Several Studies Of Measurement Errors In Semiparametric Models And Nonlinear Models

This dissertation is devoted to develop some statistical models, methods and theories in the literature of measurement errors.In the last two decades, statistical analysis on the measurement error data has become one of the most important issues because measurement error data are often encountered in many fields, such as medicine economics, engineering. As we know, when we deal with the measurement error data, the naive procedure by simply ignoring measurement errors always leads to a biased and inconsistent estimator. As such, we should solve such practical problems by choosing rel-ative measurement error models, consequently, which promotes a continuous development of statistical research on the measurement error data. In this dissertation, we focus on two types of measurement error data. One has a multiplicative fashion, which we call distortion measure-ment error models. Another one is additive measurement error models, including the classical measurement error models.In the first part of this thesis, including Chapter â…¡, â…¢ and â…£, we introduce some important semiparametric models into distortion measurement error data analysis. Specifically mentions, firstly, in Chapter â…¡, we study parameter estimation and variable selection of partial linear sin-gle index models when the response and the covariates in the linear part are measured with errors and distorted by unknown distorting functions of one commonly observable confounding variable. To solve the problem of parameter estimation, we draw from the method of mini-mum average variance estimation (MAVE, Xia et al.[80]). For variable selection, we propose a sparse principle component (SPC) analysis based on the recently developed coordinate inde-pendent sparse estimation (CISE, Chen et al.[7]). This is the first attempt to deal with variable selection problem on the distortion measurement error. We establish the asymptotic properties for the estimation procedure and variable selection procedure. We conduct numerical simula-tion experiments to examine the feasibility and efficacy of the proposed procedures, and a real dataset is analyzed for an illustration. In Chapter â…¢, in the context of one commonly confound-ing variable, we firstly introduce a sufficient dimension reduction procedure to this distortion measurement error. Together with Zhu et al.[88]'s CUME method and Cui et al.[12]'s direct-plug-in method, we proposed a dimension reduction based method to estimate the dimension reduction space when the data is measured with distortion errors. We investigate the asymp-totic properties of the proposed estimators. Moreover, we re-visit the Boston housing data and re-analyze this data with a perfect new point of view. In Chapter IV, we consider a classic non-linear regression model with multiple confounding variables. The distorting functions of these confounding variables are modeled as single index models. We adopt EFM method recently proposed by Cui et al.[13] to estimate the single index parameters in the distorting functions. We then use Cui et al.[12]'s direct-plug-in method and nonlinear least squares to estimate pa-rameters in the nonlinear models, and we also consider to use empirical likelihood to construct confidence region for these parameters. Next, we establish large-sample properties for those estimate, conduct numerical simulation to evaluate their performance. Furthermore, we apply our model and method to analyze a diabetes data.In the next two chapters, we focus on statistical inference on additive measurement error data. In Chapter V, we still consider the parameter estimation and variable selection of partial linear single index models. Some covariates in the linear part are not observed, but their ancil-lary variables are available. We can firstly estimate these unobservable covariates through their ancillary variables. Based on these estimated covariates, we proposed an estimation procedure and variable selection procedure. As we know, the profile least squares estimators proposed by Liang et al.[53] can achieve semi parametric efficient bound. At the same time, the "leave-one-component " procedure proposed by Zhu et al.[93] can make full use of the information that the norm of single index equals one. We take full advantage of these two estimation procedures, we proposed a "leave-one-component " profile least square estimation procedure. For the variable selection, we adopt Fan and Li [17]'s SCAD-penalty procedure and further proposed a penal-ized "leave-one-component " profile least square. We investigate asymptotic properties for the estimation procedure and variable selection procedure. Also, we conduct numerical simulation and apply our method to a real data for an illustration.In Chapter VI, we consider a dimension reduction procedure on a classical additive mea-surement error data. We first construct the surrogate variables for the unobserved predictors. Next, together with CUME estimation procedure proposed by Zhu et al.[88], we can use the observed response and surrogates to estimate the original dimension reduction subspace pro-duced by the observed response and the unobserved predictors. Our estimation procedure can even deal with high-dimensional predictors. In theory, we investigate the divergent rate for the dimension of predictors when consistency and asymptotic normality hold. We compare some re-lated alternative methods with our method to illustrate the efficiency of our proposal, we further analyze a real data to demonstrate the utility of our method.