A Goodness-of-Fit Test with Focus on Conditional Value at Risk

To verify whether an empirical distribution has a specific theoretical distribution, several tests have been used like the Kolmogorov-Smirnov and the Kuiper tests. These tests try to analyze if all parts of the empirical distribution has a specific theoretical shape. But, in a Risk Management framework, the focus of analysis should be on the tails of the distributions, since we are interested on the extreme returns of financial assets. This paper proposes a new goodness-of-fit hypothesis test with focus on the tails of the distribution. The new test is based on the Conditional Value at Risk measure. Then we use Monte Carlo Simulations to assess the power of the new test with different sample sizes, and then compare with the Crnkovic and Drachman, Kolmogorov-Smirnov and the Kuiper tests. Results showed that the new distance has a better performance than the other distances on small samples. We also performed hypothesis tests using financial data. We have tested the hypothesis that the empirical distribution has a Normal, Scaled Student-t, Generalized Hyperbolic, Normal Inverse Gaussian and Hyperbolic distributions, based on the new distance proposed on this paper.