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A note on orthogonal series regression function estimators

Waldemar Popiński (1999)

Applicationes Mathematicae

The problem of nonparametric estimation of the regression function f(x) = E(Y | X=x) using the orthonormal system of trigonometric functions or Legendre polynomials e k , k=0,1,2,..., is considered in the case where a sample of i.i.d. copies ( X i , Y i ) , i=1,...,n, of the random variable (X,Y) is available and the marginal distribution of X has density ϱ ∈ L 1 [a,b]. The constructed estimators are of the form f ^ n ( x ) = k = 0 N ( n ) c ^ k e k ( x ) , where the coefficients c ^ 0 , c ^ 1 , . . . , c ^ N are determined by minimizing the empirical risk n - 1 i = 1 n ( Y i - k = 0 N c k e k ( X i ) ) 2 . Sufficient conditions for...

A note on the rate of convergence of local polynomial estimators in regression models

Friedrich Liese, Ingo Steinke (2001)

Kybernetika

Local polynomials are used to construct estimators for the value m ( x 0 ) of the regression function m and the values of the derivatives D γ m ( x 0 ) in a general class of nonparametric regression models. The covariables are allowed to be random or non-random. Only asymptotic conditions on the average distribution of the covariables are used as smoothness of the experimental design. This smoothness condition is discussed in detail. The optimal stochastic rate of convergence of the estimators is established. The results...

A scale-space approach with wavelets to singularity estimation

Jérémie Bigot (2005)

ESAIM: Probability and Statistics

This paper is concerned with the problem of determining the typical features of a curve when it is observed with noise. It has been shown that one can characterize the Lipschitz singularities of a signal by following the propagation across scales of the modulus maxima of its continuous wavelet transform. A nonparametric approach, based on appropriate thresholding of the empirical wavelet coefficients, is proposed to estimate the wavelet maxima of a signal observed with noise at various scales. In...

A scale-space approach with wavelets to singularity estimation

Jérémie Bigot (2010)

ESAIM: Probability and Statistics

This paper is concerned with the problem of determining the typical features of a curve when it is observed with noise. It has been shown that one can characterize the Lipschitz singularities of a signal by following the propagation across scales of the modulus maxima of its continuous wavelet transform. A nonparametric approach, based on appropriate thresholding of the empirical wavelet coefficients, is proposed to estimate the wavelet maxima of a signal observed with noise at various scales....

Adaptive hard-thresholding for linear inverse problems

Paul Rochet (2013)

ESAIM: Probability and Statistics

A number of regularization methods for discrete inverse problems consist in considering weighted versions of the usual least square solution. These filter methods are generally restricted to monotonic transformations, e.g. the Tikhonov regularization or the spectral cut-off. However, in several cases, non-monotonic sequences of filters may appear more appropriate. In this paper, we study a hard-thresholding regularization method that extends the spectral cut-off procedure to non-monotonic sequences....

Adaptive tests for periodic signal detection with applications to laser vibrometry

Magalie Fromont, Céline Lévy-leduc (2006)

ESAIM: Probability and Statistics

Initially motivated by a practical issue in target detection via laser vibrometry, we are interested in the problem of periodic signal detection in a Gaussian fixed design regression framework. Assuming that the signal belongs to some periodic Sobolev ball and that the variance of the noise is known, we first consider the problem from a minimax point of view: we evaluate the so-called minimax separation rate which corresponds to the minimal l2-distance between the signal and zero so that the detection...

An application of nonprarametric Cox regression model in reliability analysis: a case study

Petr Volf (2004)

Kybernetika

The contribution deals with an application of the nonparametric version of Cox regression model to the analysis and modeling of the failure rate of technical devices. The objective is to recall the method of statistical analysis of such a model, to adapt it to the real–case study, and in such a way to demonstrate the flexibility of the Cox model. The goodness-of-fit of the model is tested, too, with the aid of the graphical test procedure based on generalized residuals.

