like.gms : Maximum Likelihood Estimation
This application from the biomedical area tests the hypothesis
that a population of systolic blood pressure can be separated into
three distinct groups.
Reference:
- 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.
Small Model of Type: NLP
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$title Maximum Likelihood Estimation (LIKE,SEQ=25)
$Ontext
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.
$Offtext
sets i observations / 1*31 /
g groups / one, two, three /
table data(*,i) systolic blood pressure data
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
pressure 95 105 110 115 120 125 130 135 140 145 150 155 160 165 170
frequency 1 1 4 4 15 15 15 13 21 12 17 4 20 8 17
+ 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
pressure 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 260
frequency 8 6 6 7 4 3 3 8 1 6 0 5 1 7 1 2
parameters y(i) pressure
w(i) frequency weight
c constant ;
y(i) = data("pressure",i); w(i) = data("frequency",i);
c = 1/sqrt(2*3.14159); display y, w, c;
positive variables p(g) proportion of population
m(g) population mean
s(g) population standard deviation
variable mlf maximum likelihood value
equations like, pdef, rank;
like.. mlf =e= sum(i, w(i)*log(c*sum(g, p(g)/s(g)*exp(-.5*sqr((y(i)-m(g))/s(g))))));
pdef.. sum(g, p(g)) =e= 1 ;
rank(g+1).. m(g+1) =g= m(g);
model ml1 maximum likelihood - ordered / like,pdef,rank /
ml2 maximum likelihood - unordered / like,pdef / ;
p.l(g) = 1/3; m.l(g) = 100 + 30*ord(g); s.l(g) = 15;
p.lo(g)=.1;
s.lo(g)=.1;
* reported solution below gives a nonoptimal solution
* p.fx('one') = .365;
* p.fx('two') = .475;
* p.fx('three') = .160;
option domlim = 1e3;
ml1.workfactor = 1.5;
solve ml1 maximizing mlf using nlp;
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