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Data-dependent kn-NN and kernel estimators consistent for arbitrary processes

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3 Author(s)
Kulkarni, S.R. ; Dept. of Electr. Eng., Princeton Univ., NJ, USA ; Posner, S.E. ; Sandilya, S.

Let X1, X2,... be an arbitrary random process taking values in a totally bounded subset of a separable metric space. Associated with Xi we observe Yi drawn from an unknown conditional distribution F(y|Xi=x) with continuous regression function m(x)=E[Y|X=x]. The problem of interest is to estimate Yn based on Xn and the data {(Xi, Yi)}i=1n-1. We construct appropriate data-dependent nearest neighbor and kernel estimators and show, with a very elementary proof, that these are consistent for every process X1, X2,.

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Information Theory, IEEE Transactions on  (Volume:48 ,  Issue: 10 )