This thesis addresses some aspects of regression model checking problems when the covariates are observed with measurement errors. Both classical error-in-variables models and Berkson models are investigated when validation data is available.In Tobit error-in-variables regression models, the response is truncated at a given level while the covariate is collected with errors. In this thesis we assume the density of measurement error to be unknown. Using the calibration idea, a new regression function is derived under the null hypothesis and estimated using the kernel smoothing method and validation data. Then a class of test statistics are constructed using the nonparametric residuals based on kernel regression estimators when validation data is available. The proposed class of tests is shown to be robust to the choices of parameter estimators and consistent against a large class of fixed alternatives. The asymptotic normality of these test statistics is established under the null hypothesis and under a sequence of local alternatives. A practical bandwidth selection strategy is developed. A finite sample simulation study shows superiority of a member of the proposed class of tests over the two existing tests in terms of empirical power. A real data application is presented to validate the current understanding of the data set.In Berkson models, without specifying the measurement error density, the calibrated regression function is estimated using both the primary data containing the responses and the validation data. A kernel smoothed integrated square distance is defined between the responses and the regression estimator. The parameter estimators are obtained by minimizing the integrated square distance. Further the test statistic is constructed based on the minimized distance. The consistency and asymptotic normality of these estimators are proved. The asymptotic null distribution of the proposed class of test statistics based on the corresponding minimized distances and the test consistency against certain alternatives are also established. A simulation study shows desirable behavior of a member of these minimum distance estimators and tests.
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