An Analog of the Poincarée Separation Theorem for Normal Matrices and the Gauss–Lucas Theorem

We establish an analog of the Cauchy–Poincarée separation theorem for normal matrices in terms of majorization. A solution to the inverse spectral problem (Borg type result) is also presented. Using this result, we generalize and extend the Gauss–Lucas theorem about the location of roots of a complex polynomial and of its derivative. The generalization is applied to prove old conjectures due to de Bruijn–Springer and Schoenberg.     

我国宏观经济非参数联立模型的局部线性工具变量变窗宽估计

联立方程模型在经济政策制定、经济结构分析和预测方面起重要作用，目前关于非参数计量经济模型的研究主要停留在单方程模型上，而联立方程模型的研究在国际上刚刚起步，本将非参数回归模型的局部线性估计方法与传统联立方程模型估计方法相结合，首次提出了非参数计量经济联立模型的局部线性工具变量变窗宽估计并应用于我国宏观经济非参数联立模型，结果表明：我国宏观经济非参数联立模型优于线性联立模型且线性模型将造成不必要的人为设定误差；对于非参数联立模型，局部线性工具变量变窗宽估计优于局部线性估计。     

Discretized Tikhonov–Phillips regularization for a naturally linearized parameter identification problem

The problem of identification of the diffusion coefficient in the partial differential equation is considered. We discuss a natural linearization of this problem and application of discretized Tikhonov–Phillips regularization to its linear version. Using recent results of regularization theory, we propose a strategy for the choice of regularization and discretization parameters which automatically adapts to unknown smoothness of the coefficient. The estimation of the accuracy will be given and various numerical test supporting theoretical results will be presented.     

Bayesian Monte Carlo estimation for profile hidden Markov models

Hidden Markov models are used as tools for pattern recognition in a number of areas, ranging from speech processing to biological sequence analysis. Profile hidden Markov models represent a class of so-called “left–right” models that have an architecture that is specifically relevant to classification of proteins into structural families based on their amino acid sequences. Standard learning methods for such models employ a variety of heuristics applied to the expectation-maximization implementation of the maximum likelihood estimation procedure in order to find the global maximum of the likelihood function. Here, we compare maximum likelihood estimation to fully Bayesian estimation of parameters for profile hidden Markov models with a small number of parameters. We find that, relative to maximum likelihood methods, Bayesian methods assign higher scores to data sequences that are distantly related to the pattern consensus, show better performance in classifying these sequences correctly, and continue to perform robustly with regard to misspecification of the number of model parameters. Though our study is limited in scope, we expect our results to remain relevant for models with a large number of parameters and other types of left–right hidden Markov models.     

Stahel-Donoho kernel estimation for fixed design nonparametric regression models

This paper reports a robust kernel estimation for fixed design nonparametric regression models. A Stahel-Donoho kernel estimation is introduced, in which the weight functions depend on both the depths of data and the distances between the design points and the estimation points. Based on a local approximation, a computational technique is given to approximate to the incomputable depths of the errors. As a result the new estimator is computationally efficient. The proposed estimator attains a high breakdown point and has perfect asymptotic behaviors such as the asymptotic normality and convergence in the mean squared error. Unlike the depth-weighted estimator for parametric regression models, this depth-weighted nonparametric estimator has a simple variance structure and then we can compare its efficiency with the original one. Some simulations show that the new method can smooth the regression estimation and achieve some desirable balances between robustness and efficiency.     

Gauss–Man i n Conne ction s for Arrangements, I E igenvalues

We construct a formal connection on the Aomoto complex of an arrangement of hyperplanes, and use it to study the Gauss–Manin connection for the moduli space of the arrangement in the cohomology of a complex rank one local system. We prove that the eigenvalues of the Gauss–Manin connection are integral linear combinations of the weights which define the local system.     

