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 CONFIDENCE INTERVALS FOR GENERALIZED ADDITIVE MODELS BASED ON PENALIZED REGRESSION SPLINES

Generalized additive models represented using low rank penalized regression splines, estimated by penalized likelihood maximisation and with smoothness selected by generalized cross validation or similar criteria, provide a computationally efficient general framework for practical smooth modelling. Various authors have proposed approximate Bayesian interval estimates for such models, based on extensions of the work of Wahba, G. (1983) [Bayesian confidence intervals for the cross validated smoothing spline. J. R. Statist. Soc. B 45 , 133–150] and Silverman, B.W. (1985) [Some aspects of the spline smoothing approach to nonparametric regression curve fitting. J. R. Statist. Soc. B 47 , 1–52] on smoothing spline models of Gaussian data, but testing of such intervals has been rather limited and there is little supporting theory for the approximations used in the generalized case. This paper aims to improve this situation by providing simulation tests and obtaining asymptotic results supporting the approximations employed for the generalized case. The simulation results suggest that while across‐the‐model performance is good, component‐wise coverage probabilities are not as reliable. Since this is likely to result from the neglect of smoothing parameter variability, a simple and efficient simulation method is proposed to account for smoothing parameter uncertainty: this is demonstrated to substantially improve the performance of component‐wise intervals.     

半参数纵向模型的惩罚二次推断函数估计

针对纵向数据半参数模型E(y|x,t)=XTβ+f(t),采用惩罚二次推断函数方法同时估计模型中的回归参数β和未知光滑函数f(t)。首先利用截断幂函数基对未知光滑函数进行基函数展开近似,然后利用惩罚样条的思想构造关于回归参数和基函数系数的惩罚二次推断函数,最小化惩罚二次推断函数便可得到回归参数和基函数系数的惩罚二次推断函数估计。理论结果显示,估计结果具有相合性和渐近正态性,通过数值方法也得到了较好的模拟结果。     

An improved Cp criterion for spline smoothing

Spline smoothing is a popular technique for curve fitting, in which selection of the smoothing parameter is crucial. Many methods such as Mallows’ C_p, generalized maximum likelihood (GML), and the extended exponential (EE) criterion have been proposed to select this parameter. Although C_p is shown to be asymptotically optimal, it is usually outperformed by other selection criteria for small to moderate sample sizes due to its high variability. On the other hand, GML and EE are more stable than C_p, but they do not possess the same asymptotic optimality as C_p. Instead of selecting this smoothing parameter directly using C_p, we propose to select among a small class of selection criteria based on Stein's unbiased risk estimate (SURE). Due to the selection effect, the spline estimate obtained from a criterion in this class is nonlinear. Thus, the effective degrees of freedom in SURE contains an adjustment term in addition to the trace of the smoothing matrix, which cannot be ignored in small to moderate sample sizes. The resulting criterion, which we call adaptive C_p, is shown to have an analytic expression, and hence can be efficiently computed. Moreover, adaptive C_p is not only demonstrated to be superior and more stable than commonly used selection criteria in a simulation study, but also shown to possess the same asymptotic optimality as C_p.     

Spline-backfitted kernel smoothing of partially linear additive model

A spline-backfitted kernel smoothing method is proposed for partially linear additive model. Under assumptions of stationarity and geometric mixing, the proposed function and parameter estimators are oracally efficient and fast to compute. Such superior properties are achieved by applying to the data spline smoothing and kernel smoothing consecutively. Simulation experiments with both moderate and large number of variables confirm the asymptotic results. Application to the Boston housing data serves as a practical illustration of the method.     

Asymptotic deficiency of score test for inhomogeneous Poisson processes

In this work, based on a realization of an inhomogeneous Poisson process whose intensity function depends on an unknown real parameter, we test a simple null hypothesis against a sequence of close (contiguous) one-sided alternatives. The main object is to obtain the asymptotic deficiency of the score test with respect to the Neyman–Pearson test.     

