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The problem of nonparametric regression function estimation is considered using the complete orthonormal system of trigonometric functions or Legendre polynomials $e_k$, k=0,1,..., for the observation model $y_i=f\left(x_i\right)+{\eta }_i$, i=1,...,n, where the ${\eta }_i$ are independent random variables with zero mean value and finite variance, and the observation points $x_i\in [a,b]$, i=1,...,n, form a random sample from a distribution with density $\varrho \in L^1[a,b]$. Sufficient and necessary conditions are obtained for consistency in the sense of the errors $\parallel f-{\widehat{f}}_N\parallel ,|f\left(x\right){-}_N\left(x\right)|$, $x\in [a,b]$,...

A method to analyse and filter real-valued discrete signals of finite duration s(n), n=0,1,...,N-1, where $N=2^p$, p>0, by means of time-frequency representation is presented. This is achieved by defining an invertible discrete transform representing a signal either in the time or in the time-frequency domain, which is based on decomposition of a signal with respect to a system of basic orthonormal discrete wavelet functions. Such discrete wavelet functions are defined using the Meyer generating wavelet...

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 ${\widehat{f}}_n\left(x\right)={\sum }_{k=0}^{N\left(n\right)}{\widehat{c}}_k{e}_k\left(x\right)$, where the coefficients ${\widehat{c}}_0,{\widehat{c}}_1,...,{\widehat{c}}_N$ are determined by minimizing the empirical risk $n^{-1}{\sum }_{i=1}^n{(Y_i-{\sum }_{k=0}^N{c}_k{e}_k\left(X_i\right))}^2$. Sufficient conditions for...

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.

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 $X_i\in A\subset {ℝ}^d$ have marginal distribution with density $\varrho \in L^1\left(A\right)$ and Var( Y | X = x) is bounded on A. Convergence rates of the errors $E_X{(f\left(X\right)-{\widehat{f}}_N\left(X\right))}^2$ and $\parallel f-{\widehat{f}}_N{\parallel }_{\infty }$ for the estimator ${\widehat{f}}_N\left(x\right)={\sum }_{k=1}^N{\widehat{c}}_k{e}_k\left(x\right)$, constructed using an orthonormal system $e_k$, k=1,2,..., in $L^2\left(A\right)$ are obtained.

The properties of two recursive estimators of the Fourier coefficients of a regression function $f\in L^2[a,b]$ with respect to a complete orthonormal system of bounded functions (ek) , k=1,2,..., are considered in the case of the observation model $y_i=f\left(x_i\right)+{\eta }_i$, i=1,...,n , where ${\eta }_i$ are independent random variables with zero mean and finite variance, $x_i\in [a,b]\subset R^1$, i=1,...,n, form a random sample from a distribution with density ϱ =1/(b-a) (uniform distribution) and are independent of the errors ${\eta }_i$, i=1,...,n . Unbiasedness and mean-square...

The problem of nonparametric function fitting using the complete orthogonal system of trigonometric functions $e_k$, k=0,1,2,..., for the observation model $y_i=f\left(x_{in}\right)+{\eta }_i$, i=1,...,n, is considered, where ${\eta }_i$ are uncorrelated random variables with zero mean value and finite variance, and the observation points $x_{in}\in [0,2\pi ]$, i=1,...,n, are equidistant. Conditions for convergence of the mean-square prediction error $(1/n){\sum }_{i=1}^{n}E{(f\left(x_{in}\right)-{\widehat{f}}_{N\left(n\right)}\left(x_{in}\right))}^2$, the integrated mean-square error $E\Vert f-{\widehat{f}}_{N\left(n\right)}{\Vert }^2$ and the pointwise mean-square error $E{\left(f\left(x\right){-}_{N\left(n\right)}\left(x\right)\right)}^2$ of the estimator ${\widehat{f}}_{N\left(n\right)}\left(x\right)={\sum }_{k=0}^{N\left(n\right)}{\widehat{c}}_k{e}_k\left(x\right)$ for f ∈ C[0,2π] and...

The problem of nonparametric estimation of a bounded regression function $f\in L²\left({[a,b]}^d\right)$, [a,b] ⊂ ℝ, d ≥ 1, using an orthonormal system of functions $e_k$, k=1,2,..., is considered in the case when the observations follow the model $Y_i=f\left(X_i\right)+{\eta }_i$, i=1,...,n, where $X_i$ and ${\eta }_i$ are i.i.d. copies of independent random variables X and η, respectively, the distribution of X has density ϱ, and η has mean zero and finite variance. The estimators are constructed by proper truncation of the function $f̂ₙ\left(x\right)={\sum }_{k=1}^{N\left(n\right)}c{̂}_k{e}_k\left(x\right)$, where the coefficients $ĉ₁,...,c{̂}_{N\left(n\right)}$ are determined...

This paper is concerned with general conditions for convergence rates of nonparametric orthogonal series estimators of the regression function. The estimators are obtained by the least squares method on the basis of an observation sample $Y_i=f\left(X_i\right)+{\eta }_i$, i=1,...,n, where $X_i\in A\subset {ℝ}^d$ are independently chosen from a distribution with density ϱ ∈ L¹(A) and ${\eta }_i$ are zero mean stationary errors with long-range dependence. Convergence rates of the error $n^{-1}{\sum }_{i=1}^n(f\left(X_i\right)-f{̂}_N\left(X_i\right))²$ for the estimator $f{̂}_N\left(x\right)={\sum }_{k=1}^{N}c{̂}_k{e}_k\left(x\right)$, constructed using an orthonormal system $e_k$, k=1,2,...,...

Asymptotic properties of the Discrete Fourier Transform spectrum of a complex monochromatic oscillation with frequency randomly distorted at the observation times t=0,1,..., n-1 by a series of independent and identically distributed fluctuations is investigated. It is proved that the second moments of the spectrum at the discrete Fourier frequencies converge uniformly to zero as n → ∞ for certain frequency fluctuation distributions. The observed effect occurs even for frequency fluctuations with...

The problem of nonparametric function fitting using the complete orthogonal system of Whittaker cardinal functions $s_k$, k = 0,±1,..., for the observation model $y_j=f\left(u_j\right)+{\eta }_j$, j = 1,...,n, is considered, where f ∈ L²(ℝ) ∩ BL(Ω) for Ω > 0 is a band-limited function, $u_j$ are independent random variables uniformly distributed in the observation interval [-T,T], ${\eta }_j$ are uncorrelated or correlated random variables with zero mean value and finite variance, independent of the observation points. Conditions for convergence...

In this work the problem of characterization of the Discrete Fourier Transform (DFT) spectrum of an original complex-valued signal $o_t$, t=0,1,...,n-1, modulated by random fluctuations of its amplitude and/or phase is investigated. It is assumed that the amplitude and/or phase of the signal at discrete times of observation are distorted by realizations of uncorrelated random variables or randomly permuted sequences of complex numbers. We derive the expected values and bounds on the variances of such...