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# Case of rejection Notes | EduRev

## : Case of rejection Notes | EduRev

``` Page 1

Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and
Design Design of Experiments of Experiments- -I I
MODULE MODULE – – II II
g g p p
GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS
LECTURE  LECTURE  - -7 7
GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS
AND ANAL YSIS OF VARIANCE AND ANAL YSIS OF VARIANCE
Dr. Shalabh
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
Page 2

Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and
Design Design of Experiments of Experiments- -I I
MODULE MODULE – – II II
g g p p
GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS
LECTURE  LECTURE  - -7 7
GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS
AND ANAL YSIS OF VARIANCE AND ANAL YSIS OF VARIANCE
Dr. Shalabh
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
C 2 Tt f bt f t h d
0
:12 Hk ßß <
2
2
Case 2: Test of asubset of parameters when and
are unknown
0
0
: , 1, 2,..,
kk
Hk rp ßß = = <
12
, ,...,
rr p
ß ßß
++
2
s
In case 1, the test of hypothesis was developed when all ’s are considered in the sense that we test for each ß In case 1, the test of hypothesis was developed when all s are considered in the sense that we test for each
Now consider another situation, in which the interest is to test only a subset of i.e., not all but
only a few parameters. This type of test of hypothesis can be used e.g., in the following situation. Suppose five levels of
voltage are applied to check the rotations per minute (rpm) of a fan at 160 volts, 180 volts, 200 volts, 220 volts and 240 volts.
ß
0
,1,2,...,.
ii
ip ßß ==
12
, ,..., ,
p
ßßß
It can be realized in practice that when the voltage is low, then the rpm at 160, 180 and 200 volts can be observed easily. At
220 and 240 volts, the fan rotates at the full speed and there is not much difference in the rotations per minute at these
voltages. So the interest of the experimenter lies in testing the hypothesis related to only first three effects, viz., , for 160
volts, for 180 volts and for 200 volts.
1
ß
2
ß
3
ß
00 0 0
01 1 2 2 3 3
:, , H ß ßß ßß ß == =
The null hypothesis in this  case can be written as:
when  and        are unknown.
45
, ß ß
2
s
Note that under  case 1, the null hypothesis will be
00 0 0 0 0
0 1 1 2 23 3 4 45 5
:, , , , . H ß ßß ßß ßß ßß ß == = = =
Page 3

Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and
Design Design of Experiments of Experiments- -I I
MODULE MODULE – – II II
g g p p
GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS
LECTURE  LECTURE  - -7 7
GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS
AND ANAL YSIS OF VARIANCE AND ANAL YSIS OF VARIANCE
Dr. Shalabh
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
C 2 Tt f bt f t h d
0
:12 Hk ßß <
2
2
Case 2: Test of asubset of parameters when and
are unknown
0
0
: , 1, 2,..,
kk
Hk rp ßß = = <
12
, ,...,
rr p
ß ßß
++
2
s
In case 1, the test of hypothesis was developed when all ’s are considered in the sense that we test for each ß In case 1, the test of hypothesis was developed when all s are considered in the sense that we test for each
Now consider another situation, in which the interest is to test only a subset of i.e., not all but
only a few parameters. This type of test of hypothesis can be used e.g., in the following situation. Suppose five levels of
voltage are applied to check the rotations per minute (rpm) of a fan at 160 volts, 180 volts, 200 volts, 220 volts and 240 volts.
ß
0
,1,2,...,.
ii
ip ßß ==
12
, ,..., ,
p
ßßß
It can be realized in practice that when the voltage is low, then the rpm at 160, 180 and 200 volts can be observed easily. At
220 and 240 volts, the fan rotates at the full speed and there is not much difference in the rotations per minute at these
voltages. So the interest of the experimenter lies in testing the hypothesis related to only first three effects, viz., , for 160
volts, for 180 volts and for 200 volts.
1
ß
2
ß
3
ß
00 0 0
01 1 2 2 3 3
:, , H ß ßß ßß ß == =
The null hypothesis in this  case can be written as:
when  and        are unknown.
45
, ß ß
2
s
Note that under  case 1, the null hypothesis will be
00 0 0 0 0
0 1 1 2 23 3 4 45 5
:, , , , . H ß ßß ßß ßß ßß ß == = = =
Let be the p parameters.
We can divide them into two parts such that out of and we are interested in testing
12
, ,...,
p
ß ßß
12 1
, ,..., , ,..., ,
rr p
ß ßßß ß
+
3
the hypothesis of a subset of it.
