Function Reference: gpfit

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

Estimate parameters and confidence intervals for the generalized Pareto distribution.

paramhat = gpfit (x) returns the maximum likelihood estimates of the parameters of the generalized Pareto distribution given the data in x. paramhat(1) is the shape parameter, k, and paramhat(2) is the scale parameter, sigma. Other functions for the generalized Pareto, such as gpcdf, allow a location parameter, mu. However, gpfit does not estimate a location parameter, and it must be assumed known, and subtracted from x before calling gpfit.

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

[…] = gpfit (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.

[…] = gpfit (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.TolBnd = 1e-6
  • options.TolFun = 1e-6
  • options.TolX = 1e-6

When k = 0 and mu = 0, the Generalized Pareto CDF is equivalent to the exponential distribution. When k > 0 and mu = k / k the Generalized Pareto is equivalent to the Pareto distribution. The mean of the Generalized Pareto is not finite when k >= 1 and the variance is not finite when k >= 1/2. When k >= 0, the Generalized Pareto has positive density for x > mu, or, when mu < 0, for 0 <= (x - mu) / sigma <= -1 / k.

Further information about the generalized Pareto distribution can be found at https://en.wikipedia.org/wiki/Generalized_Pareto_distribution

See also: gpcdf, gpinv, gppdf, gprnd, gplike, gpstat

Source Code: gpfit

Example: 1

 

 ## Sample 2 populations from different generalized Pareto distibutions
 ## Assume location parameter is known
 mu = 0;
 rand ("seed", 5);    # for reproducibility
 r1 = gprnd (1, 2, mu, 20000, 1);
 rand ("seed", 2);    # for reproducibility
 r2 = gprnd (3, 1, mu, 20000, 1);
 r = [r1, r2];

 ## Plot them normalized and fix their colors
 hist (r, [0.1:0.2:100], 5);
 h = findobj (gca, "Type", "patch");
 set (h(1), "facecolor", "r");
 set (h(2), "facecolor", "c");
 ylim ([0, 1]);
 xlim ([0, 5]);
 hold on

 ## Estimate their α and β parameters
 k_sigmaA = gpfit (r(:,1));
 k_sigmaB = gpfit (r(:,2));

 ## Plot their estimated PDFs
 x = [0.01, 0.1:0.2:18];
 y = gppdf (x, k_sigmaA(1), k_sigmaA(2), mu);
 plot (x, y, "-pc");
 y = gppdf (x, k_sigmaB(1), k_sigmaB(2), mu);
 plot (x, y, "-sr");
 hold off
 legend ({"Normalized HIST of sample 1 with k=1 and σ=2", ...
          "Normalized HIST of sample 2 with k=2 and σ=2", ...
          sprintf("PDF for sample 1 with estimated k=%0.2f and σ=%0.2f", ...
                  k_sigmaA(1), k_sigmaA(2)), ...
          sprintf("PDF for sample 3 with estimated k=%0.2f and σ=%0.2f", ...
                  k_sigmaB(1), k_sigmaB(2))})
 title ("Three population samples from different generalized Pareto distibutions")
 text (2, 0.7, "Known location parameter μ = 0")
 hold off

                    
plotted figure