STAT 420
Fall 2007
Homework #2
1.
Sometimes it is known in advance that the leastsquares regression line must go
through the origin, i.e., the regression model is of the form
Y
i
=
β
x
i
+
ε
i
,
i
= 1, 2, … ,
n
, where
ε
i
’s are i.i.d.
N
(
0,
σ
2
)
,
and the equation of the regression line is
y
ˆ
=
°
ˆ
x
. In this case, finding the
leastsquares line reduces to finding the value
°
ˆ
that minimizes the expression
(
)
[
]
°
=

=
n
i
i
i
x
y
f
1
2
°
°
.
Use the derivative of
f
with respect to
β
to derive the formula for the slope of
the leastsquares regression line in this case.
We want to minimize
( )
[
]
°
=

=
n
i
i
i
x
y
b
f
1
2
°
.
( )
[
]
(
)
°
°
°
=
=
=
⋅

⋅
=

⋅

⋅
=
n
i
i
i
n
i
i
n
i
i
i
i
y
x
x
x
x
y
b
f
1
1
2
1
2
2
2
°
°
'
.
To find the extremum points:
(
)
0
°
'
=
f
. Therefore,
°
°
=
=
=
n
i
i
n
i
i
i
x
y
x
1
2
1
°
ˆ
.
(
)
°
=
⋅
=
n
i
i
x
f
1
2
2
°
"
> 0.
±
(
)
°
f
has minimum at
°
ˆ
.
OR
( )
[
]
[
]
°
°
=
=
+

=

=
n
i
i
i
i
i
n
i
i
i
x
y
x
y
x
y
b
f
1
2
2
2
1
2
°
°
°
2
=
2
2
1
2
°
°
1
1
2
⋅
°
=
⋅
°
=
²
²
³
´
µ
µ
¶
·
+
²
²
³
´
µ
µ
¶
·
⋅

²
²
³
´
µ
µ
¶
·
°
=
n
i
i
x
n
i
i
y
i
x
y
n
i
i
– parabola.
The vertex (minimum) is at
°
°
=
=
=
=

n
i
i
n
i
i
i
x
y
x
a
b
1
2
1
2
°
ˆ
.
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2.
It has been proposed that the brightness measured in some unit of color or a
commercial product is proportional to the time it is in a certain chemical reaction
during the production process, or
Y
i
=
β
x
i
+
ε
i
,
i
= 1, 2, … ,
n
, where
ε
i
’s are i.i.d.
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 Spring '07
 STEPANOV
 Least Squares, Linear Regression, Regression Analysis, Yi, Errors and residuals in statistics, leastsquares regression line

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