STRONG CONVERGENCE RATES OF SEVERAL ESTIMATORS IN SEMIPARAMETRIC VARYING-COEFFICIENT PARTIALLY LINEAR MODELS

This article is concerned with the estimating problem of semiparametric varyingcoefficient partially linear regression models. By combining the local polynomial and least squares procedures Fan and Huang （2005） proposed a profile least squares estimator for the parametric component and established its asymptotic normality. We further show that the profile least squares estimator can achieve the law of iterated logarithm. Moreover, we study the estimators of the functions characterizing the non-linear part as well as the error variance. The strong convergence rate and the law of iterated logarithm are derived for them, respectively.

Quantile Regression of Ultra-high Dimensional Partially Linear Varying-coefficient Model with Missing Observations

In this paper,we focus on the partially linear varying-coefficient quantile regression with missing observations under ultra-high dimension,where the missing observations include either responses or covariates or the responses and part of the covariates are missing at random,and the ultra-high dimension implies that the dimension of parameter is much larger than sample size.Based on the B-spline method for the varying coefficient functions,we study the consistency of the oracle estimator which is obtained only using active covariates whose coefficients are nonzero.At the same time,we discuss the asymptotic normality of the oracle estimator for the linear parameter.Note that the active covariates are unknown in practice,non-convex penalized estimator is investigated for simultaneous variable selection and estimation,whose oracle property is also established.Finite sample behavior of the proposed methods is investigated via simulations and real data analysis.

Research on the nonlinear spherical percolation model with quadratic pressure gradient and its percolation characteristics

For bottom water reservoir and the reservoir with a thick oil formation, there exists partial penetration completion well and when the well products the oil flow in the porous media takes on spherical percolation. The nonlinear spheri-cal flow equation with the quadratic gradient term is deduced in detail based on the mass conservation principle, and then it is found that the linear percolation is the approximation and simplification of nonlinear percolation. The nonlinear spherical percolation physical and mathematical model under different external boundaries is established, considering the ef-fect of wellbore storage. By variable substitu-tion, the flow equation is linearized, then the Laplace space analytic solution under different external boundaries is obtained and the real space solution is also gotten by use of the nu-merical inversion, so the pressure and the pressure derivative bi-logarithmic nonlinear spherical percolation type curves are drawn up at last. The characteristics of the nonlinear spherical percolation are analyzed, and it is found that the new nonlinear percolation type curves are evidently different from linear per-colation type curves in shape and characteris-tics, the pressure curve and pressure derivative curve of nonlinear percolation deviate from those of linear percolation. The theoretical off-set of the pressure and the pressure derivative between the linear and the nonlinear solution are analyzed, and it is also found that the in-fluence of the quadratic pressure gradient is very distinct, especially for the low permeabil-ity and heavy oil reservoirs. The influence of the non-linear term upon the spreading of pressure is very distinct on the process of percolation, and the nonlinear percolation law stands for the actual oil percolation law in res-ervoir, therefore the research on nonlinear per-colation theory should be strengthened and reinforced.

Penalized total least squares method for dealing with systematic errors in partial EIV model and its precision estimation

When the total least squares(TLS)solution is used to solve the parameters in the errors-in-variables(EIV)model,the obtained parameter estimations will be unreliable in the observations containing systematic errors.To solve this problem,we propose to add the nonparametric part(systematic errors)to the partial EIV model,and build the partial EIV model to weaken the influence of systematic errors.Then,having rewritten the model as a nonlinear model,we derive the formula of parameter estimations based on the penalized total least squares criterion.Furthermore,based on the second-order approximation method of precision estimation,we derive the second-order bias and covariance of parameter estimations and calculate the mean square error(MSE).Aiming at the selection of the smoothing factor,we propose to use the U curve method.The experiments show that the proposed method can mitigate the influence of systematic errors to a certain extent compared with the traditional method and get more reliable parameter estimations and its precision information,which validates the feasibility and effectiveness of the proposed method.

Analytic solutions of a class of nonlinear partial differential equations

An approach is presented for computing the adjoint operator vector of a class of nonlinear （that is, partial-nonlinear） operator matrices by using the properties of conjugate operators to generalize a previous method proposed by the author. A unified theory is then given to solve a class of nonlinear （partial-nonlinear and including all linear） and non-homogeneous differential equations with a mathematical mechanization method. In other words, a transformation is constructed by homogenization and triangulation, which reduces the original system to a simpler diagonal system. Applications are given to solve some elasticity equations.

Numerical Solution of Nonlinear System of Partial Differential Equations by the Laplace Decomposition Method and the Pade Approximation

In this paper, Laplace decomposition method (LDM) and Pade approximant are employed to find approximate solutions for the Whitham-Broer-Kaup shallow water model, the coupled nonlinear reaction diffusion equations and the system of Hirota-Satsuma coupled KdV. In addition, the results obtained from Laplace decomposition method (LDM) and Pade approximant are compared with corresponding exact analytical solutions.

