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

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

2011-05-04

arxiv.org/abs/1105.0852v2

Mathematics

Statistics Theory

27 pages

Scientific paper

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.

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