Independent rule in classification of multivariate binary data

We consider the performance of the independent rule in classification of multivariate binary data. In this article, broad studies are presented including the performance of the independent rule when the number of variables, dd, is fixed or increased with the sample size, nn. The latter situation includes the case of d=O(nτ)d=O(nτ) for τ>0τ>0 which cover “the small sample and the large dimension”, namely d≫nd≫n when τ>1τ>1. Park and Ghosh [J. Park, J.K. Ghosh, Persistence of plug-in rule in classification of high dimensional binary data, Journal of Statistical Planning and Inference 137 (2007) 3687–3707] studied the independent rule in terms of the consistency of misclassification error rate which is called persistence under growing numbers of dimensions, but they did not investigate the convergence rate. We present asymptotic results in view of the convergence rate under some structured parameter space and highlight that variable selection is necessary to improve the performance of the independent rule. We also extend the applications of the independent rule to the case of correlated binary data such as the Bahadur representation and the logit model. It is emphasized that variable selection is also needed in correlated binary data for the improvement of the performance of the independent rule.