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Nonparametric confidence intervals for the integral of a function of an unknown density

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  • Christopher Withers
  • Saralees Nadarajah

Abstract

Given a random sample of size n from an unknown distribution function F on ℝ with finite derivatives and density f, we wish to estimate for a smooth function L. Examples are t f2, the differential entropy and the Kullback–Leibler distance. We estimate f using a kernel estimate [fcirc] based on a kernel of order p, say. We show that {[fcirc](xi), i=1, …, s} satisfies the Cornish–Fisher assumption with respect to m=nh. It follows that the corresponding estimate θˆ has a bias of magnitude O(hq+m−1), where p≤q≤2p depends on L. We show that the variance of θˆ has magnitude O(n−1) for a suitable bandwidth. For the regular case, we give one-sided and two-sided confidence intervals for θ with errors of magnitude O(M−1/2) and O(M−1), where M=nh2. We present simulation studies to show the practical values of the results.

Suggested Citation

  • Christopher Withers & Saralees Nadarajah, 2011. "Nonparametric confidence intervals for the integral of a function of an unknown density," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(4), pages 943-966.
    • Handle: RePEc:taf:gnstxx:v:23:y:2011:i:4:p:943-966
    DOI: 10.1080/10485252.2011.576762
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    References listed on IDEAS

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    1. Tenreiro, Carlos, 2003. "On the asymptotic normality of multistage integrated density derivatives kernel estimators," Statistics & Probability Letters, Elsevier, vol. 64(3), pages 311-322, September.
    2. Tchetgen, Eric & Li, Lingling & Robins, James & van der Vaart, Aad, 2008. "Minimax estimation of the integral of a power of a density," Statistics & Probability Letters, Elsevier, vol. 78(18), pages 3307-3311, December.
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