Sequential Shrinkage Estimattion Methods On Some Statistical Regression Models

When cost of samples is considered in many fields of studies such as quality con-trol, clinical trial and product testing, researchers expect to minimize sample size as much as possible. Hence, sequential sampling method is one of important and power-ful tools to determine the minimal sample size under which some pre-specified criteria achieve. Under sequential sampling approach, sample size is not a fixed constant but a random variable changing with the sampling condition. It is well-known that sample size determined by traditional sequential methods heavily depends on number of regres-sion variables. However, as measurement techniques develop, more and more variables of the studied subject are measured, where one kind of data has lots of explanatory vari-ables. Generally, among these variables there are few ones, called " effective "variables, having great contributions to the model. If we do not distinguish effective variables from noneffective ones, the traditional sequential approach often waste a large number of samples in estimating the parameters of the "noneffective" variables, which leads to an inefficient sampling.To overcome the shortcoming of traditional sequential method, we construct a se-quentially adaptive shrinkage estimate method by bringing the idea of variable selection into the procedure of sequential sampling. The proposed approach can not only detect and remove the noneffective variables, but also obtain an estimate of the effective vari-ables achieving the preset accuracy. Since influence of the noneffective variables on model is ignored during sampling procedure, the efficiency of the traditional sequential method has been greatly improved. There are few references on studying this sequen-tial shrinkage estimating method. In order to illustrate the performance of the proposed sequential sampling method, we will construct the sequential adaptive shrinkage esti-mate methods for generalized linear model, generalized linear model with measurement errors in the covariates and Cox regression model, respectively. And we will establish the corresponding sequential sampling procedures based on the daptive shrinkage esti-mates.This article is composed of four chapters:Firstly, we introduce our research background of this article, including sequential method especially the history of sequential estimation and the development of general-ized linear models as well as the Cox regression model, then based on history results we will describe our works.Secondly, under the generalized linear regression model, we construct a adaptive shrinkage estimate (ASE) of the regression parameters based on the maximum quasi- likelihood estimation and shrinkage estimation. And we prove the asymptotic properties of ASE by using the Last time method. Then, we establish the sequential sampling strategy by using adaptive shrinkage estimate and construct the stopping rules and the confidence sets under both of the fixed and adaptive designs. Numerical simulation results show that performance of the proposed sequential adaptive shrinkage estimate is better than the traditional sequential sampling strategy in terms of saving samples. Especially when the number of effective variables is far less than that of the noneffective variables, the proposed method can save lots of samples.Furthermore, we also consider a sequential sampling strategy under the generalized linear model with measurement errors in the covariates. Existence of the measurement error brings some difficulties to construct the sequential adaptive shrinkage estimate. However, we can establish the adaptive shrinkage estimate of the generalized linear model regression parameters under a certain condition. We prove asymptotic properties of the adaptive shrinkage estimate (ASE), and study the properties of the sequential sampling strategies based on ASE. Simulation studies show that the proposed method is more efficient than the traditional sequential method.Finally, under Cox proportional hazards model, we present a partial likelihood of the regression parameters, obtain asymptotic properties as well as the rate of conver-gence. We construct the adaptive shrinkage estimate (ASE) based on the partial like-lihood. Then, we build a sequential sampling approach based on the ASE, study the asymptotic properties of the proposed method. We also evaluate the performance of power of the proposed method by the numerical simulation.