```{include} /../core/uncertModeler/edf_tests.md
```

This part is introducing comparison tests, sometimes called "goodness of fit" tests, which are used 
as test hypothesis. This idea is to check, when considering a certain {{AttVar}}, whether it is 
following a predefined law among the list of implemented ones: normal, lognormal and uniform ones. 
To do so, there are three different tests implemented in {{uranie}}. If one calls $F_n(x)$ the 
*Empirical Distribution Function* of the law $F(x)$ (*i.e.* the distribution that we'd like to test) 
and $F_{0}(x)$ the reference law one wants to compare to, then, for $n$ the number of data in the 
EDF, these tests are defined as:
<variablelist>

**Kolmogorov-Smirnov ($D$)** {cite}`Kolmogorov33`
: $$D = {\rm sup} |F_0(X_i) - F_n(X_i)|_{i=1,\ldots, n}$$

**Anderson-Darling ($A^2$)** {cite}`Anderson52`
: $$A^2 = n \int \frac{|F_0(x) - F_n(x)|^2}{F_0(x)(1-F_0(x))} dF_0(x) = -n -\frac{1}{n} \sum_{i=1}^{n} 
(2i-1) \times [\log(F_n(X_i))+\log(F_n(X_{n+1-i})) ]$$

**Cramer-VonMises ($W^2$)** {cite}`CramervonM62`
: $$W^2 = n \int |F_0(x) - F_n(x)|^2 dF_0(x) = \frac{1}{12n} \sum_{i=1}^{n} \left(F_0(X_i) - 
\frac{2i-1}{2n} \right)^2$$

In these three formulas, the $(X_i)_{i=1,\dots, n}$ set represents the ordered data of the random 
variable $x$ which comes usually as the CDF distribution for convenience.