Semi- and non-parametric inference for multivariate data: theory and application

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Multivariate data with thousands of interdependent measurements are the new norm in many fields of statistical applications. As a consequence, modeling dependence in multivariate setups has become a critical component in modern statistical data analysis. Copulas are flexible and efficient tools to model dependence structures among random variables with applications in numerous areas of research.The current proposal aims at advancing theory and applications of copula in several directions. Assume that a copula function C belongs to a parametric family indexed by parameters ?. Two approaches are common to estimate ?: Pseudo-maximum likelihood and Method of Moments' estimators. Both methods suffer from serious drawbacks that limit their applicability in practical settings. Alternatively, I propose: (I) a simulated method of moments by constructing unbiased nonparametric estimators for the k-th order moments of the random variable T~C(U), and then deduce an estimate of ? as a solution of the system of equations of empirical and theoretical moments. However, for a large number of copulas, an explicit form for the theoretical moments is not available, which considerably limits the scope of the method. I suggest a version where the theoretical moments are simulated. This approach is expected to be efficient, flexible and, in principle, applicable to any model of copula; (II) Similarly, I will develop a simulated maximum likelihood estimator; (III) Also, I will investigate a simulated pairwise likelihood type estimator. After treating the problem of estimating the copula parameter, I will turn my attention to hypothesis testing such as a goodness-of-fit test, test the equality between two copula, change points in copula models and test for exchangeability in an arbitrary dimension. All the proposed tests will be based on the Bernstein empirical copula, its associated density or the more recently  introduced beta empirical copula. Logistic regressions are efficient tools in modeling probabilities of events as functions of covariates. Recently, I have developed a copula based link functions in binary regression models. I will extend this method to multinomial regression and multivariate outcomes for high dimensional and mixed covariates by using vine copula or hierarchical Archimedean copulas. Furthermore, I will investigate this approach in the context of quantile regressions for count data. The proposed methodology will be used to develop procedures for classification and variables selection for high dimensional data. I believe that these approaches can be adapted to the case of censored and cured data. In parametric and semiparametric regression models that use copulas, one major obstacle is the copula misspecification. To overcome this issue, I propose using a nonparametric copula based approach to estimate the quantile and expectile regression functions for cont data. Particular attention will be given to censored data.