Abstract
Let (X1,Y1),…,(Xn,Yn) be independent and identically distributed random elements taking values in ℱ×ℋ, where ℱ is a semi-metric space and ℋ is a separable Hilbert space. We investigate the rates of strong (almost sure) convergence of the k-nearest neighbor estimate. We give two convergence results assuming a finite moment condition and exponential tail condition on the noises respectively, with the latter requiring less stringent conditions on k for convergence.
Highlights
Let (F, d(., .)) be a semi-metric space, (H, · ) a separable Hilbert space, and let (X, Y ), (X1, Y1), (X2, Y2), . . . , (Xn, Yn) be independent identically distributed F × H-valued random pairs
The k-nearest neighbor (k-NN) method addresses this problem by using adaptive neighborhood size based on the distance of a point from its neighbors [5, 4, 13]
With the increasing interest at the present moment in many fields of statistics in which the observations are curves, such as speech recordings, weather data, commodity prices, functional regression analysis as an extension of classical setting has risen to the center stage of statistical research
Summary
Let (F , d(., .)) be a semi-metric space, (H, · ) a separable Hilbert space, and let (X, Y ), (X1, Y1), (X2, Y2), . . . , (Xn, Yn) be independent identically distributed F × H-valued random pairs. Usually an estimate of the function m(x) = E(Y |X = x) is being sought using n pairs of data points. The parametric modeling approach was masterfully documented in the monograph [19], and the nonparametric approach was proposed in the pioneering work [9] and popularized by the book [11] Another nonparametric approach is based on the reproducing kernel Hilbert spaces framework [18, 15]. In this note we investigate the convergence rates of functional k-NN estimate when the regression output takes values in a general separable Hilbert space H. This work can be regarded as an extension of [3] where k-NN method in functional regression with scalar responses is studied. They studied inferences using bootstrap and we did not investigate the inference problems here
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