the sum of the residuals from the least squares fit is zero.
(f) True or False. E(y) = Xb is always estimable.
(g) True or False. If cov(y) = σ^2 I, then y′PXy/σ^2 ∼ χ^2 r (λ), where λ = (Xb)′Xb/ 2 σ^2.