数据随机缺失下分位数回归模型的诱导光滑估计法

在数据随机缺失的分位数回归模型中,运用诱导光滑思想构造光滑的估计方程,得到了回归参数的诱导光滑估计及渐近协方差估计。接着证明了诱导光滑估计的渐近正态性质,并给出诱导光滑估计及其渐近协方差估计的算法。模拟研究表明新方法在有限样本中表现出色。     

Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion

Many different methods have been proposed to construct nonparametric estimates of a smooth regression function, including local polynomial, (convolution) kernel and smoothing spline estimators. Each of these estimators uses a smoothing parameter to control the amount of smoothing performed on a given data set. In this paper an improved version of a criterion based on the Akaike information criterion (AIC), termed AIC_C, is derived and examined as a way to choose the smoothing parameter. Unlike plug-in methods, AIC_C can be used to choose smoothing parameters for any linear smoother, including local quadratic and smoothing spline estimators. The use of AIC_C avoids the large variability and tendency to undersmooth (compared with the actual minimizer of average squared error) seen when other 'classical' approaches (such as generalized cross-validation (GCV) or the AIC) are used to choose the smoothing parameter. Monte Carlo simulations demonstrate that the AIC_C-based smoothing parameter is competitive with a plug-in method (assuming that one exists) when the plug-in method works well but also performs well when the plug-in approach fails or is unavailable.     

Multiple Kernel Procedure: an Asymptotic Support

ABSTRACT. This paper deals with kernel non-parametric estimation. The multiple kernel method, as proposed by Berlinet (1993), consists in choosing both the smoothing parameter and the order of the kernel function. In this paper we follow this general idea, and the selection is carried out by a combination of plug-in and cross-validation techniques. In a first attempt we give an asymptotic optimality theorem which is stated in a general unifying setting that includes many curve estimation problems. Then, as an illustration, it will be seen how this behaves in both special cases of kernel density and kernel regression estimation.     

Trend smoothness achieved by penalized least squares with the smoothing parameter chosen by optimality criteria

This work presents a study about the smoothness attained by the methods more frequently used to choose the smoothing parameter in the context of splines: Cross Validation, Generalized Cross Validation, and corrected Akaike and Bayesian Information Criteria, implemented with Penalized Least Squares. It is concluded that the amount of smoothness strongly depends on the length of the series and on the type of underlying trend, while the presence of seasonality even though statistically significant is less relevant. The intrinsic variability of the series is not statistically significant and its effect is taken into account only through the smoothing parameter.     

Nonparametric estimation of density derivatives of dependent data

We study here the kernel type, nonparametric estimation of the derivatives of the density function associated with a strongly mixing time series. The consistency and asymptotic normality properties are studied and a method for the selection of the smoothing parameter by means of the modification of the least-squares cross-validation procedure is proposed.     

Multiscale Adaptive Regression Models for Neuroimaging Data

Neuroimaging studies aim to analyze imaging data with complex spatial patterns in a large number of locations (called voxels) on a two-dimensional (2D) surface or in a 3D volume. Conventional analyses of imaging data include two sequential steps: spatially smoothing imaging data and then independently fitting a statistical model at each voxel. However, conventional analyses suffer from the same amount of smoothing throughout the whole image, the arbitrary choice of smoothing extent, and low statistical power in detecting spatial patterns. We propose a multiscale adaptive regression model (MARM) to integrate the propagation-separation (PS) approach (Polzehl and Spokoiny, 2000, 2006) with statistical modeling at each voxel for spatial and adaptive analysis of neuroimaging data from multiple subjects. MARM has three features: being spatial, being hierarchical, and being adaptive. We use a multiscale adaptive estimation and testing procedure (MAET) to utilize imaging observations from the neighboring voxels of the current voxel to adaptively calculate parameter estimates and test statistics. Theoretically, we establish consistency and asymptotic normality of the adaptive parameter estimates and the asymptotic distribution of the adaptive test statistics. Our simulation studies and real data analysis confirm that MARM significantly outperforms conventional analyses of imaging data.     

Restricted likelihood ratio lack-of-fit tests using mixed spline models

Summary.  Penalized regression spline models afford a simple mixed model representation in which variance components control the degree of non-linearity in the smooth function estimates. This motivates the study of lack-of-fit tests based on the restricted maximum likelihood ratio statistic which tests whether variance components are 0 against the alternative of taking on positive values. For this one-sided testing problem a further complication is that the variance component belongs to the boundary of the parameter space under the null hypothesis. Conditions are obtained on the design of the regression spline models under which asymptotic distribution theory applies, and finite sample approximations to the asymptotic distribution are provided. Test statistics are studied for simple as well as multiple-regression models.     

