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This paper provides some useful tests for fitting a parametric single-indexregression model when covariates are measured with error and validation data isavailable. We propose two tests whose consistency rates do not depend on thedimension of the covariate vector when an adaptive-to-model strategy isapplied. One of these tests has a bias term that becomes arbitrarily large withincreasing sample size but its asymptotic variance is smaller, and the other isasymptotically unbiased with larger asymptotic variance. Compared with theexisting local smoothing tests, the new tests behave like a classical localsmoothing test with only one covariate, and still are omnibus against generalalternatives. This avoids the difficulty associated with the curse ofdimensionality. Further, a systematic study is conducted to give an insight onthe effect of the values of the ratio between the sample size and the size ofvalidation data on the asymptotic behavior of these tests. Simulations areconducted to examine the performance in several finite sample scenarios.

An Adaptive-to-Model Test for Parametric Single-Index Errors-in-Variables Models