Simultaneous Confidence Bands For Nonparametric Component In Semiparametric Models

The semiparametric models combine the advantages of parametric models and nonpara-metric models, and these models are better to fit the real data. In recent years, the studies onsemiparametric models become a hot research direction for many statisticians. With the rapiddevelopment of technology and computer sciences, the studies of the panel data play more andmore important role in many fields, such as economy, medicine, finance, econometrics and soon. For the studies of the nonparametric components in semiparametric models under indepen-dent data or panel data, most of the literature concerned with the pointwise confidence bandsof the nonparametric components. However, the simultaneous confidence bands (SCBs) canvisually test the shape of the nonparametric component and consistently reflect the changes ofthe nonparametric component. Thus, the simultaneous confidence bands can be used to makevarious statistical inference tasks, and then a lot of investigative effort had been dedicated to thesimultaneous confidence bands in the literature by many statisticians. Therefore, the simultane-ous confidence bands of nonparametric functions are very important in both theory and practicalapplications. This dissertation mainly studies the construction of the simultaneous confidencebands for nonparametric component in semiparametric partially linear models under indepen-dent data and panel data, respectively.The main works of this dissertation includes the following two aspects:1. For semiparametric partially linear models with independent data, we consider the si-multaneous confidence band (SCB) for the nonparametric component. We establish the asymp-totic distribution of their maximum absolute deviation between the estimated nonparametriccomponent and the true nonparametric component under some suitable conditions, and hencethe result can be used to construct the simultaneous confidence band of the nonparametric com-ponent for various statistical inference tasks. Simulation studies is used to illustrate the proposedmethod under the limited samples. In addition, the proposed method is applied to the study offemale labor supply data collected in the east Germany in1994.2. For semiparametric partially linear fixed effects models with panel data, we constructthe simultaneous confidence band for the nonparametric component. Although the fixed effectsmodels are able to capture the individual heterogeneity in the data more flexibly, this also bringsmany challenges for the statistical inference of the parametric or nonparametric components inmodel. Based on the idea of least-squares dummy-variable approach in panel data parametricmodels and the nonparametric local linear regression technique, we remove the fixed effects, andfurther obtain the estimators of parametric and nonparametric components, which do not dependon the fixed effects. We establish the asymptotic distribution of their maximum absolute devi- ation between the estimated nonparametric component and the true nonparametric componentunder some suitable conditions, and hence the result can be used to construct the simultaneousconfidence band of the nonparametric component. Based on the asymptotic distribution, it be-comes difficult for the construction of the simultaneous confidence band. The reason is that theasymptotic distribution involves the estimators of the asymptotic bias and conditional variance,and the choice of the bandwidth for estimating the second derivative of nonparametric function.Clearly, these will cause computational burden and accumulative errors. To overcome theseproblems, we propose a Bootstrap method to construct the simultaneous confidence band. Sim-ulation studies indicate that the proposed Bootstrap method exhibits better performance underthe limited samples.Finally, in conclusion and prospect, we summarize the main research achievements andinnovation acquired in this dissertation and point out the further research question and direction.