Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from...
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![Page 1: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/1.jpg)
Inference for the mean vector
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Univariate InferenceLet x1, x2, … , xn denote a sample of n from the normal distribution with mean and variance 2.Suppose we want to test
H0: = 0 vsHA: ≠ 0
The appropriate test is the t test:The test statistic:
Reject H0 if |t| > t/2
0xt ns
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The multivariate TestLet denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix .Suppose we want to test
1 2, , , nx x x
0 0
0
: vs:A
HH
![Page 4: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/4.jpg)
Roy’s Union- Intersection PrincipleThis is a general procedure for developing a multivariate test from the corresponding univariate test.
1
i.e. observation vector
p
XX
X
1. Convert the multivariate problem to a univariate problem by considering an arbitrary linear combination of the observation vector.
1 1 p pU a X a X a X
arbitrary linear combination of the observations
![Page 5: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/5.jpg)
2. Perform the test for the arbitrary linear combination of the observation vector.
3. Repeat this for all possible choices of
1
p
aa
a
4. Reject the multivariate hypothesis if H0 is rejected for any one of the choices for
5. Accept the multivariate hypothesis if H0 is accepted for all of the choices for
6. Set the type I error rate for the individual tests so that the type I error rate for the multivariate test is .
.a
.a
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Let denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix .Suppose we want to test
1 2, , , nx x x
0 0
0
: vs:A
HH
Application of Roy’s principle to the following situation
1 1Let i i i p piu a x a x a x
Then u1, …. un is a sample of n from the normal distribution with mean and variance .a a aΣ
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to test
0 0
0
: vs
:
a
aA
H a a
H a a
we would use the test statistic:
0a
u
u at ns
1 1
1 1Now n n
i ii i
u u a xn n
1 1
1 1n n
i ii i
a x a x a xn n
![Page 8: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/8.jpg)
and
222
1 1
1 11 1
n n
u i ii i
s u u a x a xn n
2
1
11
n
ii
a x xn
1
11
n
i ii
a x x x x an
1
11
n
i ii
a x x x x a a an
S
![Page 9: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/9.jpg)
Thus
00
a a x a nt n a xa aa a
SS
We will reject 0 0:aH a a
if 0 / 2a nt a x t
a a
S
2
2 0 2/ 2
or an a x
t ta a
S
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We will reject
0 0 0: in favour of :AH H
Using Roy’s Union- Intersection principle:
2
2 0 2/ 2
if for at least one an a x
t t aa a
S
We accept 0 0:H
2
2 0 2/ 2
if for all an a x
t t aa a
S
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We reject
0 0:H
i.e.
2
0 2/ 2
if max a
n a xt
a a
S
We accept 0 0:H
2
0 2/ 2
if maxa
n a xt
a a
S
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Consider the problem of finding:
2
0max max
a a
n a xh a
a a
S
where
2
0 0 0n a x a x x ah a n
a a a a
S S
0 0 0 0
2
2 20
a a x x a a x x a ah an
a a a
S S
S
0 0or a a x a x a S S
![Page 13: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/13.jpg)
thus 2
0max
opt
aopt opt
n a xh a
a a
S
1 10 0
0
or opta aa x k x a
a x
S S S
21
0 0
2 1 10 0
n k x x
k x x
S
S SS
10 0n x x S
![Page 14: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/14.jpg)
We reject 0 0:H Thus Roy’s Union- Intersection principle states:
1 20 0 / 2
if n x x t
S
We accept 0 0:H
1 20 0 / 2
if n x x t
S
2 10 0The statistic T n x x S
is called Hotelling’s T2 statistic
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We reject 0 0:H Choosing the critical value for Hotelling’s T2 statistic
2 1 20 0 / 2
if T n x x t
S
2/ 2
To determine t , we need to find the sampling
distribution of T2 when H0 is true.
It turns out that if H0 is true than
2 1
0 0 1 1
n p nn pF T x xp n p n
S
has an F distribution with 1 = p and 2 = n - p
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We reject 0 0:H
ThusHotelling’s T2 test
2 1 20 0
1, a
p nT n x x F p n p T
n p
S
2 ,1
n pF T F p n pp n
or if
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f x
Another derivation of Hotelling’s T2 statistic
Another method of developing statistical tests is the Likelihood ratio method.
Suppose that the data vector, , has joint densityx
Suppose that the parameter vector, , belongs to the set . Let denote a subset of .
