STAT 714, FALL 2008 HOMEWORK 7
1. Consider the GM model y=Xb +e, where Xis N×pwith rank r≤p,E(e) = 0
and cov(e) = σ2I. Let b
bdenote a least squares estimator of band suppose that λ0bis
estimable.
(a) Assume a0yis an unbiased estimator of λ0b. What the Gauss-Markov Theorem say
about var(a0y) when compared to var(λ0b
b)?
(b) Let b
µ=Xb
band e
µ=A0yfor a nonrandom N×Nmatrix A. Assume that
E(A0y) = Xb. Show that cov(e
µ)−cov(b
µ) is nnd.
(c) Let e
µ=A0yfor a nonrandom N×Nmatrix A. Assume that E(A0y) = Xb. Show
that N
X
i=1
var(e
µi)≥rσ2,
where e
µiis the ith component of e
µ.
2. Suppose that Vis pd. Prove that (X0X)−1X0VX(X0X)−1−(X0V−1X)−1is nnd.
3. Consider the linear model defined by
y1= 2θ+e1
y2=θ+e2,
where e1= 2z1−z2and e2=z1+ 2z2, and z1and z2are independent random variables
with zero mean and constant variance σ2.
(a) Write this model in y=Xb +eform. Find E(y) and cov(y).
(b) Compute the ordinary least squares (OLS) estimator of θ.
(c) Compute the generalized least squares (GLS) estimator of θ.
(d) Show that the OLS and GLS estimators are both unbiased.
(e) Compute the variance of both estimators and compare.
4. Suppose that y1, y2, ..., yNis an iid U(0,2θ) sample, where θ > 0. Define ei=yi−θ,
for i= 1,2, ..., N .
(a) Find the mean and covariance matrix of e= (e1, e2, ..., eN)0.
(b) Show that y= (y1, y2, ..., yN)0follows a Gauss-Markov model.
(c) Find the BLUE of θ, say b
θOLS. Give both a matrix expression and an expression in
terms of simple summary statistics.
(d) Find cso that b
θ=cy(N), where y(N)= max{y1, y2, ..., yN}, is unbiased for θand
compute the variance of b
θ.
(e) Compare the variances of b
θOLS and b
θ. Are you surprised? Explain your findings in
light of the Gauss-Markov Theorem.
5. Suppose that y1∼beta(α, β), where α=β= 2. Let y2= 1 −y1,y3=y1y2, and
y= (y1, y2, y3)0. Compute E(y0Uy), where Uis the matrix in Example 2.3 (notes, pp
61). Use Result GM.3 to answer this question.
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