A model-averaging approach for smoothing spline regression

ABSTRACT

This article considers nonparametric regression problems and develops a model-averaging procedure for smoothing spline regression problems. Unlike most smoothing parameter selection studies determining an optimum smoothing parameter, our focus here is on the prediction accuracy for the true conditional mean of Y given a predictor X. Our method consists of two steps. The first step is to construct a class of smoothing spline regression models based on nonparametric bootstrap samples, each with an appropriate smoothing parameter. The second step is to average bootstrap smoothing spline estimates of different smoothness to form a final improved estimate. To minimize the prediction error, we estimate the model weights using a delete-one-out cross-validation procedure. A simulation study has been performed by using a program written in R. The simulation study provides a comparison of the most well known cross-validation (CV), generalized cross-validation (GCV), and the proposed method. This new method is straightforward to implement, and gives reliable performances in simulations.     

The non parametric regression estimate with dependent measurement errors

ABSTRACT

Non parametric regression estimation with measurement errors data has received great attention, and deconvolution local polynomial estimators can be used to deal with the problem that the errors are independent of other variables in the literature. In this article, the copula method is applied to tackle the case that the errors may depend on covariates, and the asymptotic properties of the resulting estimators are derived. Two simulations are conducted to illustrate the performance of the proposed estimators.     

Some characteristics on the selection of spline smoothing parameter

The smoothing spline method is used to fit a curve to a noisy data set, where selection of the smoothing parameter is essential. An adaptive C_p criterion (Chen and Huang 2011 Chen, C. S., and H. C. Huang. 2011. An improved C_p criterion for spline smoothing. Journal of Statistical Planning and Inference 141:445–52.[Crossref], [Web of Science ®] , [Google Scholar]) based on the Stein’s unbiased risk estimate has been proposed to select the smoothing parameter, which not only considers the usual effective degrees of freedom but also takes into account the selection variability. The resulting fitted curve has been shown to be superior and more stable than commonly used selection criteria and possesses the same asymptotic optimality as C_p. In this paper, we further discuss some characteristics on the selection of smoothing parameter, especially for the selection variability.     

Strong consistency of non parametric kernel regression estimator for strong mixing samples

For α-mixing samples, we study Priestley–Chao kernel estimator for non parametric regression model. By using the moment inequality and the exponential inequality, the strong consistency and the uniformly strong consistency of the estimator are obtained for some weak conditions.     

Making robust the cross-validatory choice of smoothing parameter in spline smoothing regression

Cross-validation as a means of choosing the smoothing parameter in spline regression has achieved a wide popularity. Its appeal comprises of an automatic method based on an attractive criterion and along with many other methods it has been shown to minimize predictive mean square error asymptotically. However, in practice there may be a substantial proportion of applications where a cross-validation style choice may lead to drastic undersmoothing often as far as interpolation. Furthermore, because the criterion is so appealing the user may be misled by an inappropriate, automatically-chosen value. In this paper we investigate the nature of cross-validatory methods in spline smoothing regression and suggest variants which provide small sample protection against undersmoothing.     

A cubic smoothing spline based lack of fit test for nonlinear regression models

The use of a statistic based on cubic spline smoothing is considered for testing nonlinear regression models for lack of fit. The statistic is defined to be the Euclidean squared norm of the smoothed residual vector obtained from fitting the nonlinear model, The asymptotic distribution of the statistic is derived under suitable smooth local alternatives and a numerical example is presented.     

Non parametric mixture of strictly monotone regression models

In this article, we propose the non parametric mixture of strictly monotone regression models. For implementation, a two-step procedure is derived. We further establish the asymptotic normality of the resultant estimator and demonstrate its good performance through numerical examples.     

Inference in smoothing spline analysis of variance

Summary. Smoothing spline analysis of variance decomposes a multivariate function into additive components. This decomposition not only provides an efficient way to model a multivariate function but also leads to meaningful inference by testing whether a certain component equals 0. No formal procedure is yet available to test such a hypothesis. We propose an asymptotic method based on the likelihood ratio to test whether a functional component is 0. This test allows us to choose an optimal model and to compare groups of curves. We first develop the general theory by exploiting the connection between mixed effects models and smoothing splines. We then apply this to compare two groups of curves and to select an optimal model in a two-dimensional problem. A small simulation is used to assess the finite sample performance of the likelihood ratio test.     

Self‐modelling regression for longitudinal data with time‐invariant covariates

The authors propose the use of self‐modelling regression to analyze longitudinal data with time invariant covariates. They model the population time curve with a penalized regression spline and use a linear mixed model for transformation of the time and response scales to fit the individual curves. Fitting is done by an iterative algorithm using off‐the‐shelf linear and nonlinear mixed model software. Their method is demonstrated in a simulation study and in the analysis of tree swallow nestling growth from an experiment that includes an experimentally controlled treatment, an observational covariate and multi‐level sampling.     

