ISBN 978-3-8439-2563-1

978-3-8439-2563-1, Reihe Mathematik

Andreas Mändle
Assessing the goodness of fit of multidimensional data – An Anderson-Darling approach to multivariate distributional tests

171 Seiten, Dissertation Carl von Ossietzky Universität Oldenburg (2015), Softcover, B5

Zusammenfassung / Abstract

In the fields of finance and insurance we often face the problem of modeling extremal events, e.g. when measuring the risk of holding an equity portfolio or the insurance risk of possible losses in different lines of business. This involves making distributional assumptions for multivariate, sometimes even high-dimensional 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. There is often quite rightly uncertainty, if the normal assumption can still be justified.

In the univariate case a popular method for testing the assumption of normality is by using the Anderson-Darling test. The test is known for its strong power for testing normality. Its statistic puts more weight on deviations in the tails and therefore particularly considers extreme outcomes.

A possible generalization of the Anderson-Darling test can be achieved as a weighted average of the discrepancy between the hypothesized distribution function and the multivariate empirical distribution function resulting from sample data. Although some theoretical results about a multivariate extension of the Anderson-Darling test have been found before, its application as multivariate test seemed still inconvenient, as the calculation of the n-variate test statistic required the calculation of an n-dimensional integral.

Here, a calculation formula of this multivariate Anderson-Darling statistic for finite, multidimensional samples will be presented. Using this formula immensely simplifies the computation and therefore the simulation of the statistic. Thus it serves as one key ingredient to facilitate the practical use of the test and to gain further insight into the power of the test in certain situations.

In a multivariate example setting of tail dependence and/or heavy tails it will be demonstrated how the test can be applied, to test the assumption of a specific multivariate distribution. Although it will be referred to methods that allow simulating the asymptotic statistic, conveniently and preferably the sample quantiles will be computed using Monte Carlo simulations of the finite sample statistic. In order to assess the power of this new approach in comparison with other tests, it will run against widely used tests for multivariate normality, including Mardia’s, Henze-Zirkler’s and Royston’s test.