The Asymptotic Covariance Matrix of the Odds Ratio Parameter Estimator in Semiparametric Log-bilinear Odds Ratio Models

The association between two random variables is often of primary interest in statistical research. In this paper semiparametric models for the association between random vectors X and Y are considered which leave the marginal distributions arbitrary. Given that the odds ratio function comprises the whole information about the association the focus is on bilinear log-odds ratio models and in particular on the odds ratio parameter vector {\theta}. The covariance structure of the maximum likelihood estimator {\theta}^ of {\theta} is of major importance for asymptotic inference. To this end different representations of the estimated covariance matrix are derived for conditional and unconditional sampling schemes and different asymptotic approaches depending on whether X and/or Y has finite or arbitrary support. The main result is the invariance of the estimated asymptotic covariance matrix of {\theta}^ with respect to all above approaches. As applications we compute the asymptotic power for tests of linear hypotheses about {\theta} - with emphasis to logistic and linear regression models - which allows to determine the necessary sample size to achieve a wanted power.