On some goodness-of-fit tests for copulas

Abstract

Copulas have been known in the statistical literature for many years, and

have become useful tools in modeling dependence structure of multivariate

random variables, overcoming some of the drawbacks of the commonly-used

correlation measures. Goodness-of-fit tests for copulas play a very important

role in evaluating the suitability of a potential input copula model. In recent

years, many approaches have been proposed for constructing goodness-of-fit

tests for copula families. Among them, the so-called “blanket tests" do not

require an arbitrary data categorization or any strategic choice of weight function, smoothing parameter, kernel, and so on.

As preliminaries, some background and related results of copulas are firstly

presented. Three goodness-of-fit test statistics belonging to the blanket test

classification are then introduced. Since the asymptotic distributions of the

test statistics are very complicated, parametric bootstrap procedures are employed to approximate critical values of the test statistics under the null hypotheses. To assess the performance of the three test statistics in the low

dependence cases, simulation studies are carried out for three bivariate copula families, namely the Gumbel-Hougaard copula family, the Ali-Mikhail-Haq

copula family, and the Farlie-Gumbel-Morgenstern copula family. Specifically

the effect of low dependence on the empirical sizes and powers of the three

blanket tests under various combinations of null and alternative copula families are examined. Furthermore, to check the performance of the three tests

for higher dimensional copulas, the simulation studies are extended to some

three-dimensional copulas. Finally the three goodness-of-fit tests are applied

to two real data sets.