Efficient estimation of parameters in marginal in semiparametric multivariate models

Abstract

We consider a general multivariate model where univariate marginal distributions are known up to a common parameter vector and we are interested in estimating that vector without assuming anything about the joint distribution, except for the marginals. If we assume independence between the marginals and maximize the resulting quasi-likelihood, we obtain a consistent but inefficient estimate. If we assume a parametric copula (other than independence) we obtain a full MLE, which is efficient but only under correct copula specification and badly biased if the copula is misspecified. Instead we propose a sieve MLE estimator which improves over OMLE but does not suffer the drawbacks of the full MLE. We model the unknown part of the joint distribution using the Bernstein-Kantorovich polynomial copula and assess the resulting improvement over QMLE and over misspecified FMLE in terms of relative efficiency and robustness. We derive the asymptotic distribution of the new estimator and show that it reaches the semiparametric efficiency bound. Simulations suggest that the sieve MLE can be almost as efficient as FMLE relative to QMLE provided there is enough dependence between the marginals. An application using insurance company loss and expense data demonstrates empirical relevance of the estimator.