An ℓ1-oracle inequality for the Lasso in finite mixture gaussian regression models

Caroline Meynet (2013)

ESAIM: Probability and Statistics

We consider a finite mixture of Gaussian regression models for high-dimensional heterogeneous data where the number of covariates may be much larger than the sample size. We propose to estimate the unknown conditional mixture density by an ℓ1-penalized maximum likelihood estimator. We shall provide an ℓ1-oracle inequality satisfied by this Lasso estimator with the Kullback–Leibler loss. In particular, we give a condition on the regularization parameter of the Lasso to obtain such an oracle inequality....

Bias-variance decomposition in Genetic Programming

Taras Kowaliw, René Doursat (2016)

Open Mathematics

We study properties of Linear Genetic Programming (LGP) through several regression and classification benchmarks. In each problem, we decompose the results into bias and variance components, and explore the effect of varying certain key parameters on the overall error and its decomposed contributions. These parameters are the maximum program size, the initial population, and the function set used. We confirm and quantify several insights into the practical usage of GP, most notably that (a) the...

Bootstrapping the shorth for regression

Cécile Durot, Karelle Thiébot (2006)

ESAIM: Probability and Statistics

The paper is concerned with the asymptotic distributions of estimators for the length and the centre of the so-called η-shorth interval in a nonparametric regression framework. It is shown that the estimator of the length converges at the n1/2-rate to a Gaussian law and that the estimator of the centre converges at the n1/3-rate to the location of the maximum of a Brownian motion with parabolic drift. Bootstrap procedures are proposed and shown to be consistent. They are compared with the plug-in...

Convergence rates of orthogonal series regression estimators

Waldemar Popiński (2000)

Applicationes Mathematicae

General conditions for convergence rates of nonparametric orthogonal series estimators of the regression function f(x)=E(Y | X = x) are considered. The estimators are obtained by the least squares method on the basis of a random observation sample (Yi,Xi), i=1,...,n, where X i A d have marginal distribution with density ϱ L 1 ( A ) and Var( Y | X = x) is bounded on A. Convergence rates of the errors E X ( f ( X ) - f ^ N ( X ) ) 2 and f - f ^ N for the estimator f ^ N ( x ) = k = 1 N c ^ k e k ( x ) , constructed using an orthonormal system e k , k=1,2,..., in L 2 ( A ) are obtained.

Detecting atypical data in air pollution studies by using shorth intervals for regression

Cécile Durot, Karelle Thiébot (2010)

ESAIM: Probability and Statistics

To validate pollution data, subject-matter experts in Airpl (an organization that maintains a network of air pollution monitoring stations in western France) daily perform visual examinations of the data and check their consistency. In this paper, we describe these visual examinations and propose a formalization for this problem. The examinations consist in comparisons of so-called shorth intervals so we build a statistical test that compares such intervals in a nonparametric regression model. This...

Detecting atypical data in air pollution studies by using shorth intervals for regression

Cécile Durot, Karelle Thiébot (2005)

ESAIM: Probability and Statistics

To validate pollution data, subject-matter experts in Airpl (an organization that maintains a network of air pollution monitoring stations in western France) daily perform visual examinations of the data and check their consistency. In this paper, we describe these visual examinations and propose a formalization for this problem. The examinations consist in comparisons of so-called shorth intervals so we build a statistical test that compares such intervals in a nonparametric regression model. This...

Efficient robust nonparametric estimation in a semimartingale regression model

Victor Konev, Serguei Pergamenshchikov (2012)

Annales de l'I.H.P. Probabilités et statistiques

The paper considers the problem of robust estimating a periodic function in a continuous time regression model with the dependent disturbances given by a general square integrable semimartingale with an unknown distribution. An example of such a noise is a non-Gaussian Ornstein–Uhlenbeck process with jumps (see (J. R. Stat. Soc. Ser. B Stat. Methodol.63 (2001) 167–241), (Ann. Appl. Probab.18 (2008) 879–908)). An adaptive model selection procedure, based on the weighted least square estimates, is...

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