Nonparametrie recursive estimator for mean residual life and vitality function under dependence conditions

In this paper, we develop a nonparametrie recursive estimator for the vitality and mena residual life function, based on kernel density estimators under mixing dependence conditions. The consistency and asymptotic normality of the estimator are established, under suitable regularity conditions. It is also shown that the Integrated Mean Squared Error converges to zero. The paper is concluyed with some simulation results.     

The coefficient of determination and its adjusted version in linear regression models

This article presents a comparative study of the efficiency properties of the coefficient of determination and its adjusted version in linear regression models when disturbances are not necessarily normal.     

Bandwidth selection for nonparametric modal regression

In the context of estimating local modes of a conditional density based on kernel density estimators, we show that existing bandwidth selection methods developed for kernel density estimation are unsuitable for mode estimation. We propose two methods to select bandwidths tailored for mode estimation in the regression setting . Numerical studies using synthetic data and a real-life dataset are carried out to demonstrate the performance of the proposed methods in comparison with several well-received bandwidth selection methods for density estimation.     

Finite sample properties of ml and reml estimators in time series regression models with long memory noise

Finite sample properties of ML and REML estimators in time series regression models with fractional ARIMA noise are examined. In particular, theoretical approximations for bias of ML and REML estimators of the noise parameters are developed and their accuracy is assessed through simulations. The impact of noise parameter estimation on performance of t -statistics and likelihood ratio statistics for testing regression parameters is also investigated.     

The coefficient of determination and its adjusted version in linear regression models

This article presents a comparative study of the efficiency properties of the coefficient of determination and its adjusted version in linear regression models when disturbances are not necessarily normal.     

Asymptotic normality and mean consistency for the weighted estimator in nonparametric regression models

In this paper, we mainly study the asymptotic properties of weighted estimator for the nonparametric regression model based on linearly negative quadrant dependent (LNQD, for short) errors. We obtain the rate of uniformly asymptotic normality of the weighted estimator which is nearly $O\left(n^{?1∕4}\right)$ when the moment condition is appropriate. The results generalize the corresponding ones of Yang (2003) from NA samples to LNQD samples and improve or extend the corresponding one of Li et al. (2012) for LNQD samples. Moreover, we obtain some results on mean consistency, uniformly mean consistency, and the rate of mean consistency for the weighted estimator. Finally we carry out some simulations to verify the validity of our results.     

On preliminary test ridge regression estimators for linear restrictions in a regression model with non-normal disturbances

In this paper, we study the properties of the preliminary test, restricted and unrestricted ridge regression estimators of the linear regression model with non-normal disturbances. We present the estimators of the regression coefficients combining the idea of preliminary test and ridge regression methodology, when it is suspected that the regression coefficients may be restricted to a subspace and the regression error is distributed as multivariate t. Accordingly we consider three estimators, namely the Unrestricted Ridge Regression Estimator (URRRE), the Restricted Ridge Regression Estimator (RRRE) and finally the Preliminary test Ridge Regression Estimator (PTRRE). The biases and the mean square error (MSE) of the estimators are derived under the null and alternative hypotheses and compared with the usual estimators. By studying the MSE criterion, the regions of optimahty of the estimators are determined.     

Asymptotic properties of nonparametric regression for long memory random fields

The kernel estimator of spatial regression function is investigated for stationary long memory (long range dependent) random fields observed over a finite set of spatial points. A general result on the strong consistency of the kernel density estimator is first obtained for the long memory random fields, and then, under some mild regularity assumptions, the asymptotic behaviors of the regression estimator are established. For the linear long memory random fields, a weak convergence theorem is also obtained for kernel density estimator. Finally, some related issues on the inference of long memory random fields are discussed through a simulation example.     

Optimal global rates of convergence for nonparametric regression with unbounded data

Estimation of regression functions from independent and identically distributed data is considered. The _L2$L_2$ error with integration with respect to the design measure is used as an error criterion. Usually in the analysis of the rate of convergence of estimates a boundedness assumption on the explanatory variable X$X$ is made besides smoothness assumptions on the regression function and moment conditions on the response variable Y$Y$. In this article we consider the kernel estimate and show that by replacing the boundedness assumption on X$X$ by a proper moment condition the same (optimal) rate of convergence can be shown as for bounded data. This answers Question 1 in Stone [1982. Optimal global rates of convergence for nonparametric regression. Ann. Statist., 10, 1040–1053].     

Selection buasubg parameters in adaptive ridge regression estimators

Strawderman's family of regression estimators is considered. The choice of the scalars wbich characterize the biasing parameter is studied by obtaining the bias vector and the mean squared error matrix.     

