Aspects of copulas and goodness-of-fit

The goodness-of- t of a statistical model describes how well it ts a set of observations. Measuresof goodness-of- t typically summarize the discrepancy between observed values and the valuesexpected under the model in question. Such measures can be used in statistical hypothesistesting, for example to test for normality, to test whether two samples are drawn from identicaldistributions, or whether outcome frequencies follow a speci ed distribution. Goodness-of- tfor copulas is a special case of the more general problem of testing multivariate models, but iscomplicated due to the di culty of specifying marginal distributions.In this thesis, the goodness-of- t test statistics for general distributions and the tests for copulasare investigated, but prior to that an understanding of copulas and their properties is developed.In fact copulas are useful tools for understanding relationships among multivariate variables, andare important tools for describing the dependence structure between random variables. Severalunivariate, bivariate and multivariate test statistics are investigated, the emphasis being ontests for normality. Among goodness-of- t tests for copulas, tests based on the probability integraltransform, Rosenblatt's transformation, as well as some dimension reduction techniques areconsidered. Bootstrap procedures are also described. Simulation studies are conducted to rstcompare the power of rejection of the null hypothesis of the Clayton copula by four di erent teststatistics under the alternative of the Gumbel-Hougaard copula, and also to compare the powerof rejection of the null hypothesis of the Gumbel-Hougaard copula under the alternative of theClayton copula. An application of the described techniques is made to a practical data set.