A note on orthogonal series regression function estimators

• Autorzy: Waldemar Popiński
• Języki publikacji: EN

Abstrakty

EN

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 $\widehat f_n(x) = \sum_{k=0}^{N(n)}\widehat c_ke_k(x)$, where the coefficients $\widehat c_0,\widehat c_1,...,\widehat c_N$ are determined by minimizing the empirical risk $n^{-1}\sum_{i=1}^n(Y_i - \sum_{k=0}^Nc_ke_k(X_i))^2$. Sufficient conditions for consistency of the estimators in the sense of the errors $E_X\vert f(X)-\widehat f_n(X)\vert^2$ and $n^{-1}\sum_{i=1}^nE(f(X_i)-\widehat f_n(X_i))^2$ are obtained.

Słowa kluczowe

EN

consistent estimator; orthonormal system; empirical risk minimization; nonparametric regression

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Applicationes Mathematicae
• Rocznik: 1999
• Tom: 26
• Numer: 3
• Strony: 281-291

Daty

wydano

1999

otrzymano

1998-10-01

poprawiono

1999-03-04

Twórcy

autor

Waldemar Popiński

• Department of Standards, Central Statistical Office, Al. Niepodległości 208, 00-925 Warszawa, Poland

Bibliografia

• [1] A. R. Gallant and H. White, There exists a neural network that does not make avoidable mistakes, in: Proc. Second Annual IEEE Conference on Neural Networks, San Diego, California, IEEE Press, New York, 1988, 657-664.
• [2] L. Györfi and H. Walk, On the strong universal consistency of a series type regression estimate, Math. Methods Statist. 5 (1996), 332-342.
• [3] G. Lugosi and K. Zeger, Nonparametric estimation via empirical risk minimization, IEEE Trans. Inform. Theory IT-41 (1995), 677-687.
• [4] W. Popiński, On least squares estimation of Fourier coefficients and of the regression function, Appl. Math. (Warsaw) 22 (1993), 91-102.
• [5] --, Consistency of trigonometric and polynomial regression estimators, ibid. 25 (1998), 73-83.
• [6] G. Sansone, Orthogonal Functions, Interscience Publ., New York, 1959.\vadjust
• [7] V. N. Vapnik, Estimation of Dependencies Based on Empirical Data, Springer, New York, 1982.