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Truncation level selection in nonparametric regression using Pade approximation

COMMUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION 2021年第3期50卷 744-763页

作者： Aydin, Dursun Yilmaz, Ersin Mugla Sitki Kocman Univ Dept Stat Fac Sci TR-48000 Mugla Turkey

This paper introduces a Pade-type approximation for an unknown regression function in a nonparametric regression model. This newly introduced approximation provides a linear model with multi-collinearities and errors in all its variables. To deal with these issues, we used the truncated total least squares (TTLS) method. The efficient implementation of a Pade-type method using TTLS depends on choosing a truncation level. To provide an optimum truncation level for this method, we update the conventional parameter selection methods, including the generalized cross validation (GCV), improved version of the Akaike information criterion (AICc), restricted maximum likelihood (REML), Bayesian information criterion (BIC), and Mallows' C-p criterion. The primary aim of this study is to compare the performances of these level selection methods. A Monte Carlo simulation and a real data example are performed to illustrate the ideas in the paper. The results confirm that the GCV and AICc slightly outperform the other methods, especially when sample sizes are small and large, respectively.

关键词： Truncation level Selection methods Total least squares regression function

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Model selection with the Loss Rank Principle

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2010年第5期54卷 1288-1306页

作者： Hutter, Marcus Tran, Minh-Ngoc Natl Univ Singapore Dept Stat & Appl Probabil Singapore 117546 Singapore RSISE ANU Canberra ACT 0200 Australia SML NICTA Canberra ACT 0200 Australia

A key issue in statistics and machine learning is to automatically select the "right" model complexity, e.g., the number of neighbors to be averaged over in k nearest neighbor (kNN) regression or the polynomial degree in regression with polynomials. We suggest a novel principle - the Loss Rank Principle (LoRP) - for model selection in regression and classification. It is based on the loss rank, which counts how many other (fictitious) data would be fitted better. LoRP selects the model that has minimal loss rank. Unlike most penalized maximum likelihood variants (AIC, BIC, MDL), LoRP depends only on the regression functions and the loss function. It works without a stochastic noise model, and is directly applicable to any non-parametric regressor, like kNN. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Model selection Loss Rank Principle regression model selection Polynomial regression function

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Internal electric field modification method based on region segmentation for vacuum interrupters

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IET GENERATION TRANSMISSION & DISTRIBUTION 2017年第5期11卷 1195-1201页

作者： Li, Hongyang Wang, Zhongyu Jiang, Wensong Beihang Univ Sch Instrument Sci & Optoelect Engn Beijing Peoples R China

Enhancement of insulation performance has been a long-lasting problem for vacuum interrupter (VI) design. Traditional design focuses on reducing the maximal electric field strength (EFS);however, the distribution pattern of electric field also plays an important role in insulating efficiency. A method based on region segmentation is proposed to modify the electric field distribution. By dividing the electric field area, the regression function derived from orthogonal experimental design is established for each unit to reveal the relationship between the structure parameters of VI and EFS. Then the target EFS distribution is designed, and the optimal solution set is obtained by practical swarm optimisation. Fuzzy cluster selection is introduced to choose the design scheme from the solution set. Comparison of the simulation and experimental results shows the effectiveness of the proposed method.

关键词： vacuum interrupters electric fields regression analysis particle swarm optimisation fuzzy set theory pattern clustering Internal electric field modification method region segmentation vacuum interrupter insulation performance enhancement VI design maximal electric field strength reduction electric field distribution pattern regression function orthogonal experimental design EFS distribution practical swarm optimisation fuzzy cluster selection

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Recursive estimation of nonparametric regression with functional covariate

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2014年 69卷 154-172页

作者： Amiri, Aboubacar Crambes, Christophe Thiam, Baba Univ Lille 3 Univ Lille Nord France Lab EQUIPPE EA 4018 Villeneuve Dascq France Univ Montpellier 2 Inst Math & Modelisat Montpellier F-34090 Montpellier France

The main purpose is to estimate the regression function of a real random variable with functional explanatory variable by using a recursive nonparametric kernel approach. The mean square error and the almost sure convergence of a family of recursive kernel estimates of the regression function are derived. These results are established with rates and precise evaluation of the constant terms. Also, a central limit theorem for this class of estimators is established. The method is evaluated on simulations and real dataset studies. (C) 2013 Elsevier B.V. All rights reserved.

关键词： functional data Recursive kernel estimators regression function Quadratic error Almost sure convergence Asymptotic normality

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On asymptotic behavior of Nadaraya-Watson regression estimator

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2016年第19期45卷 5751-5761页

作者： Li, Jiexiang Coll Charleston Dept Math Robert Scott Small Room 339 Charleston SC 29424 USA

The random field (X-i, Y-i), is dependent and assumed to satisfy some mild mixing conditions. In this article, a Nadaraya-Watson regression estimator (N. W. R. E.) is established and investigated. We show that the dis... 详细信息

关键词： Asymptotic normality Nadaraya-Watson regression estimator Random field regression function Primary 62G08 62E20 Secondary 62M40

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Learning theory viewpoint of approximation by positive linear operators

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2010年第12期60卷 3177-3186页

作者： Lv, Shaogao Shi, Lei City Univ Hong Kong Univ Sci & Technol China Joint Adv Res Ctr Suzhou 215123 Peoples R China

