Inference for the mean vector

- one-sample (and two-samples)

– Roy’s Union-Intersection Test

& a review of the LR test

1

Univariate Inference

Let x1, x2, … , xn denote a sample of n from the normal distribution with mean m and variance s2 (unknown).

Suppose we want to test

H0: m = m0 vs

HA: m ≠ m0

The appropriate test is the t test:

The test statistic:

Reject H0 if |t| > ta/2

0xt ns

m

2

Univariate Inference

Let x1, x2, … , xn denote a sample of n from the normal distribution with mean m and variance s2 (known).

Suppose we want to test

H0: m = m0 vs

HA: m ≠ m0

The appropriate test is the t test:

The test statistic: 𝑍 = 𝑛𝑋 −𝜇0

𝜎

Reject H0 if |Z| > Za/2 3

The multivariate Test

Let denote a sample of n from the p-variate

normal distribution with mean vector and covariance

matrix S.

Suppose we want to test

1 2, , , nx x xm

0 0

0

: vs

:A

H

H

m m

m m

4

Roy’s Union- Intersection Principle

This is a general procedure for developing a

multivariate test from the corresponding univariate test.

1

i.e. observation vector

p

X

X

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

5

2. Perform the test for the arbitrary linear combination of the

observation vector.

3. Repeat this for all possible choices of

1

p

a

a

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

.a

6

Let denote a sample of n from the p-variate

normal distribution with mean vector and covariance

matrix S where S is known.

Suppose we want to test

1 2, , , nx x xm

0 0

0

: vs

:A

H

H

m m

m m

Application of Roy’s principle to the following situation

Then y1, …. yn is a sample of n from the normal

distribution with mean and variance 𝜎𝑦2 = a m a aΣ

7

Let 𝑦𝑖 = 𝑎 ′𝑥 𝑖 = 𝑎1𝑥1𝑖 + ⋯ + 𝑎𝑝𝑥𝑝𝑖

to test

0 0

0

: vs

:

a

a

A

H a a

H a a

m m

m m

we would use the test statistic:

1 1

1 1n n

i i

i i

a x a x a xn n

8

𝑍𝑎 = 𝑛 𝑦 −𝑎′𝜇0

𝜎𝑦

where 𝑦

We will reject

0 0 0: in favour of :AH Hm m m m

Using Roy’s Union- Intersection principle:

2

2 0 2

/ 2if for at least one a

n a xt t a

a a a

m

S

We accept (fail to reject) 0 0:H m m

2

2 0 2

/ 2if for all a

n a xt t a

a a a

m

S

9

Σ

Σ 𝑍

𝑍 𝑍

𝑍

We reject

0 0:H m m

i.e.

2

0 2

/ 2if max

a

n a xt

a a a

m

S

We accept 0 0:H m m

2

0 2

/ 2if max

a

n a xt

a a a

m

S

10

Σ

Σ

𝑍

𝑍

Consider the problem of finding:

2

0

max maxa a

n a xh a

a a

m

Swhere

2

0 0 0n a x a x x a

h a na a a a

m m m S S

0 0 0 0

2

2 2

0

a a x x a a x x a ah a

na a a

m m m m

S S

S

0 0or a a x a x am m S S11

Σ

Σ Σ

Σ Σ

Σ

Σ Σ

thus

2

0

maxopt

aopt opt

n a xh a

a a

m

S

1 10 0

0

or opta a

a x k x aa x

m mm

SS S

2

1

0 0

2 1 1

0 0

n k x x

k x x

m m

m m

S

S SS

10 0n x xm m

S

12

Σ Σ Σ

Σ

Σ

Σ Σ Σ

Σ

We reject 0 0:H m m

Thus Roy’s Union- Intersection principle states:

1 20 0 / 2if n x x tam m S

We accept 0 0:H m m

1 20 0 / 2if n x x tam m S

2 10 0The statistic T n x xm m

S

follows 𝜒2 with p degrees of freedom when 𝐻0 is true 13

Σ

Σ

Σ

Choosing the critical value for Hotelling’s T2 statistic

𝜒2

𝑝, 𝛼

𝜒2

𝑝, 𝛼

f x

Another derivation of the 𝜒2statistic

Another method of developing statistical tests is

the Likelihood ratio method (as we have already

learned).

Suppose that the data vector, , has joint density x

Suppose that the parameter vector, , belongs to

the set W. Let w denote a subset of W.

Finally we want to test 0 : vs

:A

H

H

w

w

14

ˆ̂

max max

ˆmaxmax

Lf x L

Lf x L

w wa

WW

The Likelihood ratio test rejects H0 if

ˆwhere the MLE of

0

ˆ̂and the MLE of when is true.H

15

The situation

Let denote a sample of n from the p-variate

normal distribution with mean vector and covariance

matrix S (known).

Suppose we want to test

1 2, , , nx x xm

0 0

0

: vs

:A

H

H

m m

m m

16

The Likelihood function is:

1

1

1

2

/ 2 / 2

1, e

2

n

i i

i

x x

np nL

m m

m

S S

S

and the Log-likelihood function is:

, ln , l Lm mS S

11

1ln 2 ln

2 2 2

n

i i

i

np nx x m m

S S

17

the unrestricted Maximum Likelihood estimators of

is

1

1ˆ n

i

i

x xn

m

18

𝜇

the Maximum Likelihood estimators of

when H 0 is true are:

19

𝜇

𝜇 = 𝜇 0