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\footnotesize \noindent Statistica Sinica {\bf 21} (2011), 841-871

\vspace{3pc} \normalsize \centerline{\large\bf FAST LARGE-SAMPLE
GOODNESS-OF-FIT TESTS} \vspace{2pt} \centerline{\large\bf  FOR
COPULAS} \vspace{.25cm}
\author{Ivan Kojadinovic, Jun Yan and Mark Holmes}
\vspace{.4cm} \centerline{\it Universit\'e de Pau et des Pays de
l'Adour,  University of Connecticut}\centerline{\it  and The
University of Auckland} \vspace{.55cm}
\begin{quotation}
\noindent {\it Abstract:} Goodness-of-fit tests are a fundamental
element in the copula-based modeling of multivariate continuous
distributions. Among the different procedures proposed in the
literature, recent large scale simulations suggest that one of the
most powerful tests is based on the empirical process comparing the
empirical copula with a parametric estimate of the copula derived
under the null hypothesis. As for most of the currently available
goodness-of-fit procedures for copula models, the null distribution
of the statistic for the latter test is obtained by means of a
parametric bootstrap. The main inconvenience of this approach is its
high computational cost, which, as the sample size increases, can be
regarded as an obstacle to its application. In this work, fast
large-sample tests for assessing goodness of fit are obtained by
means of multiplier central limit theorems. The resulting procedures
are shown to be asymptotically valid when based on popular
method-of-moment estimators. Large scale Monte Carlo experiments,
involving six frequently used parametric copula families and three
different estimators of the copula parameter, confirm that the
proposed procedures provide a valid, much faster alternative to the
corresponding parametric bootstrap-based tests. An application of
the derived tests to the modeling of a well-known insurance data set
is presented. The use of the multiplier approach instead of the
parametric bootstrap can reduce the computing time from about a day
to minutes.
\par
\vspace{9pt} \noindent {\it Key words and phrases:} Empirical
process, multiplier central limit theorem, pseudo-observation, rank,
semiparametric model.\par
\end{quotation}

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