Suppose, we want to test the null hypothesis when                         and       are unknown.
p
0
0
:,1,2,..,
kk
Hk rp ßß = =<
12
, ,...,
rr p
ß ßß
++
2
s
The alternative hypothesis under consideration is
In order to develop a test for  such a hypothesis, the linear model
under the usual assumptions can be rewritten as follows:
0
1
: 1, 2,.., . or at least one f
kk
Hkrp ßß ? =<
YX ß e =+
under the usual assumptions can be rewritten as follows:
Partition
(1)
12
(2)
(), XXX
ß
ß
ß
??
==
??
??
Th d l b itt
where
(1) 1 2 (2) 1 2
( , ,..., ) , ( , ,..., )
rrr p
ß ßß ß ß ß ß ß
++
'' ==
with the orders of matrices  as                                                       and
12 (1)
:, : ( ), : 1 Xn rX n prr ß × ×- ×
(2)
:( ) 1. pr ß - ×
The model can be rewritten as
(1)
12
(2)
()
YX
XX
ß e
ß
e
ß
=+
??
=+
??
??
(2)
1(1) 2 (2)
. XX
ß
ß ße
??
=+ +
Page 4

Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and
Design Design of Experiments of Experiments- -I I
MODULE MODULE – – II II
g g p p
GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS
LECTURE  LECTURE  - -7 7
GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS
AND ANAL YSIS OF VARIANCE AND ANAL YSIS OF VARIANCE
Dr. Shalabh
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
C 2 Tt f bt f t h d
0
:12 Hk ßß <
2
2
Case 2: Test of asubset of parameters when and
are unknown
0
0
: , 1, 2,..,
kk
Hk rp ßß = = <
12
, ,...,
rr p
ß ßß
++
2
s
In case 1, the test of hypothesis was developed when all ’s are considered in the sense that we test for each ß In case 1, the test of hypothesis was developed when all s are considered in the sense that we test for each
Now consider another situation, in which the interest is to test only a subset of i.e., not all but
only a few parameters. This type of test of hypothesis can be used e.g., in the following situation. Suppose five levels of
voltage are applied to check the rotations per minute (rpm) of a fan at 160 volts, 180 volts, 200 volts, 220 volts and 240 volts.
ß
0
,1,2,...,.
ii
ip ßß ==
12
, ,..., ,
p
ßßß
It can be realized in practice that when the voltage is low, then the rpm at 160, 180 and 200 volts can be observed easily. At
220 and 240 volts, the fan rotates at the full speed and there is not much difference in the rotations per minute at these
voltages. So the interest of the experimenter lies in testing the hypothesis related to only first three effects, viz., , for 160
volts, for 180 volts and for 200 volts.
1
ß
2
ß
3
ß
00 0 0
01 1 2 2 3 3
:, , H ß ßß ßß ß == =
The null hypothesis in this  case can be written as:
when  and        are unknown.
45
, ß ß
2
s
Note that under  case 1, the null hypothesis will be
00 0 0 0 0
0 1 1 2 23 3 4 45 5
:, , , , . H ß ßß ßß ßß ßß ß == = = =
Let be the p parameters.
We can divide them into two parts such that out of and we are interested in testing
12
, ,...,
p
ß ßß
12 1
, ,..., , ,..., ,
rr p
ß ßßß ß
+
3
the hypothesis of a subset of it.
Suppose, we want to test the null hypothesis when                         and       are unknown.
p
0
0
:,1,2,..,
kk
Hk rp ßß = =<
12
, ,...,
rr p
ß ßß
++
2
s
The alternative hypothesis under consideration is
In order to develop a test for  such a hypothesis, the linear model
under the usual assumptions can be rewritten as follows:
0
1
: 1, 2,.., . or at least one f
kk
Hkrp ßß ? =<
YX ß e =+
under the usual assumptions can be rewritten as follows:
Partition
(1)
12
(2)
(), XXX
ß
ß
ß
??
==
??
??
Th d l b itt
where
(1) 1 2 (2) 1 2
( , ,..., ) , ( , ,..., )
rrr p
ß ßß ß ß ß ß ß
++
'' ==
with the orders of matrices  as                                                       and
12 (1)
:, : ( ), : 1 Xn rX n prr ß × ×- ×
(2)
:( ) 1. pr ß - ×
The model can be rewritten as
(1)
12
(2)
()
YX
XX
ß e
ß
e
ß
=+
??
