STAT 330 Lecture 13

Reading for Today's Lecture: 4.4, 6.1, 6.2

Goals of Today's Lecture:

• Learn the method of moments.
• Learn to manipulate the Gamma distribution and density.
• Introduce the method of maximum likelihood.

Today's notes

The Gamma distribution and gamma function:

Definition:

Properties:

and, if n is an integer

Note:

Stirling's formula

The function

is called the Gamma density with scale parameter and shape parameter . (Note that and are required.)

Some sketches of Gamma densities ( )

Exponential density .

Gamma density with , .

Gamma density with .

Gamma density with large (approximately normal).

Method of moments example: example 6.12 in text. (Note: most examples lead either to trivial equations or to very difficult equations to solve or to very hard integrals to do; that's why I am duplicating the example in the text.)

Suppose are iid with density

Population moments

First moment

Make the substitution and to get

where we have used the recursive property of the Gamma function .

Second moment

Make the substitution and to get

To get method of moments estimates of and we solve the equations:

and

Divide the second equation by the square of the first to get

so that

Then

Notice the use of the bar over the top of to indicate the average of the squares; this is quite different than averaging and then squaring.

Maximum Likelihood Estimation

New treatment for disease tried until fourth success. Resulting data: SFFSSS.

Problem: estimate .

Likelihood:

which depends on the unknown parameter p.

Idea: to estimate p make L(p) as large as possible, that is choose as your estimate of p the one which assigns the largest possible probability to the data you observed.

Solution: set

and get the equation

or

or

so that

The Maximum Likelihood Estimate (MLE) of p is