Consistent regression estimation with fixed design points under dependence conditions

Abstract

For n = 1, 2,… and i integer between 1 and n, let x ni be fixed design points in a compact subset S of R p , p⩾ 1 , and let Y ni be observations taken at these points through g, an unknown continuous real-valued function defined on R p , and subject to errors ε ni ; that is, Y ni = g( x ni ) + ε ni . For any x in R p , g(x) is estimated by g n( x; x n ) = Σ n i = 1 w ni ( x; x n ) Y ni , where x n = ( x n1 ,…, x nn ) and w ni (·;·) are suitable weights. If the errors ε ni are centered at their expectations, the proposed estimate is asymptotically unbiased. It is also consistent in quadratic mean and strongly consistent, if, in addition and for each n, the random variables ε ni , i ⩾ 1, are coming from a strictly stationary sequence obeying any one of the four standard modes of mixing.