gpfit
              Estimate parameters and confidence intervals for the generalized Pareto distribution.
 paramhat = gpfit (x, theta) returns the maximum
 likelihood estimates of the parameters of the generalized Pareto distribution
 given the data in x and the location parameter theta.
 paramhat(1) is the shape parameter, k,
 paramhat(2) is the scale parameter, sigma, and
 paramhat(3) is the location parameter, theta.  Although
 theta is returned in the estimated paramhat, gpfit does
 not estimate the location parameter theta, and it must be assumed to be
 known, given as a fixed parameter in input argument theta.
 [paramhat, paramci] = gpfit (x, theta) returns
 the 95% confidence intervals for the estimated parameter k and
 sigma. The third colummn of paramci includes the location
 parameter theta without any confidence bounds.
 […] = gpfit (x, theta, 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, theta, 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 and theta = 0, the Generalized Pareto
 is equivalent to the exponential distribution.  When k > 0 and
 theta = 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 > theta, or, when
 theta < 0, for
 0 <= (x - theta) / 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
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 ## Sample 2 populations from different generalized Pareto distibutions
 ## Assume location parameter θ is known
 theta = 0;
 rand ("seed", 5);    # for reproducibility
 r1 = gprnd (1, 2, theta, 20000, 1);
 rand ("seed", 2);    # for reproducibility
 r2 = gprnd (3, 1, theta, 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), theta);
 k_sigmaB = gpfit (r(:,2), theta);
 ## Plot their estimated PDFs
 x = [0.01, 0.1:0.2:18];
 y = gppdf (x, k_sigmaA(1), k_sigmaA(2), theta);
 plot (x, y, "-pc");
 y = gppdf (x, k_sigmaB(1), k_sigmaB(2), theta);
 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 θ = 0")
 hold off
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