Finally we want to test 0 : vs
:A
H
H
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ˆ̂max max
ˆmaxmax
Lf x L
Lf x L
The Likelihood ratio test rejects H0 if
ˆwhere the MLE of
0
ˆ̂and the MLE of when is true.H
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The situationLet denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix .Suppose we want to test
1 2, , , nx x x
0 0
0
: vs:A
HH
![Page 20: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/20.jpg)
The Likelihood function is:
1
1
12
/ 2 / 2
1, e 2
n
i ii
x x
np nL
and the Log-likelihood function is:
, ln , l L
1
1
1ln 2 ln 2 2 2
n
i ii
np n x x
![Page 21: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/21.jpg)
and
the Maximum Likelihood estimators of
are
1
1ˆ n
ii
x xn
and
1
1 1ˆ n
i ii
nx x x x Sn n
![Page 22: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/22.jpg)
and the Maximum Likelihood estimators of
when H 0 is true are:
0ˆ̂ ˆ
and
0 01
1ˆ̂ n
i ii
x xn
![Page 23: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/23.jpg)
The Likelihood function is:
1
1
12
/ 2 / 2
1, e 2
n
i ii
x x
np nL
now
11 1
1 1
ˆ ˆˆ n n
ni i i in
i i
x x x x S x x
11
1
n
ni in
i
tr x x S x x
11
1
n
ni in
i
tr S x x x x
![Page 24: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/24.jpg)
11
1
n
ni in
i
tr S x x x x
1 11 = 1 = n nn ntr n I n p np
Thus 2
/ 2/ 2 1
1ˆ ˆ, 2
np
nnp nn
L eS
similarly
2/ 2
/ 2
1ˆ ˆˆ ˆ, ˆ̂2
np
nnp
L e
![Page 25: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/25.jpg)
and
/ 2 / 21 1
/ 2 / 2
0 01
ˆ ˆˆ ˆ,
ˆ ˆ ˆ, 1ˆ
n nn nn n
n nn
i ii
L S S
Lx x
n
/ 2
/ 2
0 01
1
n
nn
i ii
n S
x x
![Page 26: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/26.jpg)
Note:11 12
21 22
A A u wA
A A w V
Let
111 22 21 11 12
122 11 12 22 21
A A A A AA
A A A A A
1
1u V wwu
V u w V w
11Thus u V ww V u w V wu
![Page 27: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/27.jpg)
and1
1
1V ww
w V wuV u
/ 2
/ 2
0 01
1
n
nn
i ii
n S
x x
Now
and
2/
0 01
1 n
n
i ii
n S
x x
![Page 28: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/28.jpg)
Also
0 0 0 01 1
= n n
i i i ii i
x x x x x x x x
01 1
=n n
i i ii i
x x x x x x x
0 0 01
n
ii
x x x n x x
0 01
=n
i ii
x x x x n x x
0 01
=n
i ii
x x x x n x x
0 0= 1n S n x x
![Page 29: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/29.jpg)
Thus
2/
0 01
1 n
n
i ii
n S
x x
0 0
1
1
n S
n S n x x
0 0
1
SnS x x
n
![Page 30: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/30.jpg)
Thus 0 02/ 1 n
nS x xn
S
using 11
1V ww
w V wuV u
0
1, and
u nV S
w n x
![Page 31: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/31.jpg)
Then 10 02/ 1
1n
n x S x
n
Thus to reject H0 if < 2/i.e. n n
2/or n n
10 0and 1
1n
n x S x
n
10 0or 1 -1 nn x S x n
This is the same as Hotelling’s T2 test if
2/ 11 -1 , n p n
n T F p n pn p
![Page 32: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/32.jpg)
Example
For n = 10 students we measure scores on – Math proficiency test (x1),
– Science proficiency test (x2),
– English proficiency test (x3) and
– French proficiency test (x4)
The average score for each of the tests in previous years was 60. Has this changed?
