PROBABILITY DISTRIBUTIONS

AND

ESTIMATING A MEAN AND VARIANCE USING MLE

The purpose of this session
is to familiarize you with some of the more important probability distributions
and to take an initial step in understanding how probability distributions
relate to likelihood and log likelihood functions.

PROBABILITY DISTRIBUTIONS:
Recall that a probability distribution is a mapping from a random variable
(call it Y) to the probability of having observed that random variable (P[Y]).
Maximum Likelihood Estimation (MLE) involves specifying a probability
distribution that describes the social science experiment that generated the
random variable under investigation. Of course, the data generation process may
have been consistent with more than one probability distribution since it is
possible to derive some distributions from others.

The first part of this
assignment is to go to Microsoft Excel and call up various spreadsheets that
plot some of the more interesting univariate probability
distributions. You should have the following spreadsheets for this exercise.

Normal Distribution

Chi-Square Distribution

Bernoulli Distribution

Binomial Distribution

Negative Binomial Distribution

Poisson Distribution

Weibull Distribution

Exponential Distribution

Gamma Distribution

LogNormal Distribution

Beta Distribution

Note that there are two spreadsheets for the Gamma and Beta distributions to
illustrate the different ways in which these may be parameterized. In general,
there are many possible parameterizations of most probability distributions.

For each of these
spreadsheets, the cells marked green are areas where you may change the
parameters and observe the behavior of the resulting distributions. (Note: You
may need to resize or reposition the graphs because of the different settings
and different versions of Excel that are around.) In any case, play with each
of these spreadsheets, noting the essential features of each probability
distribution, including the range of the random variable, the range of the
parameters, and the effect of the parameters on means, modes, skew, kurtosis,
and shape. The Evans, Hastings, and Peacock text contains many more details on
each of these distributions, so you might use this reference in combination
with the spreadsheets to learn more about the probability functions. As an
exercise, you might also try on your own to graph one or more of the other
probability distributions in the Evans, Hastings, and Peacock book.

The probability distributions in these spreadsheets are univariate
distributions, meaning that there is a single random variable in the domain of
the probability function. However, social scientists are also sometimes
interested in the joint probability associated with multiple random variables.
For example, we sometimes assume a bivariate normal
distribution when there are two dependent variables, both normal, to be modeled
simultaneously. Another example, the multinomial distribution is a discrete
joint distribution with dimensions equal to the number of categories in the
multinomial variable. Excel doesn't do so well in plotting multivariate
distributions, particularly when they are continuous. However, Maple can do the
job. Use Maple to look at the bivariate normal
distribution contained in the file called Bivariate
Normal.mws. Change r (the correlation between the two random variables), s_{1},
s_{2}, m_{1}, and m_{2} to observe the effect on the
distribution.

ESTIMATING A MEAN AND
VARIANCE OF A DISTRIBUTION USING MLE: Maximum likelihood is purely and simply
an estimation technique. In practice, we specify a probability distribution
that could have generated the data, put that probability distribution into a
likelihood and log-likelihood function, and then estimate the parameters of
that distribution using MLE. One approach to implementing MLE would be through
trial and error. As an example of how this might work, go to the Excel
spreadsheet entitled MLNormal Mu.xls. In this
spreadsheet I have entered the data on page 9 of Eliason
in the second column. In the first column I have a vector of initial guesses
for the mean of the distribution. In the third column is the log-likelihood
associated with each initial guess, assuming independent draws from a normal
distribution. (Note: We could have used any of the distributions above in
computing the log-likelihoods). The MLE estimate of the mean is just the guess
that produces the largest number in column 3. The graph plots the values of the
log-likelihood function in column 3 against the vector of guesses in column 1.
We can also look at the graph to find the maximum. If we want to increase the
precision of estimation, then we can change the vector of guesses to range from
say 1.81 through 1.9. Do this to get a sense of what happens to the
log-likelihoods and graph.

Of course, trial and error
methods are very inefficient and may also be quite cumbersome when there are
multiple parameters. Thus, a better way to do estimation is using the methods
of calculus or iterative techniques. In class we show how to optimize a
function using both analytical and numerical methods. Computers are very useful
in implementing the latter. Below is a short STATA program for finding the mean
and variance of a normally distributed variable using MLE.

Implement this program to become initially familiar with STATA's ML features.
STATA has several different ML estimation routines. The one below is an example of the LNF
procedure. We shall demonstrate others
in future lessons.

/* This file demonstrates maximum likelihood
estimation of normal models. We compute the mean and variance using maximum
likelihood methods */

clear

/* The
next line will read a data file. Change the path to find data */

use "ostrom.dta", clear

/* The
next line sets the sample for observations 1 through 22 */

gen id = _n

keep if (1<=id<=22)

/* Now let's print the data and
compute descriptive statistics */

list year us ussr

summarize year us ussr,
detail

correlate year us ussr

/* Now lets
compute the mean and standard deviation of US using

maximum likelihood. First define a program to calculate the mean
*/

program define meanest

version 7

args lnf theta1 theta2

quietly replace
`lnf'=-ln(sqrt(2*_pi*`theta2'^2))-1/(2*`theta2'^2)*($ML_y1-`theta1')^2

end

/* Now call the program using ml
for estimation. */

ml model lf meanest (mean:us=) (sigma:)

ml search

ml init 140 79, copy

ml report

ml plot _cons

ml maximize

ml graph

/* The MLE of the standard
deviation (and variance) is biased.

We can get the unbiased estimate
by multiplying the variance by

n/n-k */

matrix list e(b)

scalar mean=[mean]_cons

display mean

scalar sig=[sigma]_cons

display sig

scalar sigunb=sqrt([sigma]_cons^2*(_N/(_N-1)))

display sigunb

/* Delete the program and exit
*/

program drop meanest

exit

clear