Checking nonparametric component for partial linear regression model with missing response

•Specification test for nonparametric part in PLM with missing response is studied.•Two quadratic conditional moment test methods are proposed.•Both two test statistics own limiting normal distributions under null hypothesis.•They can detect the alternative hypotheses at the optimal nonparametric rate.

The partial linear regression model is wildly used due to its well established theories, flexibility and easy interpretation. This paper aims to investigate the specification test of nonparametric component in partial linear model with response missing at random. Two quadratic conditional moment tests are proposed and both two test statistics own limiting normal distributions when nonparametric component is correctly specified. Our tests’ virtue is that pp-values can be easily determined based on limiting null distributions which are intractable for existing tests. The tests can detect the alternative hypotheses distinct from the null hypothesis at the optimal nonparametric rate for local smoothing-based methods. Simulation studies reveal that our tests can control type I error well and have excellent power performance. A HIV clinical trial real data is analyzed for illustrating our methods.