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A nearest neighbour-estimator for the score function
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  • Published: January 1986

A nearest neighbour-estimator for the score function

  • M. Csörgo1 &
  • P. Révész2 

Probability Theory and Related Fields volume 71, pages 293–305 (1986)Cite this article

  • 73 Accesses

  • 6 Citations

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Summary

Let f(·) be a strictly positive density function defined on (a,b)\( \subseteq \) R 1 with a continuous derivative f′(·) and let \(F(x) = \int\limits_a^x {f(t)} dt\),-∞≦a<x<+∞ be the corresponding distribution function. Define the quantile function Q of F by Q(y)=F −1(y)=inf{x:F(x)≧y}, 0<y<1, the score function (-1)J of the density function f by J(y)=f′(Q(y))/f(Q(y)), and the Fisher information I(f) of f by \(I(f) = \int\limits_0^1 {(J(y))^2 } dy\), assumed to be finite. Given some regularity conditions on F, we propose a sequence of nearest neighbour (N.N.) type estimators J n for J and prove that for all ε∈(0,1/5) there exists an estimator J n,ε of J such that for all δ∈(0,(5ε/18)∧(ε/12+1/40)) we have

$$\begin{gathered} \mathop {\sup }\limits_{n^{ - \delta } \leqq y \leqq - n^{ - \delta } } |J_{n,\varepsilon } (y) - J(y)| _ = ^{a.s} O(n^{ - 1/5 + \varepsilon } (\log n)^{1/2} ), \hfill \\ and I_n \left( f \right)\xrightarrow{{a.s.}}I\left( f \right), where I_n \left( f \right) = \mathop \smallint \limits_0^1 \left( {\bar J_n \left( y \right)} \right)^2 dy, with \bar J_n \left( y \right) = J_{n,\varepsilon } \left( y \right) if \hfill \\ n^{ - \delta } \leqq y \leqq 1 - n^{ - \delta } {\text{ }}and zero otherwise. \hfill \\ \end{gathered} $$

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References

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Carleton University, K 1S 5B6, Ottawa, Canada

    M. Csörgo

  2. Mathematical Institute of the Hungarian Academy of Sciences, Réaltanoda u. 13-15, 1053, Budapest, Hungary

    P. Révész

Authors
  1. M. Csörgo
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  2. P. Révész
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Additional information

This research was supported by a NSERC Canada Grant at Carleton University, Ottawa

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Csörgo, M., Révész, P. A nearest neighbour-estimator for the score function. Probab. Th. Rel. Fields 71, 293–305 (1986). https://doi.org/10.1007/BF00332313

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  • Received: 25 August 1982

  • Revised: 02 June 1985

  • Issue Date: January 1986

  • DOI: https://doi.org/10.1007/BF00332313

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Keywords

  • Distribution Function
  • Density Function
  • Stochastic Process
  • Probability Theory
  • Statistical Theory
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