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A strong law of large numbers for nonparametric regression

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  • Mukerjee, Hari

Abstract

Suppose (i1,n, ..., in,n) is permutation of (1, ..., n) for each positive integer n such that the order of the indices {1, h., n - 1} in the permutation corresponding to n - 1 is preserved. If {Zn} is a sequence of mean-zero random variables and {kn} is a sequence of positive integers with kn --> [infinity] and kn/n --> 0, we prove max1 0 a.s. under a first moment-type assumption on {Zn} and appropriate conditions on the permutations and the growth rate of {kn}. The result is applied to prove strong consistency of nonparametric estimators of regression functions with heavy-tailed error distributions using the k-nearest neighbor and the unikform kernel methods under similar moment assumptions on the conditional distributions of the regressed variable.

Suggested Citation

  • Mukerjee, Hari, 1989. "A strong law of large numbers for nonparametric regression," Journal of Multivariate Analysis, Elsevier, vol. 30(1), pages 17-26, July.
    • Handle: RePEc:eee:jmvana:v:30:y:1989:i:1:p:17-26
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    Cited by:

    1. Walk, Harro, 2008. "A universal strong law of large numbers for conditional expectations via nearest neighbors," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1035-1050, July.

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