2.2.5.11. GumbelMax law

This law describes a Gumbel max distribution depending on the mode \(\mu\) and the scale \(\beta\), as

\[f(x) = z \times\frac{e^{-z}}{\beta}, \; \mbox{where} \; z = e^{\frac{-(x - \mu)}{\beta}}\]

The scale \(\beta\) should be greater than 0.000001 times \(\mu\).

Uranie code to simulate a GumbelMax random variable is:

    TDataServer *tds = new TDataServer("tdssampler", "Sampler Uranie demo");
    tds->addAttribute( new TGumbelMaxDistribution("gm", 0.5, 2.0));
    
    TSampling *fsamp = new TSampling(tds, "lhs", 300);
    fsamp->generateSample(); // Create a representative sample
    
    tds->Draw("gm");

Figure 2.19 shows the PDF, CDF and inverse CDF generated for different sets of parameters.

../../../_images/TGumbelMaxDistribution.png — Figure 2.19 Example of PDF, CDF and inverse CDF for GumbelMax distributions.

Is it also possible to set boundaries to the infinite span of this distribution to create a truncated GumbelMax law. This can be done by calling the following method:

    tds->getAttribute("gm")->setBounds(-1.0,12.0); //truncate the law

The resulting PDF, CDF and inverse CDF, with and without truncation, can be seen, in this case, in Figure 2.20 for a given set of parameters and various boundaries.