![Page 33: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/33.jpg)
The data
Student Math Science Eng French1 81 89 73 742 73 79 73 743 61 86 81 814 55 70 76 735 61 71 61 666 52 70 56 587 56 74 56 568 65 87 73 699 54 76 69 72
10 48 71 62 63
![Page 34: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/34.jpg)
Summary Statistics60.677.368.068.6
x
S
102.044 56.689 41.222 39.48956.689 56.456 42.000 35.35641.222 42.000 75.778 65.11139.489 35.356 65.111 61.378
0.0245 -0.0255 0.0195 -0.0218-0.0255 0.0567 -0.0405 0.02670.0195 -0.0405 0.1782 -0.1783-0.0218 0.0267 -0.1783 0.2040
1
: S
Note
2 10 0 151.135T n x S x
0.05 0.05 0.05
1 4 9 4 9, 4,6 = 4.53 27.18
6 6p n
T F p n p Fn p
0
60606060
![Page 35: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/35.jpg)
Simultaneous Inference for means
Recall
2 1T n x S x
2
21max max
a a
n a x at a
a S a
(Using Roy’s Union Intersection Principle)
![Page 36: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/36.jpg)
Now 2 1P T T P n x S x T
2
1maxa
n a x aP T
a S a
2
1 for all n a x a
P T aa S a
12
for all a S aP a x a T an
1
![Page 37: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/37.jpg)
Thus1 1
for all a S a a S aP a x T a a x T an n
1
and the set of intervals
1 1
to a S a a S aa x T a x Tn n
Form a set of (1 – )100 % simultaneous confidence intervals for a
![Page 38: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/38.jpg)
Recall
,-1= p n pn p
T Fn p
1,-1
p n pn pa S aa x Fn n p
Thus the set of (1 – )100 % simultaneous confidence intervals for a
1,-1
to p n pn pa S aa x Fn n p
![Page 39: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/39.jpg)
The two sample problem
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Univariate InferenceLet x1, x2, … , xn denote a sample of n from the normal distribution with mean x and variance 2.
Let y1, y2, … , ym denote a sample of n from the normal distribution with mean y and variance 2.
Suppose we want to testH0: x = y vs
HA: x ≠ y
![Page 41: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/41.jpg)
The appropriate test is the t test:
The test statistic:
Reject H0 if |t| > t/2 d.f. = n + m -2
1 1pooled
x yts
n m
2 21 12
x ypooled
n s m ss
n m
![Page 42: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/42.jpg)
The multivariate TestLet denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix .
1 2, , , nx x x
x
0 : vs
:x y
A x y
H
H
Suppose we want to test
Let denote a sample of m from the p-variate normal distribution with mean vector and covariance matrix .
1 2, , , my y y
y
![Page 43: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/43.jpg)
Hotelling’s T2 statistic for the two sample problem
2 111 1 pooledT x y x y
n m
S
if H0 is true than
21
2n m pF Tp n m
has an F distribution with 1 = p and
2 = n +m – p - 1
1 12 2pooled x y
n mn m n m
S S S
![Page 44: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/44.jpg)
We reject 0 : x yH
ThusHotelling’s T2 test
21if , 12
n m pF T F p n m pp n m
2 11with 1 1 pooledT x y x y
n m
S
1 12 2pooled x y
n mn m n m
S S S
![Page 45: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/45.jpg)
Simultaneous inference for the two-sample problem
• Hotelling’s T2 statistic can be shown to have been derived by Roy’s Union-Intersection principle
2 11namely 1 1 pooledT x y x y
n m
S
2
2max max1 1a a
pooled
a x yt a
a an m
S
where x y
![Page 46: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/46.jpg)
Thus
211 , 12
n m pP F T F p n m pp n m
2 2, 1
1p n m
P T F p n m pn m p
2P T T
2where , 1
1p n m
T F p n m pn m p
![Page 47: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/47.jpg)
Thus
2
max 11 1a
pooled
a x yP T
a an m
S
2
or for all 11 1
pooled
a x yP T a
a an m
S
![Page 48: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/48.jpg)
Thus
2 1 1 for all 1pooledP a x y T a a an m
S
Hence
1 1pooled x yP a x y T a a a
n m
S
1 1 for all 1pooleda x y T a a an m
S
![Page 49: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/49.jpg)
Thus
form 1 – simultaneous confidence intervals for
1 1pooleda x y T a a
n m S
x ya
![Page 50: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/50.jpg)
Example Annual financial data are collected for firms approximately 2 years prior to bankruptcy and for financially sound firms at about the same point in time. The data on the four variables
• x1 = CF/TD = (cash flow)/(total debt), • x2 = NI/TA = (net income)/(Total assets), • x3 = CA/CL = (current assets)/(current liabilties, and • x4 = CA/NS = (current assets)/(net sales) are given in
the following table.
