Some adjusted marked empirical processes for model checking with missing binary data

To test the adequacy of a parametric model for the conditional expectation of a binary response variable given an explanatory variable, Kolmogorov–Smirnov and Cramér-von Mises statistics, which are based on a marked empirical process introduced by Stute can be used. Dikta, Kvesic, and Schmidt proposed a model-based resampling scheme for the bootstrap and approximated the critical values of tests formed using these statistics. However, their approach is inapplicable when the binary response variable is not always observed. Some missingness adjusted marked empirical processes are proposed under the framework that the missing binary responses are missing at random. We employ Dikta and Winkler’s modified model-based resampling scheme for the bootstrap in this set up and prove its asymptotic validity. Numerical comparison studies between the competing adjustments lead to interesting conclusions.