Abstract: This study provides a useful test for parametric single-index regression models when covariates are measured with errors and validation data are available. The proposed test is asymptotically unbiased, and its consistency rate does not depend on the dimension of the covariate vector. The proposed test behaves like a classical local smoothing test with only one covariate, and retains the omnibus property against general alternatives. This suggests that the proposed test can potentially alleviate the difficulty associated with the curse of dimensionality in this field. Furthermore, a systematic study is conducted to investigate the effect of the ratio between the sample size and the size of the validation data on the asymptotic behavior of these tests. Lastly, simulations are conducted to examine the performance in several finite sample scenarios.

Key words and phrases: Adaptive-to-model test, dimension reduction, errors-in-variables model.