Improved model checking methods for parametric models with responses missing at random

In this paper, we consider the lack-of-fit test of a parametric model when the response variable is missing at random. The popular imputation and inverse probability weighting methods are first employed to tackle the missing data. Then by employing the projection technique, we propose empirical-process-based testing methods to check the appropriateness of the parametric model. The asymptotic properties of the test statistics are obtained under the null and local alternative hypothetical models. It is shown that the proposed testing methods are consistent, and can detect local alternative hypothetical models converging to the null model at the parametric rate. To determine the critical values, a consistent bootstrap method is proposed, and its asymptotic properties are established. The simulation results show that the tests outperform the existing methods in terms of empirical sizes and powers, especially under the situation with high dimensional covariates. Analysis of a diabetes data set of Pima Indians is carried out to demonstrate the application of the testing procedures.