Goodness-of-fit tests in measurement error models with replications

Abstract

In this dissertation, goodness-of-fit tests are proposed for checking the adequacy of parametric distributional forms of the regression error density functions and the error-prone predictor density function in measurement error models, when replications of the surrogates of the latent variables are available. In the first project, we propose goodness-of-fit tests on the density function of the regression error in the errors-in-variables model. Instead of assuming that the distribution of the measurement error is known as is done in most relevant literature, we assume that replications of the surrogates of the latent variables are available. The test statistic is based upon a weighted integrated squared distance between a nonparametric estimate and a semi-parametric estimate of the density functions of certain residuals. Under the null hypothesis, the test statistic is shown to be asymptotically normal. Consistency and local power results of the proposed test under fixed alternatives and local alternatives are also established. Finite sample performance of the proposed test is evaluated via simulation studies. A real data example is also included to demonstrate the application of the proposed test. In the second project, we propose a class of goodness-of-fit tests for checking the parametric distributional forms of the error-prone random variables in the classic additive measurement error models. We also assume that replications of the surrogates of the error-prone variables are available. The test statistic is based upon a weighted integrated squared distance between a non-parametric estimator and a semi-parametric estimator of the density functions of the averaged surrogate data. Under the null hypothesis, the minimum distance estimator of the distribution parameters and the test statistics are shown to be asymptotically normal. Consistency and local power of the proposed tests under fixed alternatives and local alternatives are also established. Finite sample performance of the proposed tests is evaluated via simulation studies.