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STAT 100B HW 5 Due Friday 5pm
Problem 1:
For the simplest regression model
y
i
=
βx
i
+
±
i
,
i
= 1
,...,n
, suppose
x
i
are ﬁxed, and
E[
±
i
] = 0, Var[
±
i
] =
σ
2
, and
±
i
are independent. Consider the estimator
ˆ
β
=
∑
n
i
=1
w
i
y
i
/
∑
n
i
=1
w
i
x
i
,
where
w
i
may depends on
x
i
, but not on
y
i
.
(1) Find E[
ˆ
β
] and Var[
ˆ
β
].
(2) Show that Var[
ˆ
β
] is minimized when
w
i
∝
x
i
. Calculate the minimum.
Problem 2:
For the simple regression
y
i
=
β
0
+
β
1
x
i
+
±
i
,
i
= 1
,...,n
, deﬁne the sample variance
of (
x
i
,i
= 1
,...,n
) as
V
x
=
∑
n
i
=1
˜
x
2
i
/n
, deﬁne the sample variance of (
y
i
,i
= 1
,...,n
) as
V
y
=
∑
n
i
=1
˜
y
2
i
/n
, and deﬁne the sample covariance between (
x
i
,y
i
,i
= 1
,...,n
) as
C
xy
=
∑
n
i
=1
˜
x
i
˜
y
i
/n
,
where ˜
x
i
=
x
i

¯
x
, and ˜
y
i
=
y
i

¯
y
. Deﬁne the sample correlation as
ρ
xy
=
C
xy
/
(
√
V
x
p
V
y
).
(1) Show that
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This note was uploaded on 03/30/2011 for the course STAT 100B taught by Professor Wu during the Winter '11 term at UCLA.
 Winter '11
 Wu

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