# Graphical goodness-of-fit tests¶

We gather some graphical tools to validate whether a given sample of data is drawn from a given continuous distribution of dimension 1.

We denote by the data of dimension 1 which have been independently generated by the random variable . Let be a continuous cumulative distribution function.

We want to validate whether follows the distribution characterized by .

## QQ-plot¶

The Quantile - Quantile - Plot (QQ Plot) is based on the comparison of some quantiles between the tested distribution and the empirical ones. Let be the quantile of order of the distribution , with . It is defined by:

The empirical quantile of order built on the sample is defined by:

where denotes the integral part of and is the sample sorted in ascended order:

Thus, the smallest value of the sample is an estimate of the -quantile where , for .

The QQ-plot draws the couples . If follows the distribution , then the points should be close to the diagonal.

The following figure illustrates a QQ-plot with a sample of size . In this example, the points remain close to the diagonal and the hypothesis “ is the cumulative distribution function of ” does not seem false, even if a more quantitative analysis should be carried out to confirm this.

In this second example, the tested continuous distribution is clearly false.

## Normal probability plot (Henry’s line)¶

This test is dedicated to the normal distribution.

The following result is used in the test: if follows the distribution, then follows the distribution. Furthermore, let be the quantile of order of and let be the quantile of order of . Then we have the relation:

Then the Henri line draws the QQ-plot built from the empirical quantiles of order and the quantiles of same order of the distribution. If the sample comes from the distribution, then the points should be close to the line of equation .

The following figure illustrates the Henry’s line with a sample of size . In this example, the points remain close to a line and the hypothesis “ follows a normal distribution“ does not seem false, even if a more quantitative analysis should be carried out to confirm this.

In this second example, the hypothesis of a normal distribution seems far less plausible because of the behavior for small values of .

## Kendall plot¶

In the bivariate case, the Kendall Plot test allows one to validate whether a sample is drawn from a given copula or to check whether two samples share the same copula.

Let be a bivariate random vector with the copula and the marginal cumulative distribution functions . Let be the random vector with marginal distributions and copula.

Let a sample drawn from . We build the rank sample defined by where .

We define:

where is a bivariate random vector with marginal distributions and copula. We denote by the cumulative distribution function of .

We can get a sample of denoted by from the sample as follows:

where is the empirical cumulative distribution function of the sample . Then, we have, for all :

From the sample , we build the ordered sample .

Let be the order statistics of . Then we know that the cumulative distribution function of is the composition between the cumulative distribution function of the distribution and the distribution of :

Let be the statistic defined by:

Thus we have:

(1)

For a given copula , equation (1) is evaluated by Monte Carlo sampling: we generate samples of size from , in order to get realizations of the statistics that are used to calculate as the empirical mean of .

The Kendall Plot draws the points . If the points are on the first diagonal, the copula is validated. In particular, we can use the Kendall plot to test the independence between and by using the independent copula to calculate the values .

To test whether two samples share the same copula, the Kendall Plot test draws the points respectively associated to the first and second sample. Note that the two samples must have the same size.

In the first example, the Kendall Plot test validates the use of the Frank copula for the given sample.

In the second example, the Kendall Plot test invalidates the use of the Frank copula for the given sample.

Remark: In the case where you want to test a sample with respect to a specific copula, if the size of the sample is greater than 500, we recommend to use the second form of the Kendall plot test: generate a sample of the proper size from your copula and then test both samples. Testing this way is more efficient.