STAT 542: HOMEWORK 1
Problem 1
Since yi∼N(xt
iβ, σ2), i ∈(1 : n), the log-likelihood is:
l(ˆ
β) = nlog 1
σ√2π−Pn
i=1(yi−xiˆ
β)2
2σ2.
then,
AIC =−2l(ˆ
β)+2p
=−2nlog 1
σ√2π+Pn
i=1(yi−xiˆ
β)2
σ2+ 2p
=1
σ2(constant +RSS + 2σ2p).
Given Cp=RSS + 2σ2pand σis known, when AIC is minimized, so does Cp, and
vice versa. So, AI C criterion and Mallow’s Cpare equivalent.
Problem 2
(a) By the definition of ˆ
β∗, we have:
ˆ
β∗= arg min
β(y∗
1−xt
1β)2+
n
X
j=2
(yj−xt
jβ)2.
In order to prove ˆ
β∗=ˆ
β[1], we just need to show:
ˆ
β[1] = arg min
β(y∗
1−xt
1β)2+
n
X
j=2
(yj−xt
jβ)2.
As known,
ˆ
β[1] = arg min
β
n
X
j=2
(yj−xt
jβ)2.
1
