Sta113: Lab6 (Friday, February 28, 2003)

The task today is to graphically illustrate how the method of maximum likelihood works.

At the beginning of the lab, the TA will review the following contents: suppose we have n observations from a Exponential distribution with unknown parameter beta,

• What is the likelihood function in terms of beta?
• How to find the MLE?

% y records 10 observations from Exp(beta) where beta is unknown. 
y = [1.02, .49, .17, .64, 3.65, 1.89, .14, .18, 1.87, .24]; 
n=length(y);

subplot(1,2,1);
beta = 0.1:0.25:6; 
logL = -n.*log(beta) - sum(y)./beta; % calculate the Log Likelihood 
plot(beta, logL); % plot the Log Likelihood vs beta where beta is 
                  % ranged from 0.1 to 6. 
subplot(1,2,2);
beta = 0.5:0.01:2; % From the previous plot, we are sure that the
                   % maximum occurs when beta is between 0.5 and 
                   % 2. So we plot the Log Likelihood in this range
                   % with finer grid. 
logL = -n.*log(beta) - sum(y)./beta;
plot(beta, logL);
xlabel('beta');
ylabel('Log Likelihood');
mle=mean(y);       
logL_mle = -n*log(mle) - sum(y)/mle; hold on;
plot(mle, logL_mle, 'or');  hold off;
% mark the mle in red circle. See it does maximize the Log Likelihood. 
legend('Plot of Log likelihood Function', 'MLE');

Tail, Head, Tail, Tail, Head, Head, Head, Tail, Head, Head

• Graphically show me how the Likelihood (or Log Likelihood) of the sample looks like when p varies, that is, draw the Likelihood (or Log Likelihood) vs p. If you work with Log Likelihood, can you allow p to take values like 0 or 1?
• Based on the previous plot, guess the range of p where the maximum occurs and draw another plot of Likelihood (or Log Likelihood) vs p with p in this smaller range. Mark the mle on the curve (Recall that mle of p is equal to (y_1 + y_2 + ...+y_n)/n). Is the mle located at the peak of the curve?