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:

tds = DataServer.TDataServer("tdssampler", "Sampler Uranie demo")
tds.addAttribute(DataServer.TGumbelMaxDistribution("gm", 0.5, 2.0))

fsamp = Sampler.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.