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Some New Estimator in Linear Mixed Models with Measurement error
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Linear mixed models (LMMs) are an important tool for the analysis of a broad range of structures including longitudinal data, repeated measures data (including cross-over studies), growth and dose-response curve data, clustered (or nested) data, multivariate data, and correlated data. In many practical situations, the observation of variables is subject to measurement errors, and ignoring these in data analysis can lead to inconsistent parameter estimation and invalid statistical inference. Therefore, it is necessary to extend LMMs by taking the effect of measurement errors into account. Multicollinearity and fixed-effect variables with measurement errors are two well-known problems in the analysis of linear regression models. Although there exists a large amount of research on these two problems, there is by now no single technique superior to all other techniques for the analysis of regression models when these problems are present. In this thesis, we propose two new estimators using Nakamura's approach in LMM with measurement errors to overcome multicollinearity. We consider that prior information is available on fixed and random effects. The first estimator is the new mixed ridge estimator (NMRE) and the second estimator is the weighted mixed ridge estimator (WMRE). We investigate the asymptotic properties of these proposed estimators and compare the performance of them over the other estimators using the mean square error matrix (MSEM) criterion. Finally, a data example and a Monte Carlo simulation are also provided to show the theoretical results.