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On bandwidth selection problems in nonparametric trend estimation under martingale difference errors

BERNOULLI 2022年第1期28卷 395-423页

作者： Benhenni, Karim Girard, Didier A. Louhichi, Sana Univ Grenoble Alpes Lab Jean Kuntzmann 700 Ave Cent F-38401 St Martin Dheres France CNRS Lab Jean Kuntzmann 700 Ave Cent F-38401 St Martin Dheres France

In this paper, we are interested in the problem of smoothing parameter selection in nonparametric curve estimation under dependent errors. We focus on kernel estimation and the case when the errors form a general stationary sequence of martingale difference random variables where neither linearity assumption nor "all moments are finite" are required. We compare the behaviors of the smoothing bandwidths obtained by minimizing either the unknown average squared error, the theoretical mean average squared error, a Mallows-type criterion adapted to the dependent case and the family of criteria known as generalized cross validation (GCV) extensions of the Mallows' criterion. We prove that these three minimizers and those based on the GCV family are first-order equivalent in probability. We give also a normal asymptotic behavior of the gap between the minimizer of the average squared error and that of the Mallows-type criterion. This is extended to the GCV family. Finally, we apply our theoretical results to a specific case of martingale difference sequence, namely the Auto-Regressive Conditional Heteroscedastic (ARCH(1)) process. A Monte-Carlo simulation study, for this regression model with ARCH(1) process, is conducted.

关键词： Nonparametric trend estimation kernel nonparametric models smoothing parameter selection martingale difference sequences average squared error mean average squared error Mallows criterion cross validation generalized cross validation ARCH(1)

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Performance criteria and discrimination of extreme undersmoothing in nonparametric regression

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JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2014年 153卷 56-74页

作者： Lukas, Mark A. Murdoch Univ Murdoch WA 6150 Australia

The prediction error (average squared error) is the most commonly used performance criterion for the assessment of nonparametric regression estimators. However, there has been little investigation of the properties of the criterion itself. This paper shows that in certain situations the prediction error can be very misleading because it fails to discriminate an extreme undersmoothed estimate from a good estimate. For spline smoothing, we show, using asymptotic analysis and simulations, that there is poor discrimination of extreme undersmoothing in the following situations: small sample size or small error variance or a function with high curvature. To overcome this problem, we propose using the Sobolev error criterion. For spline smoothing, it is shown asymptotically and by simulations that the Sobolev error is significantly better than the prediction error in discriminating extreme undersmoothing. Similar results hold for other nonparametric regression estimators and for multivariate smoothing. For thin-plate smoothing splines, the prediction error's poor discrimination of extreme undersmoothing becomes significantly worse with increasing dimension. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Prediction error average squared error Sobolev error Optimality criterion Smoothing spline Kernel smoothing

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One-sided cross-validation for nonsmooth regression functions

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JOURNAL OF NONPARAMETRIC STATISTICS 2013年第4期25卷 889-904页

作者： Savchuk, Olga Y. Hart, Jeffrey D. Sheather, Simon P. SUNY Binghamton Dept Math Sci Binghamton NY 13902 USA Texas A&M Univ Dept Stat College Stn TX 77843 USA

The one-sided cross-validation (OSCV) method is shown to be robust to lack of smoothness in the regression function. Two corrections for the case where the regression function has a discontinuous first derivative are proposed. Simulation results suggest that proposed modifications of the OSCV method are efficient for regression functions whose first derivative is discontinuous at more than two points. The OSCV method and its modification outperform the cross-validation method and the Ruppert-Sheather-Wand plug-in method in a data example involving a function that, potentially, has one discontinuity in its derivative.

关键词： local linear estimator cross-validation bandwidth selection average squared error mean average squared error

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Robustness of one-sided cross-validation to autocorrelation

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JOURNAL OF MULTIVARIATE ANALYSIS 2005年第1期92卷 77-96页

作者： Hart, JD Lee, CL Texas A&M Univ Dept Stat College Stn TX 77843 USA PPD Dev Dept Biostat Austin TX 78704 USA

The effects of moderate levels of serial correlation on one-sided and ordinary cross-validation in the context of local linear and kernel smoothing is investigated. It is shown both theoretically and by simulation that one-sided cross-validation is much less adversely affected by correlation than is ordinary cross-validation. The former method is a reliable means of window width selection in the presence of moderate levels of serial correlation, while the latter is not. It is also shown that ordinary cross-validation is less robust to correlation when applied to Gasser-Muller kernel estimators than to local linear ones. (C) 2003 Elsevier Inc. All rights reserved.

关键词： nonparametric regression data-driven smoothing parameters autoregressive process average squared error

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One-Sided Cross-Validation

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Quarterly Publications of the American Statistical Association 1998年第442期93卷 620-631页

A new method of selecting the smoothing parameters of nonparametric regression estimators is introduced. The method, termed one-sided cross-validation (OSCV), has the objectivity of cross-validation and statistical properties comparable to those of a plug-in rule. The new method may be viewed as an application of the prequential model selection method of Dawid. As such, our results identify a situation in which the prequential method is a more efficient model selector than cross-validation. An example, simulations, and theoretical results demonstrate the utility of OSCV when used with local linear and kernel estimators.

关键词： average squared error Data-driven smoothing Mean average squared error Optimal bandwidths Prediction.

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Testing the hypothesis of a general linear model using nonparametric regression estimation

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Test 1993年第1-2期2卷 161-188页

作者： González-Manteiga, W. Cao, R. Facultad de Matemáticas Universidad de Santiago de Compostela Santiago de Compostela 15771 Spain

Given the model Y i =m(χ i )+e{open}i,where E(e{open} i) =0, X i ≠Ci=1, ..., n, and C is a p-dimensional compact set, we have designed a new method for testing the hypothesis that the regression function follows a g... 详细信息

Given the model Y_i =m(χ_i )+e{open}i,where E(e{open}_i) =0, X_i ≠Ci=1, ..., n, and C is a p-dimensional compact set, we have designed a new method for testing the hypothesis that the regression function follows a general linear model, m(·) ∈ {m_θ(·) =A^t (·)θ}_{θ∈Θ⊂ℛq}, with A a function from ℜ^p to ℜ^q. The statistic, denoted ΔASE, used fortesting the given hypothesis is defined to be the difference between the average squared errors (ASE) associated with the non-parametric estimator {Mathematical expression} of m and the minimum distance parametric estimator {Mathematical expression} of m. The asymptotic normality of both ΔASE and the minimum distance estimators is proved under general conditions. Alternative bootstrap versions of ΔASE are also considered. © 1993 SEIO.

关键词： average squared error Bootstrap General linear model Nonparametric regression estimation

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Invalidity of average squared error criterion in density estimation

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Canadian Journal of Statistics 1978年第2期6卷 193-200页

作者： Michael Steele, J. Department of Statistics Stanford University Stanford California 94305 United States

The average squared error has been suggested earlier as an appropriate estimate of the integrated squared error, but an example is given which shows their ratio can tend to infinity. The results of a Monte Carlo study are also presented which suggest the average squared error can seriously underestimate the errors inherent in even the simplest density estimations. Copyright © 1978 Statistical Society of Canada

关键词： AMS 1970 subject classifications: Primary 62G20 secondary 62F20 average squared error Density estimation mean integrated squared error

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