An Anderson-Darling approach for testing the goodness of fit of multivariate data

Resumen

When modeling risk, e.g. in fields like finance and insurance, we often face the problem of modelling extremal events. This involves making distributional assumptions for (sometimes  multivariate) data. It has been frequently observed that in practice tails are heavier than «normal» and extremes appear in clusters, indicating tail dependence. In such cases the assumptions of normality are violated. Therefore there is often uncertainty if the normal assumption can still be justified. 

In the univariate case a popular method of testing the assumption of normality is by using the Anderson-Darling test. It is known for its strong power, especially when there are deviations in the tails of a distribution. We will consider a possible generalization of this test to the multivariate case. Although some theoretical results about a multivariate extension of the multivariate Anderson-Darling statistic are already known, so far its application in a multivariate test seemed inconvenient, as the calculation of the n-variate test statistic required the calculation of an n-dimensional integral. 

A calculation formula of this multivariate Anderson-Darling statistic for finite, multidimensional samples will be presented. Using this formula immensely simplifies the calculation and thus serves as one key ingredient to facilitate the practical use of the test. 

The power of this new approach in comparison with the other tests has been evaluated  in a few multivariate example settings. As a practical example, the test has been applied in a collaboration with P. Ruckdeschel, B. Spangl and S. Desmettre which has been presented as «Measuring and Combinig Different Aspects of Goodness of Fit of Dynamical Extreme Value Models in Hydrology» at ERCIM 2015: we applied the test to assess the goodness of fit of four different model approaches to capture time dependence and extreme value behaviour of river discharge data. 

Andreas Mändle

Obtuvo su grado de doctor de la  Universidad Carl von Ossietzky en Oldenburgo, Alemania.
Trabajó bajo la dirección de la Dra. Angelika May y actualmente colabora con Dr. Peter Ruckdeschel en la misma universidad, donde planean establecer un centro para consultoria estadística en colaboración con la Universidad de Bremen.