Numerical Algorithms Group Manual Mark 7

We consider the estimation of the 90 and 95 percentiles of a normal distribution and also the construction of one-sided 90% and 95% -normal ranges. Three methods are proposed -the sample percentile method, and two based on kernel estimates of the density function using Fryer's method and the leaving-one-out method for choosing a smoothing parameter.

A simulation study compares the methods in terms of bias, variance and mean square error of the population percentile estimates and of the eovers of the consequent normal ranges.     

Nonparametric regression for functional data: Automatic smoothing parameter selection

We study regression estimation when the explanatory variable is functional. Nonparametric estimates of the regression operator have been recently introduced. They depend on a smoothing factor which controls its behavior, and the aim of our work is to construct some data-driven criterion for choosing this smoothing parameter. The criterion can be formulated in terms of a functional version of cross-validation ideas. Under mild assumptions on the unknown regression operator, it is seen that this rule is asymptotically optimal. As by-products of this result, we state some asymptotic equivalences for several measures of accuracy for nonparametric estimate of the regression operator. We also present general inequalities for bounding moments of random sums involving functional variables. Finally, a short simulation study is carried out to illustrate the behavior of our method for finite samples.     

Consistent bandwidth selection for kernel binary regression

The use of nonparametric regression techniques for binary regression is a promising alternative to parametric methods. As in other nonparametric smoothing problems, the choice of smoothing parameter is critical to the performance of the estimator and the appearance of the resulting estimate. In this paper, we discuss the use of selection criteria based on estimates of squared prediction risk and show consistency and asymptotic normality of the selected bandwidths. The usefulness of the methods is explored on a data set and in a small simulation study.     

Adaptive tests of regression functions via multiscale generalized likelihood ratios

Many applications of nonparametric tests based on curve estimation involve selecting a smoothing parameter. The author proposes an adaptive test that combines several generalized likelihood ratio tests in order to get power performance nearly equal to whichever of the component tests is best. She derives the asymptotic joint distribution of the component tests and that of the proposed test under the null hypothesis. She also develops a simple method of selecting the smoothing parameters for the proposed test and presents two approximate methods for obtaining its P‐value. Finally, she evaluates the proposed test through simulations and illustrates its application to a set of real data.     

Tail asymptotic of discounted aggregate claims with compound dependence under risky investment

This paper considers the tail asymptotic of discounted aggregate claims with compound dependence under risky investment. The price of risky investment is modeled by a geometric Lévy process, while claims are modeled by a one-sided linear process whose innovations further obeying a so-called upper tail asymptotic independence. When the innovations are heavy tailed, we derive some uniform asymptotic formulas. The results show that the linear dependence has significant impact on the tail asymptotic of discounted aggregate claims but the upper tail asymptotic independence is negligible.     

Parametric Inference in Stationary Time Series Models with Dependent Errors

This article is concerned with inference for the parameter vector in stationary time series models based on the frequency domain maximum likelihood estimator. The traditional method consistently estimates the asymptotic covariance matrix of the parameter estimator and usually assumes the independence of the innovation process. For dependent innovations, the asymptotic covariance matrix of the estimator depends on the fourth‐order cumulants of the unobserved innovation process, a consistent estimation of which is a difficult task. In this article, we propose a novel self‐normalization‐based approach to constructing a confidence region for the parameter vector in such models. The proposed procedure involves no smoothing parameter, and is widely applicable to a large class of long/short memory time series models with weakly dependent innovations. In simulation studies, we demonstrate favourable finite sample performance of our method in comparison with the traditional method and a residual block bootstrap approach.     

Local Adaptivity of Kernel Estimates with Plug-in Local Bandwidth Selectors

In non-parametric function estimation selection of a smoothing parameter is one of the most important issues. The performance of smoothing techniques depends highly on the choice of this parameter. Preferably the bandwidth should be determined via a data-driven procedure. In this paper we consider kernel estimators in a white noise model, and investigate whether locally adaptive plug-in bandwidths can achieve optimal global rates of convergence. We consider various classes of functions: Sobolev classes, bounded variation function classes, classes of convex functions and classes of monotone functions. We study the situations of pilot estimation with oversmoothing and without oversmoothing. Our main finding is that simple local plug-in bandwidth selectors can adapt to spatial inhomogeneity of the regression function as long as there are no local oscillations of high frequency. We establish the pointwise asymptotic distribution of the regression estimator with local plug-in bandwidth.