Strong Convergence of Partitioning Estimation for Nonparametric Regression Function under Dependence Samples

In this paper, we study the strong consistency for partitioning estimation of regression function under samples that axe φ-mixing sequences with identically distribution.Key words: nonparametric regression function; partitioning estimation; strong convergence;φ-mixing sequences.

Using Evolutionary Computation to Solve Problems in Nonparametric Regression

This paper studies evolutionary mechanism of parameter selection in the construction of weight function for Nearest Neighbour Estimate in nonparametric regression. Construct an algorithm which adaptively evolves fine weight and makes good prediction about unknown points. The numerical experiments indicate that this method is effective. It is a meaningful discussion about practicability of nonparametric regression and methodology of adaptive model-building.

Floating Car Data Based Nonparametric Regression Model for Short-Term Travel Speed Prediction

A K-nearest neighbor （K-NN） based nonparametric regression model was proposed to predict travel speed for Beijing expressway. By using the historical traffic data collected from the detectors in Beijing expressways, a specically designed database was developed via the processes including data filtering, wavelet analysis and clustering. The relativity based weighted Euclidean distance was used as the distance metric to identify the K groups of nearest data series. Then, a K-NN nonparametric regression model was built to predict the average travel speeds up to 6 min into the future. Several randomly selected travel speed data series, collected from the floating car data （FCD） system, were used to validate the model. The results indicate that using the FCD, the model can predict average travel speeds with an accuracy of above 90%, and hence is feasible and effective.

BOOTSTRAP WAVELET IN THE NONPARAMETRIC REGRESSION MODEL WITH WEAKLY DEPENDENT PROCESSES

This paper introduces a method of bootstrap wavelet estimation in a non-parametric regression model with weakly dependent processes for both fixed and random designs. The asymptotic bounds for the bias and variance of the bootstrap wavelet estimators are given in the fixed design model. The conditional normality for a modified version of the bootstrap wavelet estimators is obtained in the fixed model. The consistency for the bootstrap wavelet estimator is also proved in the random design model. These results show that the bootstrap wavelet method is valid for the model with weakly dependent processes.

Testing heteroscedasticity in nonparametric regression models based on residual analysis

The importance of detecting heteroscedasticity in regression analysis is widely recognized because efficient inference for the regression function requires that heteroscedasticity should be taken into account. In this paper, a simple test for heteroscedasticity is proposed in nonparametric regression based on residual analysis. Furthermore, some simulations with a comparison with Dette and Munk＇s method are conducted to evaluate the performance of the proposed test. The results demonstrate that the method in this paper performs quite satisfactorily and is much more powerful than Dette and Munk＇s method in some cases.

Asymptotic Property for the Estimator of Nonparametric Regression Models Under Negatively Orthant Dependent Errors

In this paper, by using some inequalities of negatively orthant dependent(NOD,in short) random variables and the truncated method of random variables, we investigate the nonparametric regression model. The complete consistency result for the estimator of g(x) is presented.

ASYMPTOTICS OF THE RESIDUALS DENSITY ESTIMATION IN NONPARAMETRIC REGRESSION UNDER m(n)-DEPENDENT SAMPLE

Let Y_i=M(X_i)+ei, where M(x)=E(Y|X=x) is an unknown realfunction on B(? R), {(X_1,Y_i)} is a stationary and m(n)-dependent sample from(X, Y), the residuals {e_i} are independent of {X_i} and have unknown common densityf(x). In [2] a nonparametric estimate f_n(x) for f(x) has been proposed on the basisof the residuals estimates. In this paper, we further obtain the asymptotic normalityand the law of the iterated logarithm of f_n(x) under some suitable conditions. Theseresults together with those in [2] bring the asymptotic theory for the residuals densityestimate in nonparametric regression under m(n)-dependent sample to completion.

Longitudinal Survey, Nonmonotone, Nonresponse, Imputation, Nonparametric Regression

The study focuses on the imputation for the longitudinal survey data which often has nonignorable nonrespondents. Local linear regression is used to impute the missing values and then the estimation of the time-dependent finite populations means. The asymptotic properties (unbiasedness and consistency) of the proposed estimator are investigated. Comparisons between different parametric and nonparametric estimators are performed based on the bootstrap standard deviation, mean square error and percentage relative bias. A simulation study is carried out to determine the best performing estimator of the time-dependent finite population means. The simulation results show that local linear regression estimator yields good properties.

Nonparametric Feature Screening via the Variance of the Regression Function

This article develops a procedure for screening variables, in ultra high-di- mensional settings, based on their predictive significance. This is achieved by ranking the variables according to the variance of their respective marginal regression functions (RV-SIS). We show that, under some mild technical conditions, the RV-SIS possesses a sure screening property, which is defined by Fan and Lv (2008). Numerical comparisons suggest that RV-SIS has competitive performance compared to other screening procedures, and outperforms them in many different model settings.

Robust Depth-Weighted Wavelet for Nonparametric Regression Models

In the nonparametric regression models, the original regression estimators including kernel estimator, Fourier series estimator and wavelet estimator are always constructed by the weighted sum of data, and the weights depend only on the distance between the design points and estimation points. As a result these estimators are not robust to the perturbations in data. In order to avoid this problem, a new nonparametric regression model, called the depth-weighted regression model, is introduced and then the depth-weighted wavelet estimation is defined. The new estimation is robust to the perturbations in data, which attains very high breakdown value close to 1/2. On the other hand, some asymptotic behaviours such as asymptotic normality are obtained. Some simulations illustrate that the proposed wavelet estimator is more robust than the original wavelet estimator and, as a price to pay for the robustness, the new method is slightly less efficient than the original method.

