Maximum Likelihood Estimation

TomSym implementation of GAMS Example (LIKE,SEQ=25)

This application from the biomedical area tests the hypothesis that a population of systolic blood pressure can be separated into three distinct groups.

Bracken, J, and McCormick, G P, Chapter 8.5. In Selected Applications of Nonlinear Programming. John Wiley and Sons, New York, 1968, pp. 90-92.

i: Observations (1-31)

g: Groups (one, two, three)

% Systolic blood pressure data
pressure = [95 105 110 115 120 125 130 135 140 145 150 155 160 165 170 ...
    175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 260]';
frequency = [1   1   4   4  15  15  15  13  21  12  17   4  20   8  17 ...
    8   6   6   7   4   3   3   8   1   6   0   5   1   7   1   2]';

y = pressure;
w = frequency;

% Constant
c = 1/sqrt(2*3.14159);

% p(g): proportion of population
% m(g): population mean
% s(g): population standard deviation

toms 3x1 p m s

% Maximum likelihood function
toms i
mlf = fsum(lookup(w,i)*log(c*sum(p./s.*exp(-.5*((lookup(y,i)-m)./s).^2))),...
    i, 1:31);

eq1 = {sum(p) == 1};

eq2 = {m(2) >= m(1); m(3) >= m(2)};

eq3 = {p >= 0.1; s >= 0.1; m >= 0};

x0 = {p == 1/3; m == 100+30*(1:3)'; s == 15};

options = struct;
options.solver = 'conopt';
solution = ezsolve(-mlf,{eq1,eq2,eq3},x0,options);

Problem type appears to be: con
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1:                                f_k    1138.410564424085400000
                                              f(x_0)   1204.299328240221300000

Solver: CONOPT.  EXIT=0.  INFORM=2.
Feasible Path GRG, CONOPT 3.14F
Normal completion : Locally optimal

FuncEv   21 GradEv   16 HessEv    9 Iter   15 
CPU time: 0.734375 sec. Elapsed time: 0.859000 sec.