Kernel spline regression

The authors propose «kernel spline regression,» a method of combining spline regression and kernel smoothing by replacing the polynomial approximation for local polynomial kernel regression with the spline basis. The new approach retains the local weighting scheme and the use of a bandwidth to control the size of local neighborhood. The authors compute the bias and variance of the kernel linear spline estimator, which they compare with local linear regression. They show that kernel spline estimators can succeed in capturing the main features of the underlying curve more effectively than local polynomial regression when the curvature changes rapidly. They also show through simulation that kernel spline regression often performs better than ordinary spline regression and local polynomial regression.     

A note on smoothing parameter selection for penalized spline smoothing

In nonparametric regression the smoothing parameter can be selected by minimizing a Mean Squared Error (MSE) based criterion. For spline smoothing one can also rewrite the smooth estimation as a Linear Mixed Model where the smoothing parameter appears as the a priori variance of spline basis coefficients. This allows to employ Maximum Likelihood (ML) theory to estimate the smoothing parameter as variance component. In this paper the relation between the two approaches is illuminated for penalized spline smoothing (P-spline) as suggested in Eilers and Marx Statist. Sci. 11(2) (1996) 89. Theoretical and empirical arguments are given showing that the ML approach is biased towards undersmoothing, i.e. it chooses a too complex model compared to the MSE. The result is in line with classical spline smoothing, even though the asymptotic arguments are different. This is because in P-spline smoothing a finite dimensional basis is employed while in classical spline smoothing the basis grows with the sample size.     

ON SPLINE SMOOTHING WITH AUTOCORRELATED ERRORS

The generalised cross-validation criterion for choosing the degree of smoothing in spline regression is extended to accommodate an autocorrelated error sequence. It is demonstrated via simulation that the minimum generalised cross-validation smoothing spline is an inconsistent estimator in the presence of autocorrelated errors and that ignoring even moderate autocorrelation structure can seriously affect the performance of the cross-validated smoothing spline. The method of penalised maximum likelihood is used to develop an efficient algorithm for the case in which the autocorrelation decays exponentially. An application of the method to a published data-set is described. The method does not require the data to be equally spaced in time.     

Hypothesis testing in smoothing spline models

Nonparametric regression models are often used to check or suggest a parametric model. Several methods have been proposed to test the hypothesis of a parametric regression function against an alternative smoothing spline model. Some tests such as the locally most powerful (LMP) test by Cox et al. (Cox, D., Koh, E., Wahba, G. and Yandell, B. (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Stat., 16, 113–119.), the generalized maximum likelihood (GML) ratio test and the generalized cross validation (GCV) test by Wahba (Wahba, G. (1990). Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.) were developed from the corresponding Bayesian models. Their frequentist properties have not been studied. We conduct simulations to evaluate and compare finite sample performances. Simulation results show that the performances of these tests depend on the shape of the true function. The LMP and GML tests are more powerful for low frequency functions while the GCV test is more powerful for high frequency functions. For all test statistics, distributions under the null hypothesis are complicated. Computationally intensive Monte Carlo methods can be used to calculate null distributions. We also propose approximations to these null distributions and evaluate their performances by simulations.     

Difference-based estimation and model identification for panel data semiparametric models with cross-section dependence

Abstract

In this article, we consider a panel data partially linear regression model with fixed effect and non parametric time trend function. The data can be dependent cross individuals through linear regressor and error components. Unlike the methods using non parametric smoothing technique, a difference-based method is proposed to estimate linear regression coefficients of the model to avoid bandwidth selection. Here the difference technique is employed to eliminate the non parametric function effect, not the fixed effects, on linear regressor coefficient estimation totally. Therefore, a more efficient estimator for parametric part is anticipated, which is shown to be true by the simulation results. For the non parametric component, the polynomial spline technique is implemented. The asymptotic properties of estimators for parametric and non parametric parts are presented. We also show how to select informative ones from a number of covariates in the linear part by using smoothly clipped absolute deviation-penalized estimators on a difference-based least-squares objective function, and the resulting estimators perform asymptotically as well as the oracle procedure in terms of selecting the correct model.     

ACCELERATED FAILURE TIME MODELS WITH NONLINEAR COVARIATES EFFECTS

As a flexible alternative to the Cox model, the accelerated failure time (AFT) model assumes that the event time of interest depends on the covariates through a regression function. The AFT model with non‐parametric covariate effects is investigated, when variable selection is desired along with estimation. Formulated in the framework of the smoothing spline analysis of variance model, the proposed method based on the Stute estimate ( Stute, 1993 [Consistent estimation under random censorship when covariables are present, J. Multivariate Anal. 45 , 89–103]) can achieve a sparse representation of the functional decomposition, by utilizing a reproducing kernel Hilbert norm penalty. Computational algorithms and theoretical properties of the proposed method are investigated. The finite sample size performance of the proposed approach is assessed via simulation studies. The primary biliary cirrhosis data is analyzed for demonstration.     