Robust Solution to Fuzzy Identification Problem with Uncertain Data by Regularization

This study considers the robust identification of the parameters describing a Sugeno type fuzzy inference system with uncertain data. The objective is to minimize the worst-case residual error using a numerically efficient algorithm. The Sugeno type fuzzy systems are linear in consequent parameters but nonlinear in antecedent parameters. The robust consequent parameters identification problem can be formulated as second-order cone programming problem. The optimal solution of this second-order cone problem can be interpreted as solution of a Tikhonov regularization problem with a special choice of regularization parameter which is optimal for robustness (Ghaoui and Lebret (1997). SAIM Journal of Matrix Analysis and Applications 18, 1035–1064). The final regularized nonlinear optimization problem allowing simultaneous identification of antecedent and consequent parameters is solved iteratively using a generalized Gauss–Newton like method. To illustrate the approach, several simulation studies on numerical examples including the modelling of a spectral data function (one-dimensional benchmark example) is provided. The proposed robust fuzzy identification scheme has been applied to approximate the physical fitness of patients with a fuzzy expert system. The identified fuzzy expert system is shown to be capable of capturing the decisions (experiences) of a medical expert.     

人口增长率的非参数自回归预测模型

针对传统的人口增长预测模型不能理想地捕获我国人口增长率数据的非线性性特征,本文基于局部线性非参数估计理论,对我国建国以来的年人口增长率建立了非参数自回归NAR(1)模型,并对2000-2003年的年人口增长率进行了预测,计算结果表明,相对于参数自回归模型而言,非参数自回归模型能够很好地解决人口增长预测这一非线性问题,预测精度较高。     

Predictors for stationary processes with finite generalized Markovian property

A theorem is proved which establishes the conditions for a Gaussian vector stationary process to be Markovian. For a stationary process with finite generalized Markov property we construct a vector Markov process whose first coordinate coincides with the given process. Applying our theorem to the vector process, we derive formulas for the linear predictor of a process with finite generalized Markov property.Translated from Statisticheskie Metody, pp. 82–90, 1980.I would like to acknowledge the helpful attention of P. N. Sapozhnikov.     

Symmetric Gauss–Lobatto and Modified Anti-Gauss Rules

The present paper is concerned with symmetric Gauss–Lobatto quadrature rules, i.e., with Gauss–Lobatto rules associated with a nonnegative symmetric measure on the real axis. We propose a modification of the anti-Gauss quadrature rules recently introduced by Laurie, and show that the symmetric Gauss–Lobatto rules are modified anti-Gauss rules. It follows that for many integrands, symmetric Gauss quadrature rules and symmetric Gauss–Lobatto rules give quadrature errors of opposite sign.     

Bayesian nonparametric estimation of first passage distributions in semi‐Markov processes

Bayesian nonparametric (BNP) models provide a flexible tool in modeling many processes. One area that has not yet utilized BNP estimation is semi‐Markov processes (SMPs). SMPs require a significant amount of computation; this, coupled with the computation requirements for BNP models, has hampered any applications of SMPs using BNP estimation. This paper presents a modeling and computational approach for BNP estimation in semi‐Markov models, which includes a simulation study and an application of asthma patients' first passage from one state of control to another.     

Semi-empirical likelihood ratio confidence intervals for the difference of two sample means

We all know that we can use the likelihood ratio statistic to test hypotheses and construct confidence intervals in full parametric models. Recently, Owen (1988,Biometrika,75, 237–249; 1990,Ann. Statist.,18, 90–120) has introduced the empirical likelihood method in nonparametric models. In this paper, we combine these two likelihoods together and use the likelihood ratio to construct confidence intervals in a semiparametric problem, in which one model is parametric, and the other is nonparametric. A version of Wilks's theorem is developed.     

Asymptotic Gauss–Jacobi quadrature error estimation for Schwarz–Christoffel integrals

Numerical conformal mapping packages based on the Schwarz–Christoffel formula have been in existence for a number of years. Various authors, for good reasons of practical efficiency, have chosen to use composite n-point Gauss–Jacobi rules for the estimation of the Schwarz–Christoffel path integrals. These implementations rely on an ad hoc, but experimentally well-founded, heuristic for selecting the spacing of the integration end-points relative to the position of the nearby integrand singularities. In the present paper we derive an explicitly computable estimate, asymptotic as n→∞, for the relevant Gauss–Jacobi quadrature error. A numerical example illustrates the potential accuracy of the estimate even at low values of n. It is apparent that the error estimate will allow the adaptive construction of composite rules in a manner that is more efficient than has been possible hitherto.     

