The Theoretical And Application Research On Nonlinear Semiparametric Model

Nonparametric and semiparametric models have flourished during the last thirty years, they have been applied widely in many fields and their theoretical properties have been studied extensively. Semiparametric model is a statistical model consisting of both parametric and nonparametric components. The evaluating of the efficiency of an estimator in a purely parametric model or a purely nonparametric model has been well-developed. But the efficiency problem for a mixed parametric-nonparametric model is relatively new. Now the research focus on semiparametric models are how to construct the estimation of the parametric and nonparametric components and what is the large-sample property of these estimations. Nonlinear parametric models have been studied deeply, and a set of basic theory have been set up, such as the measurement of the non-linearity of nonlinear models and the statistics property of nonlinear parametric estimation. The focus of this paper is to expand the estimating technique of nonlinear parametric models to semiparametric models and to expand the estimating theory of linear semiparametric parametric into nonlinear semiparametric model and setup the basic estimation theory of nonlinear semiparametric model under Least-square Principle. Because a semiparametric model consists of a nonparametric component, it can overcome the limitation of parametric model in dealing with some practical problems. In chapter 2 the theory and technique of nonlinear parametric models are discussed in detail, including the estimating principles and techniques of nonlinear parametric models. The estimating principles includes Least-Square, Maximum Likelihood, Robustness and Bayes principles. On the bases of non-linearity and its measurement, this chapter discusses the techniques of nonlinear parametric models, including approximate technique, iteration technique, direct technique and some random searching methods such as Simple Geometry technique, Simulated Annealing Algorithm and Genetic (or Evolutionary) Algorithms. Chapter 3 deals with the estimating of nonparametric models, including the constructing of estimator, evaluating the quality of estimator and the selecting smoothing parameters. As to the constructing of estimator, this part mainly discusses the Kernel, Nearest-Neighbor, Fourier series, Polynomial, Spline, Blocked-polynomial, and Wavelet estimator of nonparametricmodels. Since there are smoothing parametric selecting problems in all estimators, this chapter discusses several methods about selecting smoothing parameters, they are Cross-Validated method, General Cross-Validated method, L-Curve method and Iterative Plug-in method The effect of some techniques is tested with numerical examples.In chapter 4, the theory of linear semiparametric model is introduced. This part presents the formulas of Kernel, Nearest-Neighbor, Fourier series, Spline, Blocked-polynomial, Wavelet and Two-phase estimator under Least-Square principle of linear semiparametric model, including parametric and nonparametric components. The statistic property of some estimators are also discussed. The main innovations of this chapter include: proposes the estimating formulas of the aforementioned methods, derives the formulary for asymptotic variance of the estimation of parametric component under Least-Square principle, proves that the estimation of parametric and nonparametric components of linear semiparametric model are bias, and proposes the calculating method of bias for aforementioned methods. At the end of this chapter, a technique for solving nonlinear parametric model by semiparametric model is proposed. It expands the non-linear functions as Taylor series at an approximate point, the expanding includes 2 parts, the first order term and the remainder, and the 2 parts make up a semi-parametric model. The effect of some techniques is tested with numerical examples. In chapter 5, the estimating theory of nonlinear semiparametric models is researched, including the calculating formulae of the estimation of parametric and nonparametric components, the statistic property of the estimation of parametric component. The main innovations of this part are: gives the nonlinear estimating formulas of Kernel, Nearest-Neighbor, Nature Spline, Blocked-polynomials, Trigonometric-series, Wavelet and Two-phase Estimator under the principle of Least-Square, derives the formulae for Guass-Newton Method and some approximate nonlinear methods, derives the formula for asymptotic variance of the estimation of parametric component under Least-Square principle, proves that the coefficient matrix of normal equation is positive definite, proves that the estimation of parametric and nonparametric components of Least-Square Kernel Estimator for nonlinear semiparametric model are bias, and the bias property is suitable for the other estimating methods of nonlinear semiparametric model. At the end of this chapter simulated data are used to verify theeffectiveness of these techniques for nonlinear semiparametric model.Chapter 6 expounds the definition and property of systematic errors, studies the systematic errors in GPS and techniques to handle them. A new scheme is proposed which is using nonlinear semiparametric models to deal with systematic errors in GPS.