Copulas: a Review and Recent Developments (2007)

y Genest et al. (1993) and Shi and Louis (1995), see also Chebrian et al. (2002). Theparametric methods relies on the assumption of parametric marginal distributions.Each parametric margin is then plugged in the full likelihood which is maximized withrespect to µ. The ¯nal results depend on the right speci¯cation of all marginals. Inthe semiparametric method, the marginal empirical cumulative distribution functionscan be incorporated in the likelihood. In this case the estimation procedure su®ersfrom the loss of e±ciency, see Genest and Rivest (1993).Besides these two methods, it is also possible to estimate a copula by some nonparametricmethods using empirical distributions, see Deheuvels (1979, 1981a,b) andFermanian et al. (2004). Non-parametric estimation of copulas in context of timedependence based on kernel approach have been studied by Fermanian and Scaillet(2003). The goodness of ¯t tests for copulas are discussed by Fermanian (2005).The copula theory has an incredible evolution during the last decade, motivatedby its application in Probability theory, Biostatistics, Finance, Insurance, Economics,see Embrechts et al. (2002), the recent monographs Cherubini et al. (2004), Nelsen(2006) and references therein. The interested reader can ¯nd many research andapplied papers visiting, say, the home pages of the following institutions and theirrelated links:² Financial and Actuarial Research Group at the ETH, Zurich, Switzerland;² Financial Econometric Research Center, UK;² Actuarial Science, Katholieke Universiteit Leuven, Belgium;² Center for Research in Economics and Statistics, France;² International Center for Financial Asset Management and Eng., Switzerland;² Research Group at Credit Lyonnais, France, etc.Therefore, there is no need to demonstrate the importance of copula theory fromtheoretical and practical point of view. In spite of this, the research on relevantspeci¯cations for copulas and on their time (dynamic) dependence is still in its infancy.In Section 2 we present brie°y two extensions of the probability integral transform,Bernstein and Bertino's family of copulas, notion about quasi-copulas, several copulalikeversions related to conditional and pseudo-conditional copulas usedintimeseriescontext. We consider as well copula applications for evaluating quantile risk measures.In Section 3 we discuss three topics: order statistics copula, copulas with multivariatemarginals and a representation of copula via a local dependence measure. In Section4 is given a review of some known facts and recent author's results treating theapplications of extreme value copulas. Several conclusions are given at the end.5