Statistical Inference Based On Quantile Regression Models And Its Application With Complex Data

This thesis mainly studies statistical inference based on quantile regression models and its application with complex data (such as right censored length biased data, longitudinal data and missing data etc.) These complex data are very important in financial, economic, biological and medical etc. practical application fields, and complex data be one of the forefront of modern statistics and hot issues.The thesis is divided into seven chapters. In Chapter One, we introduce the background of this thesis from the aspects of data, model, statistical method, motivation.Length-biased data attracts many attentions because this kind of data is often encountered in many practical application fields such as epidemiological cohort studies, cancer screening trials and labor economy studies. Existing lit-eratures are based on the transformation model, Cox model, accelerate failure time model and assumed the censoring variable independent of covariate. In Chapter Two, we consider a quantile regression model with the right-censored length-biased data and modeling the censoring variable by Cox model. We con-ducted the information censoring induced by length-biased sampling, adjusted the effect of the length biased sampling and right censoring by constructing the inverse probability weighted estimating equation. The classic empirical process and semiparametric method were applied to derive the asymptotic property of quantile regression estimator. It is easy to give an estimate of the asymptotic variance by an simple re-sampling method. Simulation study and the Channing house data analysis exhibit that the proposed methods are perform very well.The constant coefficient quantile regression model can't meet the demand when the risk factors on the response variable change as a quantity such as such as time and space, or what be interested are the dynamic influence of risk expo-sure factor and the interaction between different risk factors. In Chapter Three, we modeling the right censored and length biased data by the varying-coefficient model. Our approach not only allows the direct estimation of the conditional quantiles of survival times based on a flexible additive-linear model structure, it also has the important appeal of permitting dependence between the censoring variable and the covariates. In order to improve the efficiency, we further de-velop a composite quantile regression procedure. Using local inverse probability weighted estimating equations and composite local inverse probability weighted estimating equations, local linear estimators and local linear composite estimators of the coefficients are developed, and their asymptotic properties investigated. A resampling method is developed for computing the covariance of estimates. The small sample properties of the proposed procedure are investigated in a Monte Carlo study and a real data example illustrates the application of the method in practice.In Chapter Four, we modeling the length biased and right censored data by the conditional restriction model which very generalized and include the con-ditional mean regression and conditional quantile regression. Constructe the inverse probability estimating equation and make statistical inference under the framework of GMM. It is worthy noting that proposed method could conduct the smooth and non-smooth conditional restriction models simultaneously. The consistency and asymptotic normality of GMM estimator are derived. The finite sample performance of proposed method is evaluate in simulation study part.Longitudinal data are often encountered in medical follow-up studies and economic research. Conditional mean regression and conditional quantile regres-sion are often used to fit longitudinal data. Many methods focused on the case that observation time is completely/conditionally (given the covariates) indepen-dent of response variables or equal-time span observations. Few papers considered the case that response variables depend on the observation times or observation times are random variables associating with a counting process. In the Chapter Five, a marginally conditional quantile regression is proposed for modeling lon-gitudinal data with random observing times. Estimators of conditional quantile regression are derived by constructing non-smooth estimating equations when the observation time follows a counting process. Both consistency and asymptotic normality for these estimators are established. Asymptotic variance is estimated by a resampling method.In Chapter Six. we consider the variable selection of the longitudinal data with random informative observation times by using SCAD and adaptive LASSO penalty function. Derived the oracle property of the penalized quantile regression estimator based on proposed penalized quantile regression estimating equation. Moreover, we develop the composite quantile regression(CQR) type inference procedure to improve the efficiency of quantile estimator, the large sample prop-erties also be investigated. Variable selection also be considered by construct the penalized CQR estimating equation and the oracle properties of the correspond the penalized CQR estimator are presented.It is well-known that when data are missing, direct imputation of the missing observations can lead to estimating equations that are biased, violating a fun-damental property of estimating equations inference. A recent chapter by Zhou, Wan and Wang (2008) developed an alternative kernel-based imputation proce-dure where the objects of imputation are the estimating functions and not the missing data themselves, and derived asymptotic properties of estimators based on empirical likelihood and generalized methods of moments for cases where aux-iliary information on the unknowns is available. One basic assumption of Zhou, Wan&Wang's [1] analysis is that the underlying estimating equations are all smooth, and this excludes many important statistical applications. The primary object of Chapter Seven is to extend Zhou. Wan&Wang's[1] investigation to systems of non-smooth estimating equations. This requires the development of a completely different set of proof techniques as the Taylor series expansion is no longer applicable when the underlying functions are non-smooth. Furthermore, we develop two resampling methods for obtaining the asymptotic variances of the estimators. A simulation study and a real data example illustrate that the proposed procedures work well in finite samples. The proposed methodology can be applied to a wide range of statistical problems such as quantile and rank regressions, truncated mean and robust estimation, and the estimation of ROC curves and distribution functions.