Least squares orthogonal polynomial regression estimation for irregular design. Issue 3 (1st February 2020)

• Record Type: Journal Article
• Authors: Popiński, Waldemar
• Is Part Of: Communications in statistics. Volume 49:Issue 3(2020)
• Page Start: 631
• Page End: 647
• Publication Date: 2020-02-01
• Subjects: Consistent estimator -- convergence rate -- least squares method -- orthogonal polynomials -- polynomial approximation -- regression function; 62G08 -- 62G20; Mathematical statistics -- Periodicals; Mathematics; Statistics; 519.2
• DOI: 10.1080/03610926.2018.1549244
• Languages: English
• ISSNs: 0361-0926

Abstract

Abstract: The problem of nonparametric function fitting using the complete orthogonal system of Legendre polynomials e k, k = 0, 1, …, for the observation model with errors in the independent and dependent variables y j = f ( x 0 j + ξ j ) + η j, j = 1, 2, …, n, is considered for f ∈ C [ − 1, 1 ], where x 0 j are equidistant points in [ − 1, 1 ], ξj are deterministic or random errors such that | ξ j | < 1 / n, and ηj are correlated random variables with E η η j = 0, E η η j 2 = σ η 2 < ∞ . Finite sample properties and convergence rates of the integrated mean-square error E η | | f − f ̂ N ( n ) | | 2 2, pointwise mean-square error E η ( f ( x ) − f ̂ N ( n ) ( x ) ) 2, mean supremum norm error E η | | f − f ̂ N ( n ) ‖ ∞, as well as average sample error ∑ j = 1 n E η ( f ( x 0 j + ξ j ) − f ̂ N ( n ) ( x 0 j + ξ j ) ) 2 / n of the estimator f ̂ N ( n ) ( x ) = ∑ k = 0 N ( n ) − 1 c ̂ k e k ( x ) with coefficients c ̂ k, k = 0, 1, …, N ( n ) − 1, obtained by the least squares method are studied.