Variable Selection for Semiparametric Varying-Coefficient Partially Linear Models with Missing Response at Random

In this paper, we present a variable selection procedure by combining basis function approximations with penalized estimating equations for semiparametric varying-coefficient partially linear models with missing response at random. The proposed procedure simultaneously selects significant variables in parametric components and nonparametric components. With appropriate selection of the tuning parameters, we establish the consistency of the variable selection procedure and the convergence rate of the regularized estimators. A simulation study is undertaken to assess the finite sample performance of the proposed variable selection procedure.

Empirical Likelihood Based Diagnostics for Heteroscedasticity in Semiparametric Varying-Coefficient Partially Linear Models with Missing Responses

This paper proposes an empirical likelihood based diagnostic technique for heteroscedasticity for semiparametric varying-coefficient partially linear models with missing responses. Firstly, the authors complement the missing response variables by regression method. Then, the empirical likelihood method is introduced to study the heteroscedasticity of the semiparametric varying-coefficient partially linear models with complete-case data. Finally, the authors obtain the finite sample property by numerical simulation.

Efficient Estimation of a Varying-coefficient Partially Linear Binary Regression Model

This article considers a semiparametric varying-coefficient partially linear binary regression model. The semiparametric varying-coefficient partially linear regression binary model which is a generalization of binary regression model and varying-coefficient regression model that allows one to explore the possibly nonlinear effect of a certain covariate on the response variable. A Sieve maximum likelihood estimation method is proposed and the asymptotic properties of the proposed estimators are discussed. One of our main objects is to estimate nonparametric component and the unknowen parameters simultaneously. It is easier to compute, and the required computation burden is much less than that of the existing two-stage estimation method. Under some mild conditions, the estimators are shown to be strongly consistent. The convergence rate of the estimator for the unknown smooth function is obtained, and the estimator for the unknown parameter is shown to be asymptotically efficient and normally distributed. Simulation studies are carried out to investigate the performance of the proposed method.

Testing Serial Correlation in Semiparametric Varying-Coefficient Partially Linear EV Models

This paper studies estimation and serial correlation test of a semiparametric varying-coefficient partially linear EV model of the form Y = X^Tβ ＋Z^Tα（T） ＋ε,ξ = X ＋ η with the identifying condition E[（ε,η^T）^T] =0, Cov[（ε,η^T）^T] = σ^2Ip＋1. The estimators of interested regression parameters /3 , and the model error variance σ2, as well as the nonparametric components α（T）, are constructed. Under some regular conditions, we show that the estimators of the unknown vector β and the unknown parameter σ2 are strongly consistent and asymptotically normal and that the estimator of α（T） achieves the optimal strong convergence rate of the usual nonparametric regression. Based on these estimators and asymptotic properties, we propose the VN,p test statistic and empirical log-likelihood ratio statistic for testing serial correlation in the model. The proposed statistics are shown to have asymptotic normal or chi-square distributions under the null hypothesis of no serial correlation. Some simulation studies are conducted to illustrate the finite sample performance of the proposed tests.

Influence Diagnostics in Partially Varying-Coefficient Models

When a real-world data set is fitted to a specific type of models, it is often encountered that one or a set of observations have undue influence on the model fitting, which may lead to misleading conclusions. Therefore, it is necessary for data analysts to identify these influential observations and assess their impact on various aspects of model fitting. In this paper, one type of modified Cook＇s distances is defined to gauge the influence of one or a set observations on the estimate of the constant coefficient part in partially varying- coefficient models, and the Cook＇s distances are expressed as functions of the corresponding residuals and leverages. Meanwhile, a bootstrap procedure is suggested to derive the reference values for the proposed Cook＇s distances. Some simulations are conducted, and a real-world data set is further analyzed to examine the performance of the proposed method. The experimental results are satisfactory.

Inference on Varying-Coefficient Partially Linear Regression Model

The varying-coefficient partially linear regression model is proposed by combining nonparametric and varying-coefficient regression procedures. Wong, et al. （2008） proposed the model and gave its estimation by the local linear method. In this paper its inference is addressed. Based on these estimates, the generalized like- lihood ratio test is established. Under the null hypotheses the normalized test statistic follows a x2-distribution asymptotically, with the scale constant and the degrees of freedom being independent of the nuisance param- eters. This is the Wilks phenomenon. Furthermore its asymptotic power is also derived, which achieves the optimal rate of convergence for nonparametric hypotheses testing. A simulation and a real example are used to evaluate the performances of the testing procedures empirically.

