STAT 511
Homework 4
Due Date: 11:00 A.M., Wednesday, February 8
1. Suppose w ∼ N (µ, Σ), where
2
µ = −3
1
3 −1 1
4 2
and Σ = −1
1
2 5
(a) Determine the distribution of
w1 + w2 − w3
w1 + w2 + w3
.
(b) Determine the distribution of
w12 + w22 + w32 + 2w1 w2 + 2w1 w3 + 2w2 w3
.
16
(c) Determine the distribution of
4w12 + 4w22 + 4w32 + 8w1 w2 − 8w1 w3 − 8w2 w3
.
w12 + w22 + w32 + 2w1 w2 + 2w1 w3 + 2w2 w3
2. If a random variable w ∼ N (δ, 1) and is independent of a random variable u ∼ χ2n , then
w
p
u/n
has a noncentral t-distribution with noncentrality parameter (ncp) δ and degrees of freedom (df) n.
We will denote this distribution by tn (δ). If the ncp is 0, the distribution is known as the central
t-distribution, which can be denoted by tn (0) or tn . Let tn,1−α denote the 1 − α quantile of the tn (0)
distribution. Thus, tn,1−α is the value such that
P [T ≤ tn,1−α ] = 1 − α
for T ∼ tn (0).
For the following questions, suppose the Normal Theory Gauss-Markov linear model holds, and suppose c0 β is estimable.
(a) For a fixed constant d, prove that the distribution of
c0 β̂ − d
p
σ̂ 2 c0 (X 0 X)− c
is noncentral t and derive the ncp and df.
(b) Provide a test statistic for testing the null hypothesis
H0 : c0 β = d vs. HA : c0 β 6= d
for any fixed d ∈ IR.
(c) Suppose you wish to test the hypotheses in (b) using significance level α. For what values of the
test statistic provided in (b) would you reject the null hypothesis?
(d) Derive expressions for the lower and upper limits of a 100(1 − α) confidence interval for c0 β.
3. Prove the result stated on slide 3 of the notes on estimation of the error variance under the GaussMarkov model.
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