A goodness-of-fit test for variable-adjusted models

This research provides a projection-based test to check parametric single-index regression structure in variable-adjusted models. An adaptive-to-model strategy is employed, which makes the proposed test work better on the significance level maintenance and more powerful than existing tests. With mild conditions, the proposed test asymptotically behaves like a test that is for classical regression setup without distortion errors in observations. Numerical studies with simulated and real data are conducted to examine the performance of the test in finite sample scenarios.