Third-order inference for autocorrelation in nonlinear regression models

We propose third-order likelihood-based methods to derive highly accurate p-value approximations for testing autocorrelated disturbances in nonlinear regression models. The proposed methods are particularly accurate for small- and medium-sized samples whereas commonly used first-order methods like the signed log-likelihood ratio test, the Kobayashi (1991) test, and the standardized test can be seriously misleading in these cases. Two Monte Carlo simulations are provided to show how the proposed methods outperform the above first-order methods. An empirical example applied to US population census data is also provided to illustrate the implementation of the proposed method and its usefulness in practice.     

Relative efficiency of first difference estimator in panel data regression with serially correlated error components

We consider the pooled cross-sectional and time series regression model when the disturbances follow a serially correlated one-way error components. In this context we discovered that the first difference estimator for the regression coefficients is equivalent to the generalized least squares estimator irrespective of the particular form of the regressor matrix when the disturbances are generated by a first order autoregressive process where the autocorrelation is close to unity.     

Cross validation of ridge regression estimator in autocorrelated linear regression models

ABSTRACT

In this paper, we investigated the cross validation measures, namely OCV, GCV and Cp under the linear regression models when the error structure is autocorrelated and regressor data are correlated. The best performed ridge regression estimator is obtained by getting the optimal ridge parameter so as to minimize these measures. A Monte Carlo simulation study is given to see how the optimal ridge parameter is affected by autocorrelation and the strength of multicollinearity.     

Penalized proportion estimation for non parametric mixture of regressions

Abstract

In this article, we propose a penalized local log-likelihood method to locally select the number of components in non parametric finite mixture of regression models via proportion shrinkage method. Mean functions and variance functions are estimated simultaneously. We show that the number of components can be estimated consistently, and further establish asymptotic normality of functional estimates. We use a modified EM algorithm to estimate the unknown functions. Simulations are conducted to demonstrate the performance of the proposed method. We illustrate our method via an empirical analysis of the housing price index data of United States.     

Almost sure convergence for weighted sums of WNOD random variables and its applications to non parametric regression models

In this article, some results on almost sure convergence for weighted sums of widely negative orthant dependent (WNOD) random variables are presented. The results obtained in the article generalize and improve the corresponding one of J. Lita Da Silva. [(2015), “Almost sure convergence for weighted sums of extended negatively dependent random variables.” Acta Math. Hungar. 146 (1), 56–70]. As applications, the strong convergence for the estimator of non parametric regression model are established.     

Asymptotic results for parametric estimation in inadequate two phases regression models

We study the asymptotic properties, consistency, asymptotic normality, of the least squares estimator in a non linear regression problem. The model uses a parametric class Л of functions, but we do not assume that the unknown function belongs to that class. Л is here a class of continuous functions with a discontinuity in the first derivative. The problem of making a choice between two classes of that type is also studied.     

A plug-in the number of knots selector for polynomial spline regression

A plug-in the number of interior knots (NIKs) selector is proposed for polynomial spline estimation in nonparametric regression. The existence and properties of the optimal NIKs for spline regression are established by minimising the weighted mean integrated squared error. We obtain plug-in formulae for the optimal NIKs based on the theoretical results of asymptotic optimality, and develop strategies for choosing the NIKs of the spline estimator. The proposed NIKs selection method is tested on our simulated data with quite satisfactory performance, and is illustrated by analysing a fossil data set.     

Asymptotic normality for a non parametric estimator of conditional quantile with left-truncated data

In this paper, we construct a non parametric estimator of conditional distribution function by the double-kernel local linear approach for left-truncated data, from which we derive the weighted double-kernel local linear estimator of conditional quantile. The asymptotic normality of the proposed estimators is also established. Finite-sample performance of the estimator is investigated via simulation.     

Asymptotic normality of locally modelled regression estimator for functional data

We focus on the nonparametric regression of a scalar response on a functional explanatory variable. As an alternative to the well-known Nadaraya-Watson estimator for regression function in this framework, the locally modelled regression estimator performs very well [cf. [Barrientos-Marin, J., Ferraty, F., and Vieu, P. (2010), ‘Locally Modelled Regression and Functional Data’, Journal of Nonparametric Statistics, 22, 617–632]. In this paper, the asymptotic properties of locally modelled regression estimator for functional data are considered. The mean-squared convergence as well as asymptotic normality for the estimator are established. We also adapt the empirical likelihood method to construct the point-wise confidence intervals for the regression function and derive the Wilk's phenomenon for the empirical likelihood inference. Furthermore, a simulation study is presented to illustrate our theoretical results.