Abstract
In this paper we consider a class of estimates of a bivariate density function f based on an independent sample of size n. Under the assumption that f is uniformly continuous, the uniform strong consistency of such estimates was first proved by Nadaraya (1970) for a large class of kernel functions. In this note we show that the assumption of the uniform continuity of f is necessary for this type of convergence.
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References
Nadaraya, E.A.: Remarks on non-parametric estimates for density functions and regression curves. Theor. Probab. Appl. 15, 134–136 (1970)
Kiefer, J., Wolfowitz, J.: On the deviations of the empiric distribution function of vector chance variables. Trans. Amer. math. Soc., 87, 173–186 (1958)
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Samanta, M. A note on uniform strong convergence of bivariate density estimates. Z. Wahrscheinlichkeitstheorie verw Gebiete 28, 85–88 (1974). https://doi.org/10.1007/BF00533360
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DOI: https://doi.org/10.1007/BF00533360