=+
??
??
(2)
1(1) 2 (2)
. XX
ß
ß ße
??
=+ +
The null hypothesis of interest is now
h d k
000 0
() H ßß ßß ß
ß
2
4
where and are unknown.
The complete parametric space is
000 0
0(1) (1) 1 2
: ( , ,..., )
r
H ßß ßß ß == (2)
ß
2
s
{ }
22
( , ); , 0, 1,2,..., ßs ß s O= - 8 < < 8 > =
i
ip
and sample space under  is
0
H
{ }
02 2
(1) (2)
( , , ); , 0, 1, 2,..., .
i
ir r p ?ß ß s ß s =-8<<8>=++
Th i l f lik lih d f ti d i bt i d b b tit ti th i lik lih d
The likelihood function is
2
2
22
11
(| , ) exp ( )( ) .
22
n
Ly yX yX ßsßß
ps s
?? ? ?
' =- - -
??
? ?
?? ? ?
O The maximum value of likelihood function under is obtained by substituting the maximum likelihood
estimates of   and      ,  i.e.,
O
ß
2
s
1
2
ˆ
()
1
ˆˆ
ˆ ()()
XX Xy
yX yX
ß
sß ß
-
'' =
'
as
()() yX yX
n
sß ß =- -
2
2
22
11
ˆˆ
(| , ) exp ( )( )
ˆˆ 22
n
MaxL y y X y X ßsßß
ps s
O
?? ? ?
' =- - -
??
? ?
?? ? ?
2
'
exp .
ˆˆ
2 2( )( )
n
nn
yX yX pß ß
??
??
=-
??
??
-- ??
??
Page 5

Analysis of Variance and Analysis of Variance and Analysis of Variance and Analysis of Variance and
Design Design of Experiments of Experiments- -I I
MODULE MODULE – – II II
g g p p
GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS
LECTURE  LECTURE  - -7 7
GENERAL LINEAR HYPOTHESIS GENERAL LINEAR HYPOTHESIS
AND ANAL YSIS OF VARIANCE AND ANAL YSIS OF VARIANCE
Dr. Shalabh
Department of Mathematics and Statistics
Indian Institute of Technology Kanpur
C 2 Tt f bt f t h d
0
:12 Hk ßß <
2
2
Case 2: Test of asubset of parameters when and
are unknown
0
0
: , 1, 2,..,
kk
Hk rp ßß = = <
12
, ,...,
rr p
ß ßß
++
2
s
In case 1, the test of hypothesis was developed when all ’s are considered in the sense that we test for each ß In case 1, the test of hypothesis was developed when all s are considered in the sense that we test for each
Now consider another situation, in which the interest is to test only a subset of i.e., not all but
only a few parameters. This type of test of hypothesis can be used e.g., in the following situation. Suppose five levels of
voltage are applied to check the rotations per minute (rpm) of a fan at 160 volts, 180 volts, 200 volts, 220 volts and 240 volts.
ß
0
,1,2,...,.
ii
ip ßß ==
12
, ,..., ,
p
ßßß
It can be realized in practice that when the voltage is low, then the rpm at 160, 180 and 200 volts can be observed easily. At
220 and 240 volts, the fan rotates at the full speed and there is not much difference in the rotations per minute at these
voltages. So the interest of the experimenter lies in testing the hypothesis related to only first three effects, viz., , for 160
volts, for 180 volts and for 200 volts.
1
ß
2
ß
3
ß
00 0 0
01 1 2 2 3 3
:, , H ß ßß ßß ß == =
The null hypothesis in this  case can be written as:
when  and        are unknown.
45
, ß ß
2
s
Note that under  case 1, the null hypothesis will be
00 0 0 0 0
0 1 1 2 23 3 4 45 5
:, , , , . H ß ßß ßß ßß ßß ß == = = =
Let be the p parameters.
We can divide them into two parts such that out of and we are interested in testing
12
, ,...,
p
ß ßß
12 1
, ,..., , ,..., ,
rr p
ß ßßß ß
+
3
the hypothesis of a subset of it.