CONFIDENCE INTERVALS FOR NONPARAMETRIC REGRESSION FUNCTIONS WITH MISSING DATA： MULTIPLE DESIGN CASE

This paper considers two estimators of θ= g（x） in a nonparametric regression model Y = g（x） ＋ ε（x∈ （0, 1）p） with missing responses： Imputation and inverse probability weighted esti- mators. Asymptotic normality of the two estimators is established, which is used to construct normal approximation based confidence intervals on θ.

Wavelet Detection and Estimation of Change Points in Nonparametric Regression Models under Random Design

A wavelet method of detection and estimation of change points in nonparametric regression models under random design is proposed. The confidence bound of our test is derived by using the test statistics based on empirical wavelet coefficients as obtained by wavelet transformation of the data which is observed with noise. Moreover, the consistence of the test is proved while the rate of convergence is given. The method turns out to be effective after being tested on simulated examples and applied to IBM stock market data.

Fixed Design Nonparametric Regression with Truncated and Censored Data

Abstract In this paper we consider a fixed design model in which the observations are subject to left truncation and right censoring. A generalized product-limit estimator for the conditional distribution at a given covariate value is proposed, and an almost sure asymptotic representation of this estimator is established. We also obtain the rate of uniform consistency, weak convergence and a modulus of continuity for this estimator. Applications include trimmed mean and quantile function estimators.These applications demonstrate the usefulness of the new matrix products.

Complete moment convergence for weighted sums of widely orthant-dependent random variables and its application in nonparametric regression models

We establish some results on the complete moment convergence for weighted sums of widely orthant-dependent(WOD)random variables,which improve and extend the corresponding results of Y.F.Wu,M.G.Zhai,and J.Y.Peng[J.Math.Inequal.,2019,13(1):251–260].As an application of the main results,we investigate the complete consistency for the estimator in a nonparametric regression model based on WOD errors and provide some simulations to verify our theoretical results.

Asymptotics of the“Minimum L_1-Norm”Estimates in Nonparametric Regression Models

Consider the nonparametric regression model Y=go（T）+u,where Y is real-valued, u is a random error,T ranges over a nondegenerate compact interval,say[0,1],and go（·）is an unknown regression function,which is m（m≥0）times continuously differentiable and its ruth derivative,g0（m）,satisfies a H■lder condition of order γ（m +γ】1/2）.A piecewise polynomial L1- norm estimator of go is proposed.Under some regularity conditions including that the random errors are independent but not necessarily have a common distribution,it is proved that the rates of convergence of the piecewise polynomial L1-norm estimator are o(n-2（m+γ）+1/m+γ-1/δalmost surely and o(n-2（m+γ）+1/m+γ-δ)in probability,which can arbitrarily approach the optimal rates of convergence for nonparametric regression,where δ is any number in （0, min（（m+γ-1/2）/3,γ））.

Estimation of Regression Function for Nonequispaced Samples Based on Warped Wavelets

We consider the problem of estimating an unknown density and its derivatives in a regression setting with random design. Instead of expanding the function on a regular wavelet basis, we expand it on the basis , a warped wavelet basis. We investigate the properties of this new basis and evaluate its asymptotic performance by determining an upper bound of the mean integrated squared error under different dependence structures. We prove that it attains a sharp rate of convergence for a wide class of unknown regression functions.

ON THE CONSISTENCY OF CROSS-VALIDATIONIN NONLINEAR WAVELET REGRESSION ESTIMATION

For the nonparametric regression model Y-ni = g(x(ni)) + epsilon(ni)i = 1, ..., n, with regularly spaced nonrandom design, the authors study the behavior of the nonlinear wavelet estimator of g(x). When the threshold and truncation parameters are chosen by cross-validation on the everage squared error, strong consistency for the case of dyadic sample size and moment consistency for arbitrary sample size are established under some regular conditions.

WAVELET-BASED ESTIMATORS OF MEAN REGRESSION FUNCTION WITH LONG MEMORY DATA

This paper provides an asymptotic expansion for the mean integrated squared error （MISE） of nonlinear wavelet-based mean regression function estimators with long memory data. This MISE expansion, when the underlying mean regression function is only piecewise smooth, is the same as analogous expansion for the kernel estimators.However, for the kernel estimators, this MISE expansion generally fails if the additional smoothness assumption is absent.

The M-estimate of Local Linear Regression with Variable Window Breadth

In this paper, by using the Brouwer fixed point theorem, we consider the existence and uniqueness of the solution for local linear regression with variable window breadth.

Local Polynomial Regression Estimator of the Finite Population Total under Stratified Random Sampling: A Model-Based Approach

In this paper, auxiliary information is used to determine an estimator of finite population total using nonparametric regression under stratified random sampling. To achieve this, a model-based approach is adopted by making use of the local polynomial regression estimation to predict the nonsampled values of the survey variable y. The performance of the proposed estimator is investigated against some design-based and model-based regression estimators. The simulation experiments show that the resulting estimator exhibits good properties. Generally, good confidence intervals are seen for the nonparametric regression estimators, and use of the proposed estimator leads to relatively smaller values of RE compared to other estimators.