Estimators and Tests based on Likelihood Depth for Copulas and the Weibull-Distribution

New estimators based on the likelihood depth for the examples of 2-dimensional, one-parametric Gumbel-Copula (and 2-dimensional Gauss-distribution) and in a second step for the Weibulldistribution are presented. The copula model has a variety of applications because it models dependence structures, e.g. in finance, in the analysis of credit risks. Copulas can also be used in the simulation of technical production processes to model the occurrence of coupled failures. For an introduction see Nelsen (2006). Different estimation procedures for copulas were introduced, parametric, semi-parametric and nonparametric methods are proposed, see e.g. Genest et al. (1995), Hoff (2007) or Kim et al. (2007). The Weibull-distribution is often used in Survival Analysis, see for example Lee and Wang (2003). It can model constant, deand increasing Hazard-functions. Because of this and due to the fact, that the survival-function has a rather simple form, it is used in many applications. The distribution function is one-dimensional and depends on two parameters. Most times the Maximum-LikelihoodEstimator is used for parametric estimation, it can be found e.g. in Lee and Wang (2003). We derive estimators and tests for the Gumbel copula, the two-dimensional Gauss distribution and for the Weibull-distribution via likelihood depth and simplicial likelihood depth. These are rather general notions of data depth, see Müller (2004) and Müller (2005). They extend the half space depth of Tukey, see Tukey (1975), and the simplicial depth of Liu, see Liu (1988) and Liu (1990) which lead to outlier robust generalizations of the median for multivariate data. They belong to a broad class of depth notions introduced and studied in the last 20 years. Although likelihood depth bases on a parametric approach, it can lead to distribution-free estimators and tests as Müller (2004) demonstrated for location-scale estimation and Müller (2005) for regression. Müller (2004) also showed that simplicial likelihood depth is in particular appropriate for testing since it is an U-statistic. Thereby rather general hypotheses can be tested and the resulting tests are outlier robust. However, in some cases the parameters with maximum likelihood depth are biased estimators. This is in particular the case for the Gumbel-Copula, the Gaussand for the Weibull-distribution. But the bias can be corrected leading to new robust estimators in all considered cases. Copulas are often given by distributional assumptions on the form of the copula. This distributional assumptions for the copula will be used here to define likelihood depth and simplicial likelihood depth for copulas. The approach is demonstrated for the Gaussian copula and the Gumbel copula for two dimensions which are based on one parameter only. In a next step, the likelihood depth shall be used to find estimators and tests for the two parameters of the Weibull distribution. We will show that the two parameters can be estimated and corrected step by step, so the same methods as in the case of one parameter can be used. The theory can be used also to find robust estimators for other distributions.