Abstract
We investigate the mean squared error and the asymptotic normality for a class of recursive kernel estimators based on a sample of spatially dependent observations. Our main result provides sufficient conditions for a spatial version of a recursive estimator introduced by Hall and Patil (1994) to satisfy a central limit theorem. The results are stated for strongly mixing random fields in the sense of Rosenblatt (1956) and for weakly dependent random fields in the sense of Wu (2005).
Highlights
Main resultsOne can notice that if Φ(u) = 1 for any u ∈ R fn,Φ reduces to the spatial version fn, of the recursive kernel density estimator of f introduced by Hall and Patil [20] and defined for any x ∈ RN by n n fn,1(x) = ( wsi )−1 wsj h−sjN Ksj (x, Xsj )
Which received considerable attention in nonparametric estimation for time series
Bosq, Merlevede and Peligrad [6] established a central limit theorem for the kernel density estimator fn when the sequence (Xi)i∈Z is assumed to be strongly mixing but the bandwidth parameter hn is assumed to satisfy hn ≥ Cn−1/3 log n which is stronger than the bandwidth parameter assumption in [11], [31] and [44]
Summary
One can notice that if Φ(u) = 1 for any u ∈ R fn,Φ reduces to the spatial version fn, of the recursive kernel density estimator of f introduced by Hall and Patil [20] and defined for any x ∈ RN by n n fn,1(x) = ( wsi )−1 wsj h−sjN Ksj (x, Xsj ). Theorem 2 is an extension of Theorem 2.1 in [36] where the asymptotic normality of the semi-recursive kernel regression estimator for time series (i.e. d = 1) introduced by Ahmad and Lin [2] is obtained under more restrictive conditions on the bandwidth parameter and the strong mixing coefficients.
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