Outlier detection using difference-based variance estimators in multiple regression

In this article, we propose an outlier detection approach in a multiple regression model using the properties of a difference-based variance estimator. This type of a difference-based variance estimator was originally used to estimate error variance in a non parametric regression model without estimating a non parametric function. This article first employed a difference-based error variance estimator to study the outlier detection problem in a multiple regression model. Our approach uses the leave-one-out type method based on difference-based error variance. The existing outlier detection approaches using the leave-one-out approach are highly affected by other outliers, while ours is not because our approach does not use the regression coefficient estimator. We compared our approach with several existing methods using a simulation study, suggesting the outperformance of our approach. The advantages of our approach are demonstrated using a real data application. Our approach can be extended to the non parametric regression model for outlier detection.     

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.     

Using recursive algorithms for the efficient identification of smoothing spline ANOVA models

In this paper we present a unified discussion of different approaches to the identification of smoothing spline analysis of variance (ANOVA) models: (i) the “classical” approach (in the line of Wahba in Spline Models for Observational Data, 1990; Gu in Smoothing Spline ANOVA Models, 2002; Storlie et al. in Stat. Sin., 2011) and (ii) the State-Dependent Regression (SDR) approach of Young in Nonlinear Dynamics and Statistics (2001). The latter is a nonparametric approach which is very similar to smoothing splines and kernel regression methods, but based on recursive filtering and smoothing estimation (the Kalman filter combined with fixed interval smoothing). We will show that SDR can be effectively combined with the “classical” approach to obtain a more accurate and efficient estimation of smoothing spline ANOVA models to be applied for emulation purposes. We will also show that such an approach can compare favorably with kriging.     

Guided Censored Regression

Parametrically guided non‐parametric regression is an appealing method that can reduce the bias of a non‐parametric regression function estimator without increasing the variance. In this paper, we adapt this method to the censored data case using an unbiased transformation of the data and a local linear fit. The asymptotic properties of the proposed estimator are established, and its performance is evaluated via finite sample simulations.     

Modified spline regression based on randomly right-censored data: A comparative study

ABSTRACT

In this paper, we propose modified spline estimators for nonparametric regression models with right-censored data, especially when the censored response observations are converted to synthetic data. Efficient implementation of these estimators depends on the set of knot points and an appropriate smoothing parameter. We use three algorithms, the default selection method (DSM), myopic algorithm (MA), and full search algorithm (FSA), to select the optimum set of knots in a penalized spline method based on a smoothing parameter, which is chosen based on different criteria, including the improved version of the Akaike information criterion (AICc), generalized cross validation (GCV), restricted maximum likelihood (REML), and Bayesian information criterion (BIC). We also consider the smoothing spline (SS), which uses all the data points as knots. The main goal of this study is to compare the performance of the algorithm and criteria combinations in the suggested penalized spline fits under censored data. A Monte Carlo simulation study is performed and a real data example is presented to illustrate the ideas in the paper. The results confirm that the FSA slightly outperforms the other methods, especially for high censoring levels.     

A non-iterative optimization method for smoothness in penalized spline regression

Typically, an optimal smoothing parameter in a penalized spline regression is determined by minimizing an information criterion, such as one of the C _p , CV and GCV criteria. Since an explicit solution to the minimization problem for an information criterion cannot be obtained, it is necessary to carry out an iterative procedure to search for the optimal smoothing parameter. In order to avoid such extra calculation, a non-iterative optimization method for smoothness in penalized spline regression is proposed using the formulation of generalized ridge regression. By conducting numerical simulations, we verify that our method has better performance than other methods which optimize the number of basis functions and the single smoothing parameter by means of the CV or GCV criteria.     

A semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data

We propose a flexible semiparametric stochastic mixed effects model for bivariate cyclic longitudinal data. The model can handle either single cycle or, more generally, multiple consecutive cycle data. The approach models the mean of responses by parametric fixed effects and a smooth nonparametric function for the underlying time effects, and the relationship across the bivariate responses by a bivariate Gaussian random field and a joint distribution of random effects. The proposed model not only can model complicated individual profiles, but also allows for more flexible within-subject and between-response correlations. The fixed effects regression coefficients and the nonparametric time functions are estimated using maximum penalized likelihood, where the resulting estimator for the nonparametric time function is a cubic smoothing spline. The smoothing parameters and variance components are estimated simultaneously using restricted maximum likelihood. Simulation results show that the parameter estimates are close to the true values. The fit of the proposed model on a real bivariate longitudinal dataset of pre-menopausal women also performs well, both for a single cycle analysis and for a multiple consecutive cycle analysis. The Canadian Journal of Statistics 48: 471–498; 2020 © 2020 Statistical Society of Canada     

On Efficient Estimators of the Proportion of True Null Hypotheses in a Multiple Testing Setup

We consider the problem of estimating the proportion θ of true null hypotheses in a multiple testing context. The setup is classically modelled through a semiparametric mixture with two components: a uniform distribution on interval [0,1] with prior probability θ and a non‐parametric density f . We discuss asymptotic efficiency results and establish that two different cases occur whether f vanishes on a non‐empty interval or not. In the first case, we exhibit estimators converging at a parametric rate, compute the optimal asymptotic variance and conjecture that no estimator is asymptotically efficient (i.e. attains the optimal asymptotic variance). In the second case, we prove that the quadratic risk of any estimator does not converge at a parametric rate. We illustrate those results on simulated data.     

Series Estimation in Partially Linear In‐Slide Regression Models

Abstract. The partially linear in‐slide model (PLIM) is a useful tool to make econometric analyses and to normalize microarray data. In this article, by using series approximations and a least squares procedure, we propose a semiparametric least squares estimator (SLSE) for the parametric component and a series estimator for the non‐parametric component. Under weaker conditions than those imposed in the literature, we show that the SLSE is asymptotically normal and that the series estimator attains the optimal convergence rate of non‐parametric regression. We also investigate the estimating problem of the error variance. In addition, we propose a wild block bootstrap‐based test for the form of the non‐parametric component. Some simulation studies are conducted to illustrate the finite sample performance of the proposed procedure. An example of application on a set of economical data is also illustrated.     

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.     

Partially linear models and polynomial spline approximations for the analysis of unbalanced panel data

In this paper, we study the estimation of the unbalanced panel data partially linear models with a one-way error components structure. A weighted semiparametric least squares estimator (WSLSE) is developed using polynomial spline approximation and least squares. We show that the WSLSE is asymptotically more efficient than the corresponding unweighted estimator for both parametric and nonparametric components of the model. This is a significant improvement over previous results in the literature which showed that the simply weighting technique can only improve the estimation of the parametric component. The asymptotic normalities of the proposed WSLSE are also established.     

Smoothing spline based tests for non-linearity in a partially linear model

This paper deals with testing for non-linearity in a regression model with one possibly non-linear component being estimated non-parametrically using smoothing splines. We propose two new variance–covariance based tests for detecting non-linearity applying a likelihood ratio hypothesis testing approach. The first test is for the inclusion of a possibly non-linear component and the second one is for linearity of a possibly non-linear component. The tests are based on a stochastic model in state space form given by Wahba (J. Roy. Statist. Soc. Ser. B 40 (1978) 364), Wecker and Ansley (J. Amer. Statist. Assoc. 78 (1983) 81) and de Jong and Mazzi (Modeling and smoothing unequally spaced sequence data, University of York and University of British Columbia, Unpublished paper) for which smoothing splines provide an optimal estimate. Pitrun (A smoothing spline approach to non-linear interface for time series, Department of Econometrics and Business Statistics, Monash University, Unpublished Ph.D. thesis) derived the variance–covariance structure of this model, which allows the use of a marginal likelihood approach. This leads naturally to marginal-likelihood based likelihood ratio tests for non-linearity. Small sample properties of the new tests have been investigated via Monte Carlo studies.     

Some asymptotic results on generalized penalized spline smoothing

Summary.  The paper discusses asymptotic properties of penalized spline smoothing if the spline basis increases with the sample size. The proof is provided in a generalized smoothing model allowing for non-normal responses. The results are extended in two ways. First, assuming the spline coefficients to be a priori normally distributed links the smoothing framework to generalized linear mixed models. We consider the asymptotic rates such that the Laplace approximation is justified and the resulting fits in the mixed model correspond to penalized spline estimates. Secondly, we make use of a fully Bayesian viewpoint by imposing an a priori distribution on all parameters and coefficients. We argue that with the postulated rates at which the spline basis dimension increases with the sample size the posterior distribution of the spline coefficients is approximately normal. The validity of this result is investigated in finite samples by comparing Markov chain Monte Carlo results with their asymptotic approximation in a simulation study.