We follow a learning theory viewpoint to study a family of learning schemes for regression related to positive linear operators in approximation theory. Such a learning scheme is generated from a random sample by a kernel function parameterized by a scaling parameter. The essential difference between this algorithm and the classical approximation schemes is the randomness of the sampling points, which breaks the condition of good distribution of sampling points often required in approximation theory. We investigate the efficiency of the learning algorithm in a regression setting and present learning rates stated in terms of the smoothness of the regression function, sizes of variances, and distances of kernel centers from regular grids. The error analysis is conducted by estimating the sample error and the approximation error. Two examples with kernel functions related to continuous Bernstein bases and Jackson kernels are studied in detail and concrete learning rates are obtained. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： Positive linear operator Approximation theory regression function Learning theory Bernstein and Jackson kernels

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Uniform-in-bandwidth consistency results in the partially linear additive model components estimation

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2024年第9期53卷 3383-3424页

作者： Chokri, Khalid Bouzebda, Salim Hassan II Univ Casablanca MAEGE Lab Casablanca Morocco Univ Technol Compiegne LMAC Lab Appl Math Compiegne F-60203 Compiegne France

We are mainly concerned with the partially linear additive model defined for a measurable function psi : R-q -> R, by [Graphics] where (Z(i) = Z(1,i),...,Z(p,i))(T) and X-i=(X-1,X-i,...,X-i,(d))(T )are vectors of explanatory variables, beta=(beta(1), horizontexpressionl ellipsis ,beta(p))(?) is a vector of unknown parameters, m(1),..., m(d) are unknown univariate real functions, and epsilon(1), ...,epsilon(n) are independent random errors with mean zero, finite variances re and E(epsilon|X,Z)=0 a.s. Under some mild conditions, we present a sharp uniform-in-bandwidth limit law for the nonlinear additive components of the model estimated by the marginal integration device with the kernel method. We allow the bandwidth to varying within the complete range for which the estimator is consistent. We provide the almost sure simultaneous asymptotic confidence bands for the regression functions.

关键词： Additive model regression function marginal integration non-parametric estimation density estimation regression estimation

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Relative error prediction: Strong uniform consistency for censoring time series model

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2023年第11期52卷 3709-3729页

作者： Bouhadjera, Feriel Elias, Ould Said Mohamed Riad, Remita Univ Badji Mokhtar Lab Probabilit & Stat BP 12 Annaba 23000 Algeria Univ Littoral Cote dOpale Lab Math Calais France

This article considers an adaptive method based on the relative error criteria to estimate the regression operator by a kernel smoothing. It is assumed that the variable of interest is subject to random right censoring and that the observations are from a stationary alpha-mixing process. The uniform almost sure consistency over a compact set with rate where we highlighted the covariance term is established. The simulation study indicates that the proposed approach has better performance in the presence of high level censoring and outliers in data to an existing classical method based on the least squares. An experiment prediction shows the quality of the relative error predictor.

关键词： Censored data Kernel estimate regression function relative error strong mixing uniform almost sure consistency

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Concave regression: value-constrained estimation and likelihood ratio-based inference

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MATHEMATICAL PROGRAMMING 2019年第1-2期174卷 5-39页

作者： Doss, Charles R. Univ Minnesota Sch Stat 313 Ford Hall224 Church St SE Minneapolis MN 55455 USA

We propose a likelihood ratio statistic for forming hypothesis tests and confidence intervals for a nonparametrically estimated univariate regression function, based on the shape restriction of concavity (alternatively, convexity). Dealing with the likelihood ratio statistic requires studying an estimator satisfying a null hypothesis, that is, studying a concave least-squares estimator satisfying a further equality constraint. We study this null hypothesis least-squares estimator (NLSE) here, and use it to study our likelihood ratio statistic. The NLSE is the solution to a convex program, and we find a set of inequality and equality constraints that characterize the solution. We also study a corresponding limiting version of the convex program based on observing a Brownian motion with drift. The solution to the limit problem is a stochastic process. We study the optimality conditions for the solution to the limit problem and find that they match those we derived for the solution to the finite sample problem. This allows us to show the limit stochastic process yields the limit distribution of the (finite sample) NLSE. We conjecture that the likelihood ratio statistic is asymptotically pivotal, meaning that it has a limit distribution with no nuisance parameters to be estimated, which makes it a very effective tool for this difficult inference problem. We provide a partial proof of this conjecture, and we also provide simulation evidence strongly supporting this conjecture.

关键词： Solution restricted distribution Null hypothesis regression function likelihood ratio convex programming Convexity Brownian motion Conjecture Stochastic Processes Markov chain CONCAVE Hypothesis testing least-squares estimator

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Strong consistency of the nonparametric local linear regression estimation under censorship model

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2022年第20期51卷 7056-7072页

作者： Bouhadjera, Feriel Ould Said, Elias Remita, Mohamed Riad Univ Badji Mokhtar Lab Probabilites & Stat BP 12 Annaba 23000 Algeria Univ Littoral Cote dOpale Lab Math Pures & Appl Calais France

We introduce and study a local linear nonparametric regression estimator for censorship model. The main goal of this paper is, to establish the uniform almost sure consistency result with rate over a compact set for the new estimate. To support our theoretical result, a simulation study has been done to make comparison with the classical regression estimator.

关键词： Censored data local linear estimation rate of consistency regression function survival analysis uniform almost sure consistency

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