% LMEx050201_4th.m % An example of binomial MLE fitting & a difference in the way % Mathworks & Larsen & Marx define the geometric distribution % From Larsen & Marx (2006). Introduction to Mathematical Statistics, % Fourth Edition. page 348-349 % Written by Eugene.Gallagher@umb.edu 10/28/10; revised 3/3/11 % Tom Lane on 10/29/10: "unfortunately it looks like your text and MATLAB % use different definitions for the [geometric] distribution. Our version % has positive probability on 0,1,2,.... Yours starts at 1. The version we % use is the one described in "Univariate Discrete Distributions" by % Johnson, Kotz, and Kemp. Wikipedia shows both versions. X=[3 2 1 3]; [Phat,PCI] =mle(X,'distribution','geometric') % Matlab gives the wrong answer, because it uses a different definition % of the geometric distribution, defined for k=0, 1, 2, 3 ... inf % Larsen & Marx use a form defined for positive k: k=1, 2, 3, ... inf % Larsen and Marx define the geometric distribution as the number of % trials before a success, so k=0 is not part of the domain, but Mathworks % defines the geometric distribution as the number of failures % before a success, allowing k=0 to be defined. % The correct MLE for L & M is 4/9 % 10/29 email from Tom Lane, Mathworks, says that I should call % mle using [Phat,PCI] =mle(X-1,'distribution','geometric'); format rat; fprintf('The maximum likelihood estimate for p is\n') disp(Phat) format fprintf('with 95%% CIs: [%5.3f %5.3f]\n',PCI); % Just following along the text's derivation (p. 348) of the MLE: syms p; s=diff(5*log(1-p)+4*log(p),p) solve(s,p) % find the second derivative and plot; s2=diff(diff(5*log(1-p)+4*log(p),p)) % This plot shows that the second derivative is negative for all 0