Suppose, we want to test the null hypothesis when                         and       are unknown.
p
0
0
:,1,2,..,
kk
Hk rp ßß = =<
12
, ,...,
rr p
ß ßß
++
2
s
The alternative hypothesis under consideration is
In order to develop a test for  such a hypothesis, the linear model
under the usual assumptions can be rewritten as follows:
0
1
: 1, 2,.., . or at least one f
kk
Hkrp ßß ? =<
YX ß e =+
under the usual assumptions can be rewritten as follows:
Partition
(1)
12
(2)
(), XXX
ß
ß
ß
??
==
??
??
Th d l b itt
where
(1) 1 2 (2) 1 2
( , ,..., ) , ( , ,..., )
rrr p
ß ßß ß ß ß ß ß
++
'' ==
with the orders of matrices  as                                                       and
12 (1)
:, : ( ), : 1 Xn rX n prr ß × ×- ×
(2)
:( ) 1. pr ß - ×
The model can be rewritten as
(1)
12
(2)
()
YX
XX
ß e
ß
e
ß
=+
??
=+
??
??
(2)
1(1) 2 (2)
. XX
ß
ß ße
??
=+ +
The null hypothesis of interest is now
h d k
000 0
() H ßß ßß ß
ß
2
4
where and are unknown.
The complete parametric space is
000 0
0(1) (1) 1 2
: ( , ,..., )
r
H ßß ßß ß == (2)
ß
2
s
{ }
22
( , ); , 0, 1,2,..., ßs ß s O= - 8 < < 8 > =
i
ip
and sample space under  is
0
H
{ }
02 2
(1) (2)
( , , ); , 0, 1, 2,..., .
i
ir r p ?ß ß s ß s =-8<<8>=++
Th i l f lik lih d f ti d i bt i d b b tit ti th i lik lih d
The likelihood function is
2
2
22
11
(| , ) exp ( )( ) .
22
n
Ly yX yX ßsßß
ps s
?? ? ?
' =- - -
??
? ?
?? ? ?
O The maximum value of likelihood function under is obtained by substituting the maximum likelihood
estimates of   and      ,  i.e.,
O
ß
2
s
1
2
ˆ
()
1
ˆˆ
ˆ ()()
XX Xy
yX yX
ß
sß ß
-
'' =
'
as
()() yX yX
n
sß ß =- -
2
2
22
11
ˆˆ
(| , ) exp ( )( )
ˆˆ 22
n
MaxL y y X y X ßsßß
ps s
O
?? ? ?
' =- - -
??
? ?
?? ? ?
2
'
exp .
ˆˆ
2 2( )( )
n
nn
yX yX pß ß
??
??
=-
??
??
-- ??
??
Now we find the maximum value of likelihood function under . The  model under  becomes
Th lik lih d f ti d i
0
H
0
H
0
YX X ßß
5
. The likelihood function under is
0
1(1) 2 2
YX X ßß e =+ +
0
H
2
20 0
1(1) 2 (2) 1(1) 2 (2) 22
11
(| , ) exp ( )( )
22
11
n
n
Ly y X X y X X ßs ßß ßß
ps s
?? ? ?
' = - -- --
??
? ?
?? ? ?
?? ? ?
where                          Note that   and        are the unknown parameters. This likelihood function looks like as if it is
2
s
2
2(2) 2(2) 22
11
exp ( * ) ( * )
22
yX yX ßß
ps s
?? ? ?
' =- - -
??
??
?? ? ?
(0)
1(1)
*. yyX ß =-
(2)
ß
written  for
This helps is writing the maximum likelihood estimators of  and       directly as
2
2(2)
*~ ( , ). ß s yNX
2
s
(2)
ß
'1'
(2) 22 2
ˆ
() * XX X y ß
-
=
()
2
2 (2) 2 (2)
1
ˆˆ
ˆ (* )( * ). yX yX
n
sß ß ' =- -
Note that  is a principal minor of   Since  is a positive definite matrix, so is also positive
definite. Thus exists and is unique.
'
22
XX . XX ' XX '
'
22
XX
'1
22
() XX
-
definite.  Thus  exists and is unique.
Thus the maximum value of likelihood function under is obtained as
22
() XX
0
H
2
2
2(2) 2(2) 22
11
ˆˆˆ
(*| , ) exp ( * )( * )
ˆˆ 22
n
MaxL y y X y X
?
ßs ß ß
pss
?? ? ?
' =- - -
??
? ?
?? ? ?
2
2(2) 2(2)
exp.
ˆˆ
2 2(*)'(*)
n
nn
yX yX pß ß
?? ? ?
??
??
=- ??
??
??
-- ??
??
```
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