Asymptotic expansion for nonparametric M-estimator in a nonlinear regression model with long-memory errors

We consider asymptotic expansion of the nonparametric M-estimator in a fixed-design nonlinear regression model when the errors are generated by long-memory linear processes. Under mild conditions, we show that the nonparametric M-estimator is first-order equivalent to the Nadaraya-Watson (NW) estimator, which implies that the nonparametric M-estimator has the same asymptotic distribution as that of the NW estimator. Furthermore, we study the second-order asymptotic expansion of the nonparametric M-estimator and show that the difference between the nonparametric M-estimator and the NW estimator has a limiting distribution after suitable standardization. The nature of the limiting distribution depends on the range of long-memory parameter α. We also compare the finite sample behavior of the two estimators through a numerical example when the errors are long-memory.     

Selection buasubg parameters in adaptive ridge regression estimators

Strawderman's family of regression estimators is considered. The choice of the scalars wbich characterize the biasing parameter is studied by obtaining the bias vector and the mean squared error matrix.     

Error variance function estimation in nonparametric regression models

In this article, a new class of variance function estimators is proposed in the setting of heteroscedastic nonparametric regression models. To obtain a variance function estimator, the main proposal is to smooth the product of the response variable and residuals as opposed to the squared residuals. The asymptotic properties of the proposed methodology are investigated in order to compare its asymptotic behavior with that of the existing methods. The finite sample performance of the proposed estimator is studied through simulation studies. The effect of the curvature of the mean function on its finite sample behavior is also discussed.     

Additive nonparametric regression with autocorrelated errors

A Bayesian approach is presented for nonparametric estimation of an additive regression model with autocorrelated errors. Each of the potentially non-linear components is modelled as a regression spline using many knots, while the errors are modelled by a high order stationary autoregressive process parameterized in terms of its autocorrelations. The distribution of significant knots and partial autocorrelations is accounted for using subset selection. Our approach also allows the selection of a suitable transformation of the dependent variable. All aspects of the model are estimated simultaneously by using the Markov chain Monte Carlo method. It is shown empirically that the approach proposed works well on several simulated and real examples.     

Asymptotic properties for estimates of nonparametric regression model with martingale difference errors

Consider the nonparametric regression model with martingale difference errors. Nonparametric estimator g _n (x) of regression function g(x) will be introduced, and its asymptotic properties are studied. In particular, the pointwise and uniform convergence of g _n (x) and its asymptotic normality will be investigated. This extends the earlier work on independent random errors.     

Asymptotic normality of the lengths of a class of nonparametric confidence intervals for a regression parameter

In the linear regression model, the asymptotic distributions of certain functions of confidence bounds of a class of confidence intervals for the regression parameter arc investigated. The class of confidence intervals we consider in this paper are based on the usual linear rank statistics (signed as well as unsigned). Under suitable assumptions, if the confidence intervals are based on the signed linear rank statistics, it is established that the lengths, properly normalized, of the confidence intervals converge in law to the standard normal distributions; if the confidence intervals arc based on the unsigned linear rank statistics, it is then proved that a linear function of the confidence bounds converges in law to a normal distribution.     

A weighted bootstrap approximation for comparing the error distributions in nonparametric regression

Several procedures have been proposed for testing the equality of error distributions in two or more nonparametric regression models. Here we deal with methods based on comparing estimators of the cumulative distribution function (CDF) of the errors in each population to an estimator of the common CDF under the null hypothesis. The null distribution of the associated test statistics has been approximated by means of a smooth bootstrap (SB) estimator. This paper proposes to approximate their null distribution through a weighted bootstrap. It is shown that it produces a consistent estimator. The finite sample performance of this approximation is assessed by means of a simulation study, where it is also compared to the SB. This study reveals that, from a computational point of view, the proposed approximation is more efficient than the one provided by the SB.     

Complete consistency for the estimator of nonparametric regression model based on martingale difference errors

Abstract

In this paper, we study the complete consistency for the estimator of nonparametric regression model based on martingale difference errors, and obtain the convergence rates of the complete consistency by using the inequalities for martingale difference sequence. Finally, some simulations are illustrated.     

Estimating the error variance in nonparametric regression by a covariate-matched u-statistic

For nonparametric regression models with fixed and random design, two classes of estimators for the error variance have been introduced: second sample moments based on residuals from a nonparametric fit, and difference-based estimators. The former are asymptotically optimal but require estimating the regression function; the latter are simple but have larger asymptotic variance. For nonparametric regression models with random covariates, we introduce a class of estimators for the error variance that are related to difference-based estimators: covariate-matched U-statistics. We give conditions on the random weights involved that lead to asymptotically optimal estimators of the error variance. Our explicit construction of the weights uses a kernel estimator for the covariate density.