Abstract: This paper discusses the problem of fitting a parametric model to the nonparametric component in partially linear regression models when covariates in parametric and nonparametric parts are subject to Berkson measurement errors. The proposed test is based on the supremum of a martingale transform of a certain partial sum process of calibrated residuals. The asymptotic null distribution of this transformed process is shown to be the same as that of a time transformed standard Brownian motion. Consistency of this sequence of tests against some fixed alternatives and asymptotic power under some local nonparametric alternatives are also discussed. A simulation study is conducted to assess the finite sample performance of the proposed test. A Monte Carlo power comparison with some existing tests shows some superiority of the proposed test at the chosen alternatives.
Key words and phrases: Asymptotically distribution free, consistency, local alternatives, marked empirical process.