# Application of partial consistency for the semi-parametric models

• Jingxin Zhao

Student thesis: Doctoral Thesis

### Abstract

The semi-parametric model enjoys a relatively flexible structure and keeps some of the simplicity in the statistical analysis. Hence, there are abundance discussions on semi-parametric models in the literature. The concept of partial consistency was firstly brought up in Neyman and Scott (1948). It was said the in cases where infinite parameters are involved, consistent estimators are always attainable for those "structural" parameters. The "structural' parameters are finite and govern infinite samples. Since the nonparametric model can be regarded as a parametric model with infinite parameters, then the semi-parametric model can be easily transformed into a infinite-parametric model with some "structural" parameters. Therefore, based on this idea, we develop several new methods for the estimating and model checking problems in semi-parametric models. The implementation of applying partial consistency is through the method "local average". We consider the nonparametric part as piecewise constant so that infinite parameters are created. The "structural" parameters shall be the parametric part, the model residual variance and so on. Due to the partial consistency phenomena, classical statistic tools can then be applied to obtain consistent estimators for those "structural" parameters. Furthermore, we can take advantage of the rest of parameters to estimate the nonparametric part. In this thesis, we take the varying coefficient model as the example. The estimation of the functional coefficient is discussed and relative model checking methods are presented. The proposed new methods, no matter for the estimation or the test, have remarkably lessened the computation complexity. At the same time, the estimators and the tests get satisfactory asymptotic statistical properties. The simulations we conducted for the new methods also support the asymptotic results, giving a relatively efficient and accurate performance. What's more, the local average method is easy to understand and can be flexibly applied to other type of models. Further developments could be done on this potential method. In Chapter 2, we introduce a local average method to estimate the functional coefficients in the varying coefficient model. As a typical semi-parametric model, the varying coefficient model is widely applied in many areas. The varying coefficient model could be seen as a more flexible version of classical linear model, while it explains well when the regression coefficients do not stay constant. In addition, we extend this local average method to the semi-varying coefficient model, which consists of a linear part and a varying coefficient part. The procedures of the estimations are developed, and their statistical properties are investigated. Plenty of simulations and a real data application are conducted to study the performance of the proposed method. Chapter 3 is about the local average method in variance estimation. Variance estimation is a fundamental problem in statistical modeling and plays an important role in the inferences in model selection and estimation. In this chapter, we have discussed the problem in several nonparametric and semi-parametric models. The proposed method has the advantages of avoiding the estimation of the nonparametric function and reducing the computational cost, and can be easily extended to more complex settings. Asymptotic normality is established for the proposed local average estimators. Numerical simulations and a real data analysis are presented to illustrate the finite sample performance of the proposed method. Naturally, we move to the model checking problem in Chapter 4, still taking varying coefficient models as an example. One important and frequently asked question is whether an estimated coefficient is significant or really "varying". In the literature, the relative hypothesis tests usually require fitting the whole model, including the nuisance coefficients. Consequently, the estimation procedure could be very compute-intensive and time-consuming. Thus, we bring up several tests which can avoid unnecessary functions estimation. The proposed tests are very easy to implement and their asymptotic distributions under null hypothesis have been deduced. Simulations are also studied to show the properties of the tests.

Date of Award 30 Aug 2017 English Heng PENG (Supervisor)

### User-Defined Keywords

• Estimation theory
• Mathematical models
• Nonparametric statistics

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