Testing for additivity with B-splines

Regression splines are often used for fitting nonparametric functions, and they work especially well for additivity models. In this paper, we consider two simple tests of additivity: an adaptation of Tukey's one degree of freedom test and a nonparametric version of Rao's score test. While the Tukey-type test can detect most forms of the local non-additivity at the parametric rate of O(n-1/2), the score test is consistent for all alternative at a nonparametric rate. The asymptotic distribution of these test statistics is derived under both the null and local alternative hypotheses. A simulation study is conducted to compare their finite-sample performances with some existing kernel-based tests. The score test is found to have a good overall performance.     

Model Specification Tests in Nonparametric Stochastic Regression Models

In this paper, we consider testing for additivity in a class of nonparametric stochastic regression models. Two test statistics are constructed and their asymptotic distributions are established. We also conduct a small sample study for one of the test statistics through a simulated example.     

Nonparametric Regression with Singular Design

Theories of nonparametric regression are usually based on the assumption that the design density exists. However, in some applications such as those involving high-dimensional or chaotic time series data, the design measure may be singular and may be likely to have a fractal (nonintegral) dimension. In this paper, the popular Nadaraya–Watson estimator is studied under the general setup that the continuity of the design measure is governed by the local or pointwise dimension. It will be shown in the iid setup that the nonparametric regression estimator achieves a convergence rate which is dependent only on the pointwise dimension. The case of time series data is also studied. For the latter case, a new mixing condition is introduced, and an assumption of marginal or joint density is completely avoided. Three examples, a fractal regression and two applications for predicting chaotic time series, are used to illustrate the implications of the obtained results.     

Nonasymptotic Bounds on the L 2 Error of Neural Network Regression Estimates

The estimation of multivariate regression functions from bounded i.i.d. data is considered. The L ₂ error with integration with respect to the design measure is used as an error criterion. The distribution of the design is assumed to be concentrated on a finite set. Neural network estimates are defined by minimizing the empirical L ₂ risk over various sets of feedforward neural networks. Nonasymptotic bounds on the L ₂ error of these estimates are presented. The results imply that neural networks are able to adapt to additive regression functions and to regression functions which are a sum of ridge functions, and hence are able to circumvent the curse of dimensionality in these cases.     

随机设计下非参数回归模型方差变点Ratio检验

研究随机设计下非参数回归模型方差变点Ratio检验.首先用局部多项式方法估计回归曲线得到残差序列,其次基于残差的平方序列构造Ratio检验统计量并推导检验统计量的极限分布.最后数值模拟与实例分析结果表明方法的有效性.     

The Additivity Theorem in K-Theory

We present a method for converting Theorem B style proofs in algebraic K-theory to Theorem A style proofs and apply it to the additivity theorem.     

Generalized Smooth Monotonic Regression in Additive Modeling

Common approaches to monotonic regression focus on the case of a unidimensional covariate and continuous response variable. Here a general approach is proposed that allows for additive structures where one or more variables have monotone influence on the response variable. In addition the approach allows for response variables from an exponential family, including binary and Poisson distributed response variables. Flexibility of the smooth estimate is gained by expanding the unknown function in monotonic basis functions. For the estimation of coefficients and the selection of basis functions a likelihood-based boosting algorithm is proposed which is simple to implement. Stopping criteria and inference are based on AIC-type measures. The method is applied to several datasets.     

Nonparametric Estimation of Regression Functions in Point Process Models

We prove that the empirical L ²-risk minimizing estimator over some general type of sieve classes is universally, strongly consistent for the regression function in a class of point process models of Poissonian type (random sampling processes). The universal consistency result needs weak assumptions on the underlying distributions and regression functions. It applies in particular to neural net classes and to radial basis function nets. For the estimation of the intensity functions of a Poisson process a similar technique yields consistency of the sieved maximum likelihood estimator for some general sieve classes. This revised version was published online in August 2006 with corrections to the Cover Date.     

Nonparametric Comparison of Multiple Regression Curves in Scale-Space

This article concerns testing the equality of multiple curves in a nonparametric regression context. The proposed test forms an ANOVA type test statistic based on kernel smoothing and examines the ratio of between- and within-group variations. The empirical distribution of the test statistic is derived using a permutation test. Unlike traditional kernel smoothing approaches, the test is conducted in scale-space so that it does not require the selection of an optimal smoothing level, but instead considers a wide range of scales. The proposed method also visualizes its testing results as a color map and graphically summarizes the statistical differences between curves across multiple locations and scales. A numerical study using simulated and real examples is conducted to demonstrate the finite sample performance of the proposed method.     

Penalized Likelihood-type Estimators for Generalized Nonparametric Regression

We consider the asymptotic analysis of penalized likelihood type estimators for generalized nonparametric regression problems in which the target parameter is a vector-valued function defined in terms of the conditional distribution of a response given a set of covariates. A variety of examples including ones related to generalized linear models and robust smoothing are covered by the theory. Linear approximations to the estimator are constructed using Taylor expansions in Hilbert spaces. An application which is treated is upper bounds on rates of convergence for the penalized likelihood-type estimators.     

