Fixed Design Regression Under Association
George G. Roussas · Lecture notes-monograph series · 2001
For n = 1,..., 7i, let x n i,i = 1,..., n, be points in a compact subset in Sft d ,d > 1, at which observations Y n { are taken.It is assumed that these observations have the structure Y n i = g(x n i) + ε n i> where g is a real-valued unknown function, and the errors (e n i, ^nn) coincide with the segment (£χ,... ,f n ) of a strictly stationary sequence of random variables ξi, &»-F°r each x G 5R d , the function g(x) is estimated by g n (x]Xn) = i2?=i w ni(x\x n )Yni, where x n = (x nli ... ,x nn ) and Wnϊ( ; •) are weight functions.Under suitable conditions on the underlying stochastic process £1,62, and the weights w n i( ; •), it is shown that the estimate g n (x\Xn) is asymptotically unbiased, and consistent in quadratic mean.By adding the assumption of (positive or negative) association of the sequence £i,&» • •> it is shown that ρ n (^;^n), properly normalized, is also asymptotically normal.