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Maximum Likelihood Estimation

- Multivariate Normal distribution

The Method of Maximum Likelihood

- Suppose that the data x1, , xn has joint

density function - f(x1, , xn q1, , qp)
- where q (q1, , qp) are unknown parameters

assumed to lie in W (a subset of p-dimensional

space). - We want to estimate the parametersq1, , qp

Definition The Likelihood function

- Suppose that the data x1, , xn has joint

density function - f(x1, , xn q1, , qp)
- Then given the data the Likelihood function is

defined to be - L(q1, , qp)
- f(x1, , xn q1, , qp)
- Note the domain of L(q1, , qp) is the set W.

Definition Maximum Likelihood Estimators

- Suppose that the data x1, , xn has joint

density function - f(x1, , xn q1, , qp)
- Then the Likelihood function is defined to be
- L(q1, , qp)
- f(x1, , xn q1, , qp)
- and the Maximum Likelihood estimators of the

parameters q1, , qp are the values that

maximize - L(q1, , qp)

- i.e. the Maximum Likelihood estimators of the

parameters q1, , qp are the values

Such that

Note

is equivalent to maximizing

the log-likelihood function

The Multivariate Normal Distribution

- Maximum Likelihood Estiamtion

denote a sample (independent)

Let

from the p-variate normal distribution

with mean vector

and covariance matrix

Note

The matrix

is called the data matrix.

The vector

is called the data vector.

The mean vector

The vector

is called the sample mean vector

note

also

In terms of the data vector

where

Graphical representation of sample mean vector

The sample mean vector is the centroid of the

data vectors.

The Sample Covariance matrix

The sample covariance matrix

where

There are different ways of representing sample

covariance matrix

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Maximum Likelihood Estimation

- Multivariate Normal distribution

denote a sample (independent)

Let

from the p-variate normal distribution

with mean vector

and covariance matrix

Then the joint density function of

is

The Likelihood function is

and the Log-likelihood function is

To find the Maximum Likelihood estimators of

we need to find

to maximize

or equivalently maximize

Note

thus

hence

Now

Now

Summary the Maximum Likelihood estimators of

are

and

Sampling distribution of the MLEs

Note

is

The joint density function of

This distribution is np-variate normal with mean

vector

Thus the distribution of

is p-variate normal with mean vector

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Summary

- The sampling distribution of
- is p-variate normal with

The sampling distribution of the sample

covariance matrix S and

The Wishart distribution

- A multivariate generalization of the c2

distribution

Definition the p-variate Wishart distribution

be k independent random p-vectors

- Let

Each having a p-variate normal distribution with

Then U is said to have the p-variate Wishart

distribution with k degrees of freedom

The density ot the p-variate Wishart distribution

- Suppose

Then the joint density of U is

where Gp() is the multivariate gamma function.

It can be easily checked that when p 1 and S

1 then the Wishart distribution becomes the c2

distribution with k degrees of freedom.

Theorem

- Suppose

then

Corollary 1

Corollary 2

Proof

Theorem

- Suppose

are independent, then

Theorem

are independent and

Suppose

then

Theorem

Let

be a sample from

then

Theorem

Let

be a sample from

then

Theorem

Proof

etc

Theorem

Let

be a sample from

then

is independent of

Proof

be orthogonal

Then

Note H is also orthogonal

Properties of Kronecker-product

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This the distribution of

is np-variate normal with mean vector

Thus the joint distribution of

is np-variate normal with mean vector

Thus the joint distribution of

is np-variate normal with mean vector

Summary Sampling distribution of MLEs for

multivatiate Normal distribution

Let

be a sample from

then

and

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