Tutorial 8: suggested solutions
1.
suppose we have sample (
x
i
, y
i
)
, i
= 1
, ..., n
. we estimate the regression model
y
i
=
g
(
x
i
) +
ε
i
, where
g
(
x
) is a spline function
of the form
g
(
x
) =
J
+4
X
j
=1
β
j
B
j
(
x
)
(a)
Estimate the derivative
g
0
(
x
) of
g
(
x
)
(b)
ﬁnd the 95% conﬁdence band for
g
0
(
x
).
Let
X
=
B
1
(
x
1
)
B
2
(
x
1
)
... B
J
+4
(
x
1
)
B
1
(
x
2
)
B
2
(
x
2
)
... B
J
+4
(
x
2
)
...
B
1
(
x
1
)
B
2
(
x
1
)
... B
J
+4
(
x
1
)
Y
=
Y
1
Y
2
...
Y
n
and
ˆ
β
1
ˆ
β
2
...
ˆ
β
n
= (
X
>
X
)

1
X
>
Y
The estimated funciton is
ˆ
g
(
x
) =
J
+4
X
j
=1
ˆ
β
j
B
j
(
x
)
(a)
g
0
(
x
) can be estimated by
ˆ
g
0
(
x
) =
J
+4
X
j
=1
ˆ
β
j
B
0
j
(
x
) =
J
+4
X
j
=2
ˆ
β
j
B
0
j
(
x
)
where
B
1
(
x
) = 1
, B
2
(
x
) =
x, B
3
(
x
) =
x
2
, B
4
(
x
) =
x
3
, B
0
j
+4
(
x
) = (
x

t
j
)
3
+
, j
= 1
, ..., J
thus
B
0
2
(
x
) = 1
, B
0
3
(
x
) = 2
x, B
0
4
(
x
) = 3
x
2
, B
0
j
+4
(
x
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 Fall '09
 XIAYingcun
 Regression Analysis, Yi, 4 g, 4 j

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