![Page 51: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/51.jpg)
The data are given in the following table: Bankrupt Firms Nonbankrupt Firms x1 x2 x3 x4 x1 x2 x3 x4 Firm CF/TD NI/TA CA/CL CA/NS Firm CF/TD NI/TA CA/CL CA/NS 1 -0.4485 -0.4106 1.0865 0.4526 1 0.5135 0.1001 2.4871 0.5368 2 -0.5633 -0.3114 1.5314 0.1642 2 0.0769 0.0195 2.0069 0.5304 3 0.0643 0.0156 1.0077 0.3978 3 0.3776 0.1075 3.2651 0.3548 4 -0.0721 -0.0930 1.4544 0.2589 4 0.1933 0.0473 2.2506 0.3309 5 -0.1002 -0.0917 1.5644 0.6683 5 0.3248 0.0718 4.2401 0.6279 6 -0.1421 -0.0651 0.7066 0.2794 6 0.3132 0.0511 4.4500 0.6852 7 0.0351 0.0147 1.5046 0.7080 7 0.1184 0.0499 2.5210 0.6925 8 -0.6530 -0.0566 1.3737 0.4032 8 -0.0173 0.0233 2.0538 0.3484 9 0.0724 -0.0076 1.3723 0.3361 9 0.2169 0.0779 2.3489 0.3970 10 -0.1353 -0.1433 1.4196 0.4347 10 0.1703 0.0695 1.7973 0.5174 11 -0.2298 -0.2961 0.3310 0.1824 11 0.1460 0.0518 2.1692 0.5500 12 0.0713 0.0205 1.3124 0.2497 12 -0.0985 -0.0123 2.5029 0.5778 13 0.0109 0.0011 2.1495 0.6969 13 0.1398 -0.0312 0.4611 0.2643 14 -0.2777 -0.2316 1.1918 0.6601 14 0.1379 0.0728 2.6123 0.5151 15 0.1454 0.0500 1.8762 0.2723 15 0.1486 0.0564 2.2347 0.5563 16 0.3703 0.1098 1.9914 0.3828 16 0.1633 0.0486 2.3080 0.1978 17 -0.0757 -0.0821 1.5077 0.4215 17 0.2907 0.0597 1.8381 0.3786 18 0.0451 0.0263 1.6756 0.9494 18 0.5383 0.1064 2.3293 0.4835 19 0.0115 -0.0032 1.2602 0.6038 19 -0.3330 -0.0854 3.0124 0.4730 20 0.1227 0.1055 1.1434 0.1655 20 0.4875 0.0910 1.2444 0.1847 21 -0.2843 -0.2703 1.2722 0.5128 21 0.5603 0.1112 4.2918 0.4443 22 0.2029 0.0792 1.9936 0.3018 23 0.4746 0.1380 2.9166 0.4487 24 0.1661 0.0351 2.4527 0.1370 25 0.5808 0.0371 5.0594 0.1268
![Page 52: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/52.jpg)
Hotelling’s T2 test
A graphical explanation
![Page 53: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/53.jpg)
Hotelling’s T2 statistic for the two sample problem
2 111 1 pooledT x y x y
n m
S
1 1where 2 2pooled x y
n mn m n m
S S S
![Page 54: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/54.jpg)
2
2 2max max1 1a a
pooled
a x yT t a
a an m
S
: 1 1
pooled
a x a yt aa a
n m
Note
S
is the test statistic for testing:
0 : vs :x y A x yH a a a H a a a
![Page 55: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/55.jpg)
Popn A
Popn B
X1
X2
Hotelling’s T2 test
![Page 56: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/56.jpg)
Popn A
Popn B
X1
X2
Univariate test for X1
![Page 57: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/57.jpg)
Popn A
Popn B
X1
X2
Univariate test for X2
![Page 58: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/58.jpg)
Popn A
Popn B
X1
X2
Univariate test for a1X1 + a2X2
![Page 59: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/59.jpg)
Mahalanobis distance
A graphical explanation
![Page 60: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/60.jpg)
22
1
,p
i ii
d a b a b a b a b
Euclidean distance
a
points equidistantfrom a
![Page 61: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/61.jpg)
2 ,Md a b a b a b
Mahalanobis distance: , a covariance matrix
a
points equidistantfrom a
![Page 62: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/62.jpg)
Hotelling’s T2 statistic for the two sample problem
2 1 21 1 , ,pooled M pooledT x y x y d x yn m
S S
2 111 1 pooledT x y x y
n m
S
1pooled
nm x y x yn m
S
2 , ,M pooledn md x ynm
S
![Page 63: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/63.jpg)
Popn A
Popn B
X1
X2
Case I
![Page 64: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/64.jpg)
Popn A
Popn B
X1
X2
Case II
![Page 65: Inference for the mean vector. Univariate Inference Let x 1, x 2, …, x n denote a sample of n from the normal distribution with mean and variance](https://reader034.fdocuments.net/reader034/viewer/2022052606/5a4d1ad27f8b9ab059971e5f/html5/thumbnails/65.jpg)
Popn A
Popn B
X1
X2
Case IPopn A
Popn B
X1
X2
Case II
In Case I the Mahalanobis distance between the mean vectors is larger than in Case II, even though the Euclidean distance is smaller. In Case I there is more separation between the two bivariate normal distributions