协整回归残量平稳性的小波检验

本文研究基于小波方法的协整回归残量平稳性检验。将文献[6]中的小波方法推广到了协整关系的检验,给出了对线性协整模型OLS回归后得到的残量进行平稳性检验的小波方法,得到了在原假设非协整下检验统计量的渐近分布。模拟实验和实例分析都验证了新方法的有效性。     

一元线性回归检验理论的一种证明

利用正交变换法,给出一元线性回归假设检验定理的一种直接证明.这种证明方法可供在数理统计教学中作参考.     

Sufficient Dimension Reduction and Graphics in Regression

In this article, we review, consolidate and extend a theory for sufficient dimension reduction in regression settings. This theory provides a powerful context for the construction, characterization and interpretation of low-dimensional displays of the data, and allows us to turn graphics into a consistent and theoretically motivated methodological body. In this spirit, we propose an iterative graphical procedure for estimating the meta-parameter which lies at the core of sufficient dimension reduction; namely, the central dimension-reduction subspace.     

测量误差模型参数变点的非参数检验

本讨论测量误差参数变点的检测问题,利用秩统计量,给出了模型只有一个变点的检验统计量,运用检验统计量渐近分布的性质,给出了一个计算检验淅近临界值的公式,由此我们可以较为客易计算检验的临界值。     

A Bootstrap Approach to Nonparametric Regression for Right Censored Data

In this paper a two-stage bootstrap method is proposed for nonparametric regression with right censored data. The method is applied to construct confidence intervals and bands for a conditional survival function. Its asymptotic validity is established using counting process techniques and martingale central limit theory. The performance of the bootstrap method is investigated in a Monte Carlo study. An illustration is given using a real data.     

Spatially Adaptive Bayesian Penalized Regression Splines (P-splines)

In this article we study penalized regression splines (P-splines), which are low-order basis splines with a penalty to avoid undersmoothing. Such P-splines are typically not spatially adaptive, and hence can have trouble when functions are varying rapidly. Our approach is to model the penalty parameter inherent in the P-spline method as a heteroscedastic regression function. We develop a full Bayesian hierarchical structure to do this and use Markov chain Monte Carlo techniques for drawing random samples from the posterior for inference. The advantage of using a Bayesian approach to P-splines is that it allows for simultaneous estimation of the smooth functions and the underlying penalty curve in addition to providing uncertainty intervals of the estimated curve. The Bayesian credible intervals obtained for the estimated curve are shown to have pointwise coverage probabilities close to nominal. The method is extended to additive models with simultaneous spline-based penalty functions for the unknown functions. In simulations, the approach achieves very competitive performance with the current best frequentist P-spline method in terms of frequentist mean squared error and coverage probabilities of the credible intervals, and performs better than some of the other Bayesian methods.     

A Nonparametric Procedure to Detect Jumps in Regression Surfaces

This article proposes a local smoothing procedure for detecting jump location curves of regression surfaces. This procedure simplifies the computation of some existing jump detectors in the statistical literature. It also generalizes the Sobel edge detector in the image processing literature such that more observations can be used to smooth away random noise in the data. The problem to evaluate the performance of jump detectors is discussed and a new measurement of jump detection performance is suggested.     

Bayesian Nonparametric Modeling for Multivariate Ordinal Regression

Univariate or multivariate ordinal responses are often assumed to arise from a latent continuous parametric distribution, with covariate effects that enter linearly. We introduce a Bayesian nonparametric modeling approach for univariate and multivariate ordinal regression, which is based on mixture modeling for the joint distribution of latent responses and covariates. The modeling framework enables highly flexible inference for ordinal regression relationships, avoiding assumptions of linearity or additivity in the covariate effects. In standard parametric ordinal regression models, computational challenges arise from identifiability constraints and estimation of parameters requiring nonstandard inferential techniques. A key feature of the nonparametric model is that it achieves inferential flexibility, while avoiding these difficulties. In particular, we establish full support of the nonparametric mixture model under fixed cut-off points that relate through discretization the latent continuous responses with the ordinal responses. The practical utility of the modeling approach is illustrated through application to two datasets from econometrics, an example involving regression relationships for ozone concentration, and a multirater agreement problem. Supplementary materials with technical details on theoretical results and on computation are available online.     

非参数模型均值函数结构变点的Bootstrap检测

本文检测非参数回归模型均值函数结构变点,针对均值函数跃度的长期均值为零时,基于残量的CUSUM统计量对均值函数结构变点检验无效的问题,本文提出了一种基于均值函数的核估计的检验统计量,得到统计量在原假设和备择假设下的极限分布,并构造Bootstrap方法对非参数回归模型均值函数结构变点进行检验,证明了检验和估计的一致性;模拟结果表明本文方法明显优于已有方法。     

一类非参数回归函数估计的性质

设Z_(11),z_(12),…,Z_是在固定点(x_i,y_1),1≤≤n_1,1≤j≤n_2,的n_1n_2个观察值,适合模型 Z_(ij)=g(x_i,y_j)+ε_(ij),1≤i≤n_1,1≤j≤n_2。(1) 本文给出了g的一种估计并讨论了估计的性质。