Statistics: gevfit

### Function Reference: `gevfit`

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

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

### Arguments

`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 (…, options)` specifies control parameters for the iterative algorithm used to compute the maximum likelihood estimates. options is a structure with the following field and its default value:

- `options.Display = "off"`
- `options.MaxFunEvals = 1000`
- `options.MaxIter = 500`
- `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

### References

1. Rolf-Dieter Reiss and Michael Thomas. Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields. Chapter 1, pages 16-17, Springer, 2007.

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

Source Code: gevfit

### Example: 1

     ## Sample 2 populations from 2 different exponential distibutions
     rand ("seed", 1);   # for reproducibility
     r1 = gevrnd (-0.5, 1, 2, 5000, 1);
     rand ("seed", 2);   # for reproducibility
     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));

     ## 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 ξ=-0.5, σ=1, μ=2", ...
              "Normalized HIST of sample 2 with ξ=0, σ=1, μ=-4",
         sprintf("PDF for sample 1 with estimated ξ=%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 ξ=%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 distibutions")
     hold off