Maximum likelihood fitting for the r Largest Order Statistics

MATLAB Community,

I using the MLE to finding Gumble distribution parameters for the r Largest Order Statistics.

I already have the likelihood function,which is:

L=@(mu,sigma) prod( exp(-exp( - ( data2 (:,size ( data 2,2 ) ) -mu ./ sigma ))) .*prod( sigma^( -1).*exp( -( data2 - mu )./sigma),2))

where data2 is a 40*5 matrix, meaning I picking 1st to 5th largest data in one year for total 40 years.

(if data2 is a 40*1 matrix, L become the Gumble distribution likelihood function.)

The matlab code mle() can only input vector, is there any way to solve this problem?

Answer 1

[Sudarsanan A K]

Hello Yi Lun,

To estimate the parameters of a Gumbel distribution using MATLAB's "mle" function for multidimensional data, such as your 40x5 matrix, you can follow this streamlined approach:

• Reshape Your Data: Convert your matrix into a vector since "mle" is designed for vector inputs. This treats each element in your matrix as an independent observation.

data2 = randn(40, 5); % Example data, replace with your actual data
dataVector = reshape(data2, [], 1); % Reshape your 40x5 matrix into a vector

• Define the Likelihood Function: Adjust the likelihood function for the Gumbel distribution to work with your reshaped vector data. The likelihood function for the Gumbel distribution, with parameters μ (location) and σ (scale), is given by:

• Define the Negative Log-Likelihood Function: For the Gumbel distribution, define a custom negative log-likelihood function. It's important to include "varargin" in your function definition to handle any additional arguments passed by "mle".

% Define the custom negative log-likelihood function for the Gumbel distribution
nloglf = @(params, data, varargin) -sum(log((1./params(2)) .* exp(-(data-params(1))./params(2) - exp(-(data-params(1))./params(2)))));

• Estimate Parameters Using "mle": With your data vector and the negative log-likelihood function ready, use "mle" to estimate the parameters. You'll need to provide initial guesses for the parameters.

% Specify initial parameter guesses
initialParams = [mean(dataVector), std(dataVector)]; % Example initial guesses

% Use mle to estimate parameters, specifying the custom nloglf and initial guesses
paramsEst = mle(dataVector, 'nloglf', nloglf, 'start', initialParams);

% Extract the estimated parameters
mu_est = paramsEst(1);
sigma_est = paramsEst(2);
fprintf('Estimated mu: %f\n', mu_est);

Estimated mu: -0.514545

fprintf('Estimated sigma: %f\n', sigma_est);

Estimated sigma: 1.041869

To validate the estimates, comparison between the empirical Cumulative Distribution Function (CDF) of your original data and the theoretical CDF of a Gumbel distribution using parameters estimated from that data can be done as follows:

% mu_est, sigma_est are your estimated parameters
% Generate Gumbel distributed values using estimated parameters
gumbelVals = mu_est - sigma_est * log(-log(rand(size(dataVector))));

% Plot empirical CDF
[f_empirical, x_empirical] = ecdf(dataVector);
plot(x_empirical, f_empirical, 'b-', 'LineWidth', 1.5); hold on;

% Plot theoretical CDF using estimated parameters
[f_theoretical, x_theoretical] = ecdf(gumbelVals);
plot(x_theoretical, f_theoretical, 'r--', 'LineWidth', 1.5);

legend('Empirical CDF', 'Theoretical CDF', 'Location', 'best');
xlabel('Value');
ylabel('Cumulative Probability');
title('Empirical vs Theoretical Gumbel Distribution');
grid on; hold off;

For further understanding of the critical functions used in this example, refer to the following documentations:

• "reshape": https://mathworks.com/help/matlab/ref/reshape.html
• "mle": https://mathworks.com/help/stats/mle.html
• "ecdf": https://mathworks.com/help/stats/ecdf.html

I hope this helps!