Randomly censored partially linear single-index models

This paper proposes a method for estimation of a class of partially linear single-index models with randomly censored samples. The method provides a flexible way for modelling the association between a response and a set of predictor variables when the response variable is randomly censored. It presents a technique for “dimension reduction” in semiparametric censored regression models and generalizes the existing accelerated failure-time models for survival analysis. The estimation procedure involves three stages: first, transform the censored data into synthetic data or pseudo-responses unbiasedly; second, obtain quasi-likelihood estimates of the regression coefficients in both linear and single-index components by an iteratively algorithm; finally, estimate the unknown nonparametric regression function using techniques for univariate censored nonparametric regression. The estimators for the regression coefficients are shown to be jointly root-n consistent and asymptotically normal. In addition, the estimator for the unknown regression function is a local linear kernel regression estimator and can be estimated with the same efficiency as all the parameters are known. Monte Carlo simulations are conducted to illustrate the proposed methodology.     

Robust estimation for partially linear models with large-dimensional covariates

We are concerned with robust estimation procedures to estimate the parameters in partially linear models with large-dimensional covariates. To enhance the interpretability, we suggest implementing a nonconcave regularization method in the robust estimation procedure to select important covariates from the linear component. We establish the consistency for both the linear and the nonlinear components when the covariate dimension diverges at the rate of o(n1/2), where n is the sample size. We show that the robust estimate of linear component performs asymptotically as well as its oracle counterpart which assumes the baseline function and the unimportant covariates were known a priori. With a consistent estimator of the linear component, we estimate the nonparametric component by a robust local linear regression. It is proved that the robust estimate of nonlinear component performs asymptotically as well as if the linear component were known in advance.Comprehensive simulation studies are carried out and an application is presented to examine the fnite-sample performance of the proposed procedures.     

On Hsu's theorem in multivariate regression

The paper deals with optimal quadratic unbiased estimation of the unknown dispersion matrix in multivariate regression models without assuming normality of the errors. We show that Hsu's theorem for univariate regression models continues to multivariate models with no additional assumptions. Furthermore optimal quadratic plus linear estimating functions for regression coefficients are considered, and we investigate whether the ordinary linear estimates are the best. This leads to a new theorem which is similar to that of Hsu.     

Automatic model selection for partially linear models

We propose and study a unified procedure for variable selection in partially linear models. A new type of double-penalized least squares is formulated, using the smoothing spline to estimate the nonparametric part and applying a shrinkage penalty on parametric components to achieve model parsimony. Theoretically we show that, with proper choices of the smoothing and regularization parameters, the proposed procedure can be as efficient as the oracle estimator [J. Fan, R. Li, Variable selection via nonconcave penalized likelihood and its oracle properties, Journal of American Statistical Association 96 (2001) 1348–1360]. We also study the asymptotic properties of the estimator when the number of parametric effects diverges with the sample size. Frequentist and Bayesian estimates of the covariance and confidence intervals are derived for the estimators. One great advantage of this procedure is its linear mixed model (LMM) representation, which greatly facilitates its implementation by using standard statistical software. Furthermore, the LMM framework enables one to treat the smoothing parameter as a variance component and hence conveniently estimate it together with other regression coefficients. Extensive numerical studies are conducted to demonstrate the effective performance of the proposed procedure.     

Confidence bands in nonparametric regression with length biased data

In this paper we deduce a confidence bands construction for the nonparametric estimation of a regression curve from length biased data, where a result from Bickel and Rosenblatt (1973,The Annals of Statistics,1, 1071–1095) is adapted to this new situation. The construction also involves the estimation of the variance of the local linear estimator of the regression, where we use a finite sample modification in order to improve the performance of these confidence bands in the case of finite samples.     

非参数理论综述

对非参数理论进行了系统地综述.非参数理论中一个比较重要的内容是估计方法,常见的非参数估计方法有核估计、局部多项式估计、近邻估计等.光滑参数的选取、"维数灾难"与边界点问题也是与非参数理论有关的重要内容,也对这些方面进行综述.最后,文章还综述了非参数技术在时间序列模型中的有关应用问题.     

Strong law of large numbers for countable nonhomogeneous Markov chains

The aim of this paper is to establish a strong law of large numbers for the bivariate functions of countable nonhomogeneous Markov chains under the condition of uniform convergence in the Cesàro sense which differs from my previous results. As corollaries, we generalize one of the Liu and Liu’s results for the univariate functions case and obtain another Shannon–McMillan–Breiman theorem for this Markov chains.