2.2.5.14. GenPareto law

This law describes a generalised Pareto distribution depending on the location \(\mu\), the scale \(\sigma\) and a shape \(\xi\), as

\[f(x) = \frac{1}{\sigma}\times \left(1 + \xi\left(\frac{x-\mu}{\sigma}\right)\right)^{-(1/\xi +1)}\]

In this formula, \(\sigma\) should be greater than 0.

Uranie code to simulate a GenPareto random variable is:

    TDataServer *tds = new TDataServer("tdssampler", "Sampler Uranie demo");
    tds->addAttribute( new TGenParetoDistribution("gpa", 1.0, 1.0, 0.3));
    
    TSampling *fsamp = new TSampling(tds, "lhs", 300);
    fsamp->generateSample(); // Create a representative sample
    
    tds->Draw("gpa");

Figure 2.24 shows the PDF, CDF and inverse CDF generated for different sets of parameters.

../../../_images/TGenParetoDistribution.png — Figure 2.24 Example of PDF, CDF and inverse CDF for GenPareto distributions.

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

    tds->getAttribute("gpa")->setBounds(1.4,4.0); //truncate the law

The resulting PDF, CDF and inverse CDF, with and without truncation, can be seen, in this case, in Figure 2.25 for a given set of parameters and various boundaries.