变系数部分非线性模型的分位数回归估计

研究纵向数据缺失下变系数部分非线性分位数回归模型的估计问题.利用逆概率加权法结合分位数回归给出参数估计和非参估计;在一定条件下,证明了所给估计量的渐近正态性;通过数值模拟,验证了所提方法的有效性.

Efficient Statistical Inference for Partially Nonlinear Errors-in-Variables Models

In this paper, we consider the partially nonlinear errors-in-variables models when the non- parametric component is measured with additive error. The profile nonlinear least squares estimator of unknown parameter and the estimator of nonparametric component are constructed, and their asymptotic properties are derived under general assumptions. Finite sample performances of the proposed statistical inference procedures are illustrated by Monte Carlo simulation studies.

Statistical estimation in partially nonlinear models with random effects

In this article, a partially nonlinear model with random effects is proposed and its new estimation procession is provided. In order to estimate the link function, we propose generalised leastsquare estimate and B-splines estimate methods. Further, we also use the Gauss–Newton methodto construct the estimates of unknown parameters. Finally, we also consider the estimation forthe variance components. The consistency and the asymptotic normality of the estimator will beproved. Simulated and real examples are given to illustrate our proposed methodology, whichshows that our methods give effective estimation.

Shrinkage Estimation of Semiparametric Model with Missing Responses for Cluster Data

This paper simultaneously investigates variable selection and imputation estimation of semiparametric partially linear varying-coefficient model in that case where there exist missing responses for cluster data. As is well known, commonly used approach to deal with missing data is complete-case data. Combined the idea of complete-case data with a discussion of shrinkage estimation is made on different cluster. In order to avoid the biased results as well as improve the estimation efficiency, this article introduces Group Least Absolute Shrinkage and Selection Operator (Group Lasso) to semiparametric model. That is to say, the method combines the approach of local polynomial smoothing and the Least Absolute Shrinkage and Selection Operator. In that case, it can conduct nonparametric estimation and variable selection in a computationally efficient manner. According to the same criterion, the parametric estimators are also obtained. Additionally, for each cluster, the nonparametric and parametric estimators are derived, and then compute the weighted average per cluster as finally estimators. Moreover, the large sample properties of estimators are also derived respectively.

Medical X-Ray Image Enhancement Based on Kramer's PDE Model

The purpose of this study is to present an application of a novel enhancement technique for enhancing medical images generated from X-rays. The method presented in this study is based on a nonlinear partial differential equation （PDE） model, Kramer＇s PDE model. The usefulness of this method is investigated by experimental results. We apply this method to a medical X-ray image. For comparison, the X-ray image is also processed using classic Perona-Malik PDE model and Catte PDE model. Although the Perona-Malik model and Catte PDE model could also enhance the image, the quality of the enhanced images is considerably inferior compared with the enhanced image using Kramer＇s PDE model. The study suggests that the Kramer＇s PDE model is capable of enhancing medical X-ray images, which will make the X-ray images more reliable.

A higher order lattice BGK model for simulating some nonlinear partial differential equations

In this paper, we consider a one-dimensional nonlinear partial differential equation that has the form ut + αuux + βunux - γuxx + δuxxx = F(u). A higher order lattice Bhatnager-Gross-Krook (BGK) model with an amending-function is proposed. With the Chapman-Enskog expansion, different kinds of nonlinear partial differential equations are recovered correctly from the continuous Boltzmann equation. The numerical results show that this method is very effective.

Identification of Non-Varying Coefficients in Varying-Coefficient Models

A partially varying-coefficient model is one of the useful modelling tools.In this model, some coefficients of a linear model are kept to be constant whilst the others areallowed to vary with another factor. However, rarely can the analysts know a priori whichcoefficients can be assumed to be constant and which ones are varying with the given factor.Therefore, the identification problem of the constant coefficients should be solved before thepartially varying-coefficient model is used to analyze a real-world data set. In this article, asimple test method is proposed to achieve this task, in which the test statistic is constructed asthe sample variance of the estimates of each coefficient function in a well-knownvarying-coefficient model. Moreover two procedures, called F-approximation and three-moment χ~2approximation, are employed to derive the p-value of the test. Furthermore, some simulations areconducted to examine the performance of the test and the results are satisfactory.

Asymptotic Properties of Wavelet Estimators in Partially Linear Errors-in-variables Models with Long-memory Errors

While the random errors are a function of Gaussian random variables that are stationary and long dependent, we investigate a partially linear errors-in-variables（EV） model by the wavelet method. Under general conditions, we obtain asymptotic representation of the parametric estimator, and asymptotic distributions and weak convergence rates of the parametric and nonparametric estimators. At last, the validity of the wavelet method is illuminated by a simulation example and a real example.