ISyE 7405, Fall-2012
Instructor : A.Shapiro
Midterm Exam
1. Let X∼Nm(µ,Σ) and Aand Bbe two m×msymmetric matrices. Find the covariance
Cov(X0AX,X0BX).
2. Consider the linear model Y=Xβ +ε, with ε∼N(0, σ2In).
(i) Find ML estimators of βand σ2.
(ii) Let abe a given p×1 nonzero vector and ˆ
βbe the least squares estimator of β. Consider the
hypothesis H0:a0β= 0, and the statistic
T=a0ˆ
β
S[a0(X0X)−1a]1/2,
where S2=SSE/(n−p). Show that if H0is true, then Thas t-distribution with n−pdegrees of
freedom.
3. Consider the linear regression model Yi=β0+β1Xi1+... +βkXik +εi,i= 1, ..., n. Suppose that
the corresponding n×(k+1) design matrix Xhas full rank k+1, that the errors εiare uncorrelated,
E(εi) = 0 and Var(εi) = σ2,i= 1, ..., n. Suppose that an additional regressor is added and hence the
response variable is analyzed by the model Yi=γ0+γ1Xi1+... +γkXik +γk+1Xi,k+1 +εi,i= 1, ..., n
(that is, an additional column is added to the design matrix). Show that Var( ˆγj)≥Var( ˆ
βj),
j= 0, ..., k, where ˆ
βjand ˆγjare the corresponding least squares estimators. When does the equal-
ity hold there?
4. Let Xbe an n×mrandom matrix, Σbe an m×mpositive definite matrix and Pbe an n×n
symmetric idempotent matrix (i.e., P2=P) of rank k≥m. Show that if vec(X0)∼N(0,P⊗Σ),
then X0X∼Wm(k, Σ).
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