How to generate a Frechet distribution using Methods of Moments?

How could I generate extreme value type II distribution for maximum using the method of moments? I think of something similar to this

https://www.mathworks.com/matlabcentral/answers/315084-how-can-i-fit-a-gumbel-maximum-distribution-using-both-maximum-likelihood-estimates-and-method-of-mo

Answer 1

[Jeff Miller]

Here is how you might do it if with Cupid :

%% Example for the 2-parameter Frechet distribution (minimum known to be 0)

% Make some example data to illustrate fitting commands.
sampleSize = 500;
observedVal = Frechet2(10,2).Random(sampleSize,1);

figure;
histogram(observedVal,'Normalization','pdf')
xvals = linspace(min(observedVal),max(observedVal),100);

% Fit with maximum likelihood:
fitDist = Frechet2(9,1);   % Guess parameters 9,1
fitDist.EstML(observedVal)
fittedPDF = fitDist.PDF(xvals);
hold on
plot(xvals,fittedPDF)

% Fit with method of moments:
fitDist = Frechet2(9,1);   % Guess parameters 9,1
obsMean = mean(observedVal);
obsVar = var(observedVal);
fitDist.EstMom([obsMean,obsVar])

fittedPDF = fitDist.PDF(xvals);
hold on
plot(xvals,fittedPDF)

legend('observed','MLE','Moment')

%% Example for the 3-parameter Frechet distribution (minimum a free paramenter)

% Make some example data to illustrate fitting commands.
% 
sampleSize = 1000;
trueMin = -10;
observedVal = Frechet(10,2,trueMin).Random(sampleSize,1);

figure;
histogram(observedVal,'Normalization','pdf')
xvals = linspace(min(observedVal),max(observedVal),100);

% Fit with maximum likelihood:
fitDist = Frechet(9,1,-8);   % Guess parameters 9,1,-2
fitDist.EstML(observedVal)
fittedPDF = fitDist.PDF(xvals);
hold on
plot(xvals,fittedPDF)

% Fit with method of moments
fitDist = Frechet(9,1,-8);   % Guess parameters 9,1,-2
obsMean = mean(observedVal);
obsVar = var(observedVal);
obs3rdMom = mean( (observedVal - obsMean).^3 );
fitDist.EstMom([obsMean,obsVar,obs3rdMom])
fittedPDF = fitDist.PDF(xvals);
hold on
plot(xvals,fittedPDF)

legend('observed','MLE','Moment')

% Note that the parameter estimates will be unreasonable if the initial guesses
% for parameter values are too far wrong.