A universal strong law of large numbers for conditional expectations via nearest neighbors

Authors:
Harro Walk
Affiliations:
Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany
Venue:
Journal of Multivariate Analysis
Year:
2008

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Abstract

For k"n-nearest neighbor estimates of a regression Y on X (d-dimensional random vector X, integrable real random variable Y) based on observed independent copies of (X,Y), strong universal pointwise consistency is shown, i.e., strong consistency P"X-almost everywhere for general distribution of (X,Y). With tie-breaking by indices, this means validity of a universal strong law of large numbers for conditional expectations E(Y|X=x).