An empirical-error-guided bandwidth selector in nonparametric classification with incomplete data

The problem of bandwidth selection in nonparametric classification is considered when a large number of class variables may be missing, but not necessarily missing at random. This setup is generally acknowledged to be more challenging than the so-called missing at random setup. Our proposed approach starts by constructing a family of kernel-based classifiers where the members of the family are indexed by the kernel bandwidth h as well as the nonignorability parameter of the selection probability mechanism. Next, a search is performed to find the member of a finite cover of this family that has the smallest empirical misclassification error. To assess the performance of the resulting classifiers, we derive exponential performance bounds on the deviations of their errors from that of the theoretically optimal classifier. These bounds are then used to study strong convergence properties of the proposed classifiers. Our theoretical findings are further confirmed by numerical studies.