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STA 414S/2104S
: Homework #1
Due Feb.11, 2010 at 1 pm
Late homework is penalized at 20% deduction per day. You are welcome to discuss your
work on this homework with your classmates. You are required to write up the work on your
own, using your own words, and to provide your own computer code.
Answers to the computational questions must be submitted in two parts. The ﬁrst part
presents your conclusions and supporting evidence in a report, written in paragraphs and
sentences (not point form)
that does not include computer code
. This part may include
tables and ﬁgures. The second part is a complete, and annotated, ﬁle showing the computer
code that you used to obtain the results discussed in the ﬁrst part. It is important to include
readable code, since everyone’s answers will be based on diﬀerent training and test samples.
1.
Likelihood and Bayesian inference in the linear model:
Suppose that the
n
×
1 vector
Y
follows a normal distribution with mean
Xβ
and
variance
σ
2
I
:
Y
∼
N
(
Xβ,σ
2
I
)
i.e. that
f
(
y

β,σ
2
) =
1
(
√
2
πσ
)
n
exp
{
1
2
σ
2
(
y

Xβ
)
T
(
y

Xβ
)
}
.
(a) The maximum likelihood estimates (
ˆ
β,
ˆ
σ
2
) are deﬁned to be the values of
β
and
σ
2
that simultaneously maximize the likelihood function, or more conveniently
the loglikelihood function
‘
(
β,σ
2
) = log
f
(
y

β,σ
2
)
.
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 Spring '09

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