Displaying similar documents to “Least-squares trigonometric regression estimation”

Convergence rates of orthogonal series regression estimators

Waldemar Popiński (2000)

Applicationes Mathematicae

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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 ⊂ ℝ 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. ...

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 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...

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.

Sufficient conditions for the strong consistency of least squares estimator with α-stable errors

João Tiago Mexia, João Lita da Silva (2007)

Discussiones Mathematicae Probability and Statistics

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Let Y i = x i T β + e i , 1 ≤ i ≤ n, n ≥ 1 be a linear regression model and suppose that the random errors e₁, e₂, ... are independent and α-stable. In this paper, we obtain sufficient conditions for the strong consistency of the least squares estimator β̃ of β under additional assumptions on the non-random sequence x₁, x₂,... of real vectors.