Convergence of functional k-nearest neighbor regression estimate with functional responses

Abstract

Let (X₁,Y₁),…,(X_n,Y_n) 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.