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On arbitrarily slow rates of global convergence in density estimation

Summary

Let a density f on R d be estimated by f n(x, X 1, ..., X n) where xR d, f n is a Borel measurable function of its arguments, and X 1, ..., X n are independent random vectors with common density f. Let p≧1 be a constant. One of the main results of this note is that for every sequence f n, and for every positive number sequence a n satisfying lim a n=0, there exists an f such that

$$E\left( {\smallint |f_n \left( x \right) - f\left( x \right)|^p dx} \right) > a_n$$

infinitely often.

Here it suffices to look at all the f that are bounded by 2 and vanish outside [0, 1]d. For p=1, f can always be restricted to the class of infinitely many times continuously differentiable densities with all derivatives absolutely bounded and absolutely integrable.

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Research carried out while the authors was visiting the Applied Research Laboratories, University of Texas at Austin

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Devroye, L. On arbitrarily slow rates of global convergence in density estimation. Z. Wahrscheinlichkeitstheorie verw Gebiete 62, 475–483 (1983). https://doi.org/10.1007/BF00534199

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Keywords

  • Stochastic Process
  • Probability Theory
  • Measurable Function
  • Slow Rate
  • Density Estimation