Abstract
Starting from the classical theorem of Weierstrass (and its modifications) on approximation of continuous functions by means of Bernstein polynomials a smoothed histogram type estimator is developed for estimating probability densities and its derivatives. Consistency results are obtained in form of various strong laws. In particular, one gets estimates for the rates for pointwise and uniform strong convergence of estimators for the derivatives. Moreover, for approximating the density itself the exact order of consistency is established. This is done by a law of iterated logarithm for pointwise approximation and by a law of logarithm in case of uniform approximation.
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This paper contains parts of the author's “Habilitationsschrift” written at the Department of Mathematics of the University of Ulm.
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Gawronski, W. Strong laws for density estimators of bernstein type. Period Math Hung 16, 23–43 (1985). https://doi.org/10.1007/BF01855801
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DOI: https://doi.org/10.1007/BF01855801