Convergence rates of orthogonal series regression estimators

Waldemar Popiński

Displaying similar documents to “Convergence rates of orthogonal series regression estimators”

A note on orthogonal series regression function estimators

Waldemar Popiński (1999)

Applicationes Mathematicae

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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 ${\hat{f}}_{n} (x) = \sum_{k = 0}^{N (n)} {\hat{c}}_{k} e_{k} (x)$ , where the coefficients ${\hat{c}}_{0}, {\hat{c}}_{1}, . . ., {\hat{c}}_{N}$ are determined by minimizing the empirical risk $n^{- 1} \sum_{i = 1}^{n} {(Y_{i} - \sum_{k = 0}^{N} c_{k} e_{k} (X_{i}))}^{2}$ . Sufficient conditions...

Least-squares trigonometric regression estimation

Waldemar Popiński (1999)

Applicationes Mathematicae

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The problem of nonparametric function fitting using the complete orthogonal system of trigonometric functions $e_{k}$ , k=0,1,2,..., for the observation model $y_{i} = f (x_{i n}) + η_{i}$ , i=1,...,n, is considered, where $η_{i}$ are uncorrelated random variables with zero mean value and finite variance, and the observation points $x_{i n} \in [0, 2 π]$ , i=1,...,n, are equidistant. Conditions for convergence of the mean-square prediction error $(1 / n) \sum_{i = 1}^{n} E {(f (x_{i n}) - {\hat{f}}_{N (n)} (x_{i n}))}^{2}$ , the integrated mean-square error $E ‖ f - {\hat{f}}_{N (n)} ‖^{2}$ and the pointwise mean-square error $E {(f (x) -_{N (n)} (x))}^{2}$ of the estimator ${\hat{f}}_{N (n)} (x) = \sum_{k = 0}^{N (n)} {\hat{c}}_{k} e_{k} (x)$ for f ∈...

Orthogonal series regression estimators for an irregularly spaced design

Waldemar Popiński (2000)

Applicationes Mathematicae

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Nonparametric orthogonal series regression function estimation is investigated in the case of a fixed point design where the observation points are irregularly spaced in a finite interval [a,b]i ⊂ ℝ. Convergence rates for the integrated mean-square error and pointwise mean-square error are obtained in the case of estimators constructed using the Legendre polynomials and Haar functions for regression functions satisfying the Lipschitz condition.

On least squares estimation of Fourier coefficients and of the regression function

Waldemar Popiński (1993)

Applicationes Mathematicae

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Regression quantiles and trimmed least squares estimator under a general design

Jana Jurečková (1984)

Kybernetika

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Consistency of linear and quadratic least squares estimators in regression models with covariance stationary errors

František Štulajter (1991)

Applications of Mathematics

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The least squres invariant quadratic estimator of an unknown covariance function of a stochastic process is defined and a sufficient condition for consistency of this estimator is derived. The mean value of the observed process is assumed to fulfil a linear regresion model. A sufficient condition for consistency of the least squares estimator of the regression parameters is derived, too.

Estimators in the location model with gradual changes

Marie Hušková (1998)

Commentationes Mathematicae Universitatis Carolinae

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A number of papers has been published on the estimation problem in location models with abrupt changes (e.g., Cs" orgő and Horváth (1996)). In the present paper we focus on estimators in location models with gradual changes. Estimators of the parameters are proposed and studied. It appears that the limit behavior (both the rate of consistency and limit distribution) of the estimators of the change point in location models with abrupt changes and gradual changes differ substantially. ...

Robust estimation and forecasting for beta-mixed hierarchical models of grouped binary data.

Maxim A. Pashkevich, Yurij S. Kharin (2004)

SORT

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The paper focuses on robust estimation and forecasting techniques for grouped binary data with misclassified responses. It is assumed that the data are described by the beta-mixed hierarchical model (the beta-binomial or the beta-logistic), while the misclassifications are caused by the stochastic additive distorsions of binary observations. For these models, the effect of ignoring the misclassifications is evaluated and expressions for the biases of the method-of-moments estimators...