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Function Reference: gevfit

statistics: paramhat = gevfit (x)

statistics: [paramhat, paramci] = gevfit (x)

statistics: [paramhat, paramci] = gevfit (x, alpha)

statistics: [paramhat, paramci] = gevfit (x, alpha, freq)

statistics: [paramhat, paramci] = gevfit (x, alpha, options)

statistics: [paramhat, paramci] = gevfit (x, alpha, freq, options)

Estimate parameters and confidence intervals for the generalized extreme value (GEV) distribution.

paramhat = gevfit (x) returns the maximum likelihood estimates of the parameters of the GEV distribution given the data in x. paramhat(1) is the shape parameter, k, and paramhat(2) is the scale parameter, sigma, and paramhat(3) is the location parameter, mu.

[paramhat, paramci] = gevfit (x) returns the 95% confidence intervals for the parameter estimates.

[…] = gevfit (x, alpha) also returns the 100 * (1 - alpha) percent confidence intervals for the parameter estimates. By default, the optional argument alpha is 0.05 corresponding to 95% confidence intervals. Pass in [] for alpha to use the default values.

[…] = gevfit (params, x, freq) accepts a frequency vector, freq, of the same size as x. freq must contain non-negative integer frequencies for the corresponding elements in x. By default, or if left empty, freq = ones (size (x)).

[paramhat, paramci] = gevfit (x, alpha, options) specifies control parameters for the iterative algorithm used to compute ML estimates with the fminsearch function. options is a structure with the following fields and their default values:

  • options.Display = "off"
  • options.MaxFunEvals = 400
  • options.MaxIter = 200
  • options.TolX = 1e-6

When k < 0, the GEV is the type III extreme value distribution. When k > 0, the GEV distribution is the type II, or Frechet, extreme value distribution. If W has a Weibull distribution as computed by the wblcdf function, then -W has a type III extreme value distribution and 1/W has a type II extreme value distribution. In the limit as k approaches 0, the GEV is the mirror image of the type I extreme value distribution as computed by the evcdf function.

The mean of the GEV distribution is not finite when k >= 1, and the variance is not finite when k >= 1/2. The GEV distribution has positive density only for values of x such that k * (x - mu) / sigma > -1.

Further information about the generalized extreme value distribution can be found at https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution

See also: gevcdf, gevinv, gevpdf, gevrnd, gevlike, gevstat

Source Code: gevfit

Sample 2 populations from 2 different exponential distributions

 rng (42);
 r1 = gevrnd (-0.5, 1, 2, 5000, 1);
 r2 = gevrnd (0, 1, -4, 5000, 1);
 r = [r1, r2];

Plot them normalized and fix their colors

 hist (r, 50, 5);
 h = findobj (gca, 'Type', 'patch');
 set (h(1), 'facecolor', 'c');
 set (h(2), 'facecolor', 'g');
 hold on

Estimate their k, sigma, and mu parameters

 k_sigma_muA = gevfit (r(:,1));
 k_sigma_muB = gevfit (r(:,2));
warning: matrix singular to machine precision, rcond = 1.25501e-18
warning: called from
    polyfit at line 178 column 3
    gevfit at line 176 column 3
    __eval_demo__ at line 84 column 9
    __demo_notebook__ at line 43 column 3
    __build_demos__ at line 79 column 7
    function_texi2html at line 138 column 5
    package_texi2html at line 350 column 9

Plot their estimated PDFs

 x = [-10:0.5:20];
 y = gevpdf (x, k_sigma_muA(1), k_sigma_muA(2), k_sigma_muA(3));
 plot (x, y, '-pr');
 y = gevpdf (x, k_sigma_muB(1), k_sigma_muB(2), k_sigma_muB(3));
 plot (x, y, '-sg');
 ylim ([0, 0.7])
 xlim ([-7, 5])
 legend ({'Normalized HIST of sample 1 with k=-0.5, σ=1, μ=2', ...
          'Normalized HIST of sample 2 with k=0, σ=1, μ=-4',
     sprintf("PDF for sample 1 with estimated k=%0.2f, σ=%0.2f, μ=%0.2f", ...
                 k_sigma_muA(1), k_sigma_muA(2), k_sigma_muA(3)), ...
     sprintf("PDF for sample 3 with estimated k=%0.2f, σ=%0.2f, μ=%0.2f", ...
                 k_sigma_muB(1), k_sigma_muB(2), k_sigma_muB(3))})
 title ('Two population samples from different exponential distributions')
 hold off
plotted figure