View by Category

Loading...

PPT – Maximum Likelihood Estimation PowerPoint presentation | free to view - id: 1ae90e-Zjc2Z

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

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

(No Transcript)

(No Transcript)

(No Transcript)

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

(No Transcript)

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

(No Transcript)

(No Transcript)

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

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Home About Us Terms and Conditions Privacy Policy Contact Us Send Us Feedback

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Maximum Likelihood Estimation" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!