Regression in a bivariate copula model

Abstract

We consider a bivariate regression model in which the error structure is arbitrary, except for equality of its marginal distributions. Two simple but inefficient approaches to analysis are (i) to work with the differences between the two components of each response and (ii) to derive working independence estimators as if the two component responses were independent. The former strategy does well if the explanatory variables are balanced over pairs and the correlation between the two components of the response is high; the latter strategy does well when the response correlation is low or the covariates are highly unbalanced. We show that combining the two estimators can give a procedure with high efficiency in all situations. We show that an adjusted marginal score can be calculated that is uncorrelated with the within-pair score, so that separate intrapair and interpair estimators can be calculated. For bivariate normal regression models our approach is identical to the usual procedure for combining intra- and interblock information. In general the approaches are not the same. Detailed calculations and a numerical example are given for a bivariate extreme value model due to Gumbel that is also important in survival analysis. Extensions from bivariate to general multivariate responses are indicated.