Goodness-of-Fit Tests for a Heavy Tailed Distribution

In this article the goodness-of-fit tests for heavy tailed distributions are evaluated regarding their relation to the underlying distribution, power, type I error and the sample size. The Kolmogorov-Smirnov test, Berk-Jones test, estimated score test, their corresponding quadratic tests and four different tests based on test measures of R´enyi are evaluated in simulations and on Realized Variance data from the DAX stock index between 2000 and 2017. Samples with different sample sizes are simulated from different heavy tailed and non-heavy tailed distributions to compare the relation to the underlying distributions of the described test measures. The results indicate that the critical values of the test measures have a low relation to the underlying distribution if around 5% of the ordered observations are used. Furthermore, critical values should be simulated for different sample sizes for all test measures as critical values are dependent on both the proportion of the ordered distribution used and the number of observations. If this is impossible or impractical the quadratic Berk-Jones test can be used, as it is the least dependent on the sample size. Moreover, the first R´enyi measure has the Type I error closest to the corresponding critical value for different distributions and percentages of the ordered observations used. On the other hand, the quadratic estimated score test has the highest power. If the relation between the power and the type I error is the focus then the Berk-Jones test has the most added value. Of the measures based on R´enyi, the fourth one performs generally the best due to lower type I errors and higher powers.