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

Abstract

The problem of nonparametric function fitting with the observation model y i = f (x i ) + η i , i = 1, . . . , n, is considered, where η i are independent random variables with zero mean value and finite variance, andform a random sample from a distribution with density ∈ L 1 [a, b] and are independent of the errors η i , i = 1, . . . , n. The asymptotic properties of the estimator f N (n) b] and c N (n) = ( c 1 , . . . , c N (n) ) T obtained by the least squares method as well as the limits in probability of the estimators c k , k = 1, . . . , N , for fixed N , are studied in the case when the functions e k , k = 1, 2, . . . , forming a complete orthonormal system in L 2 [a, b] are analytic.1991 Mathematics Subject Classification: Primary 62G07, 62F12.

Orthogonal series regression estimation under long-range dependent errors

“…They extend the results of [11], [13] where we have only investigated consistency in the sense of that error for estimators constructed using orthonormal systems of univariate analytic functions and i.i.d. observation errors.…”

“…The above facts will be used to prove the results of this work, which continues the investigations of asymptotic properties of series type regression estimators, started by the author in [11], [13].…”

“…As proved in [11], in the case when we use multivariate orthonormal systems constructed as tensor products of univariate analytic functions forming an orthonormal system in L 2 ([a, b]) (e.g. multivariate trigonometric functions or multivariate polynomials), the normal equations matrix G n is almost surely positive definite for any density and N ≤ n. Thus, in that case the estimators considered are uniquely defined with probability one.…”

A note on orthogonal series regression function estimators

“…This follows from the author's results (see Lemma 2.2 of [4]) yielding that the matrices G n are almost surely positive definite for N + 1 ≤ n, when X i , i = 1, . .…”

“…. , n. Hence, the present work is also intended to complement and extend the results concerning the consistency of the least squares trigonometric and polynomial regression function estimators, obtained by the author in [4], [5]. A similar approach but restricted to less general regression function classes is presented by Vapnik in the monograph [7].…”