Why does gevcdf with shape of zero result in different answer than evcdf?

From what I have read online, the evcdf function returns the Type I Gumbel distribution function.

And the gevcdf function returns the Generalized Extreme Value (GEV) distribution function with a shape parameter as an input. If the shape parameter is zero, then the GEV distribution becomes the Gumbel distribution.

So, why doesn't gevcdf with a shape parameter of zero give the same answer as evcdf?

(Note that the documentation shows the gevcdf and evcdf have the mu and sigma parameters switched which is just confusing...but I think I have accounted for it properly here).

%X variable
xgrid = linspace(1,10,100);

% Compute the GEV CDF for the range of block maxima
shapek = 0; %shape parameter
scaleSigma = 0.5; %sigma parameter
locMu = 1; %mu parameter

%Gumbel distribution using evcdf
pgumb = evcdf(xgrid,locMu,scaleSigma);

%Gumbel distribution using GEV
%shapek = 0 for Gumbel
pgev = gevcdf(xgrid, shapek,scaleSigma,locMu);

%logarithm of return period (see Coles, 2001)
ypgev = -log(1-pgev);
ypgumb = -log(1-pgumb);

figure(2)
plot(ypgev,xgrid,'-','LineWidth',1.5); hold on
plot(ypgumb,xgrid,'--r','LineWidth',1.5);
title(['\sigma = ',num2str(scaleSigma),', \mu = ',num2str(locMu)])
legend('Type 1 GEV., k=0','EVCDF')
xlabel('Return Period')
ylabel('Value')
grid on

Answer 1

[Darshak]

Hi Darcy,

The discrepancy arises because the “evcdf” function is designed to model minima, while the “gevcdf” function with (k = 0) models maxima by default. This difference in modelling leads to the observed variance in results. For this, there is a workaround for either modelling minima with both or maxima with both as follows:

• Modeling Minima - The given code modifies the use of “gevcdf” to model minima by inverting the “xgrid” and location parameter (-“xgrid” and –“locMu”). This allows the GEV distribution to represent minima, aligning with how “evcdf” inherently models minima.

% Parameters
  
xgrid = linspace(1, 10, 100);
  
shapek = 0; % Shape parameter for GEV
  
scaleSigma = 0.5; % Scale parameter (sigma)
  
locMu = 1; % Location parameter (mu)
  
 
  
% Minima using evcdf
  
pgumb = evcdf(xgrid, locMu, scaleSigma);
  
 
  
% Minima using gevcdf
  
pgev = 1 - gevcdf(-xgrid, shapek, scaleSigma, -locMu);
  
 
  
% Plotting
  
figure;
  
plot(-log(1 - pgev), xgrid, '-', 'LineWidth', 1.5); hold on;
  
plot(-log(1 - pgumb), xgrid, '--r', 'LineWidth', 1.5);
  
title(['\sigma = ', num2str(scaleSigma), ', \mu = ', num2str(locMu)])
  
legend('GEV Minima', 'EVCDF Minima')
  
xlabel('Return Period')
  
ylabel('Value')
  
grid on;

• Modeling Maxima - The given code introduces an alternative approach to model maxima using “evcdf” by flipping the “xgrid” and location parameter (-“xgrid” and –“locMu”). This adjustment makes the use of “evcdf” consistent with modeling maxima, similar to how “gevcdf” operates by default.

% Parameters

xgrid = linspace(1, 10, 100);

shapek = 0; % Shape parameter for GEV

scaleSigma = 0.5; % Scale parameter (sigma)

locMu = 1; % Location parameter (mu)



% Maxima using evcdf

pgumb_max = 1 - evcdf(-xgrid, -locMu, scaleSigma);



% Maxima using gevcdf

pgev_max = gevcdf(xgrid, shapek, scaleSigma, locMu);



% Plotting

figure;

plot(-log(1 - pgev_max), xgrid, '-', 'LineWidth', 1.5); hold on;

plot(-log(1 - pgumb_max), xgrid, '--r', 'LineWidth', 1.5);

title(['\sigma = ', num2str(scaleSigma), ', \mu = ', num2str(locMu)])

legend('GEV Maxima', 'EVCDF Maxima')

xlabel('Return Period')

ylabel('Value')

grid on;

I hope this helps with your issue.