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.
Multivariate empirical distribution plot and goodness-of-fit test