Metrika

, Volume 76, Issue 3, pp 357–379 | Cite as

Density estimates of low bias

Article
First Online:
Received:

Abstract

Two methods are given for adapting a kernel density estimate to obtain an estimate of a density function with bias O(h p ) for any given p, where h = h(n) is the bandwidth and n is the sample size. The first method is standard. The second method is new and involves use of Bell polynomials. The second method is shown to yield smaller biases and smaller mean squared errors than classical kernel density estimates and those due to Jones et al. (Biometrika 82:327–338, 1995).

Keywords

Bias reduction Density estimates Kernel 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aldrovandi R (2001) Special matrices of mathematical physics. World Scientific, SingaporeMATHCrossRefGoogle Scholar
  2. Beirlant J, Goegebeur Y, Teugels J, Segers J (2004) Statistics of extremes: theory and applications. Wiley, ChichesterMATHCrossRefGoogle Scholar
  3. Bernardoff P (2006) Which multivariate gamma distributions are infinitely divisible?. Bernoulli 12: 169–189MathSciNetMATHGoogle Scholar
  4. Braun WJ, Hall P (2001) Data sharpening for nonparametric inference subject to constraints. J Comput Graph Stat 10: 786–806MathSciNetCrossRefGoogle Scholar
  5. Choi E, Hall P, Rousson V (2000) Data sharpening methods for bias reduction in nonparametric regression. Ann Stat 28: 1339–1355MathSciNetMATHCrossRefGoogle Scholar
  6. Comtet L (1974) Advanced combinatorics. Reidel, DordrechtMATHCrossRefGoogle Scholar
  7. Devroye L (1989) The double kernel method in density estimation. Ann Inst Henri Poincaré 25: 553–580MathSciNetGoogle Scholar
  8. Di Marzio M, Taylor CC (2004) Boosting kernel density estimates: a bias reduction technique?. Biometrika 91: 226–233MathSciNetMATHCrossRefGoogle Scholar
  9. Fisher RA, Cornish EA (1960) The percentile points of distributions having known cumulants. Technometrics 2: 209–242MATHCrossRefGoogle Scholar
  10. Hagmann M, Scaillet O (2007) Local multiplicative bias correction for asymmetric kernel density estimators. J Econom 141: 213–249MathSciNetCrossRefGoogle Scholar
  11. Hall P (1992) The bootstrap and Edgeworth expansion. Springer, New YorkGoogle Scholar
  12. Hall P, Minnotte MC (2002) High order data sharpening for density estimation. J R Stat Soc B 64: 141–157MathSciNetMATHCrossRefGoogle Scholar
  13. Hazelton ML (1998) Bias annihilating bandwidths for kernel density estimation at a point. Stat Probab Lett 38: 305–309MathSciNetMATHCrossRefGoogle Scholar
  14. Hazelton ML, Turlach BA (2007) Reweighted kernel density estimation. Comput Stat Data Anal 51: 3057–3069MathSciNetMATHCrossRefGoogle Scholar
  15. Jones MC, Linton O, Nielsen JP (1995) A simple bias reduction method for density estimation. Biometrika 82: 327–338MathSciNetMATHCrossRefGoogle Scholar
  16. Karunamuni RJ, Alberts T (2005) On boundary correction in kernel density estimation. Stat Methodol 2: 191–212MathSciNetMATHCrossRefGoogle Scholar
  17. Korzeniowski A (2006) Binomial tails domination for random graphs via Bell polynomials. J Probab Stat Sci 4: 99–105MathSciNetGoogle Scholar
  18. Marron JS, Wand MP (1992) Exact mean integrated squared error. Ann Stat 20: 712–736MathSciNetMATHCrossRefGoogle Scholar
  19. Mynbaev K, Martins-Filho C (2010) Bias reduction in kernel density estimation via Lipschitz condition. J Nonparametr Stat 22: 219–235MathSciNetMATHCrossRefGoogle Scholar
  20. Naito K (2001) On a certain class of nonparametric density estimators with reduced bias. Stat Probab Lett 51: 71–78MathSciNetMATHCrossRefGoogle Scholar
  21. Nielsen JP, Tanggaard C, Jones MC (2009) Local linear density estimation for filtered survival data, with bias correction. Statistics 43: 167–186MathSciNetCrossRefGoogle Scholar
  22. Nishiyama Y, Robinson PM (2005) The bootstrap and the Edgeworth correction for semiparametric averaged derivatives. Econometrica 73: 903–948MathSciNetMATHCrossRefGoogle Scholar
  23. Parzen E (1962) On estimation of a probability density function and mode. Ann Math Stat 33: 1065–1076MathSciNetMATHCrossRefGoogle Scholar
  24. Pitman J (2006) Combinatorial stochastic processes. Lectures from the 32nd summer school on probability theory held in saint-flour, July 7–24, 2002. With a foreword by Jean Picard. Lecture notes in Mathematics 1875. Springer, BerlinGoogle Scholar
  25. Rukhin AL (2010) Joint distribution of pattern frequencies and multivariate Polya-Aeppli law. Theory Probab Appl 54: 246–260MathSciNetCrossRefGoogle Scholar
  26. Sain SR, Scott DW (2002) Zero-bias locally adaptive density estimators. Scand J Stat 29: 441–460MathSciNetMATHCrossRefGoogle Scholar
  27. Schucany WR, Sommers JP (1977) Improvement of kernel type density estimators. J Am Stat Assoc 72: 420–423MathSciNetMATHCrossRefGoogle Scholar
  28. Silverman BW (1986) Density estimation for statistics and data analysis. Chapman and Hall, LondonMATHGoogle Scholar
  29. Stuetzle W, Mittal Y (1979) Some comments on the asymptotic behavior of robust smoothers. In: Gasser T, Rosenblatt Ms (eds) Smoothing techniques for curve estimation. Springer, Berlin, pp 191–195CrossRefGoogle Scholar
  30. Su LJ, Ullah A (2008) Nonparametric prewhitening estimators for conditional quantiles. Stat Sin 18: 1131–1152MathSciNetMATHGoogle Scholar
  31. Wendl MC (2005) Probabilistic assessment of clone overlaps in DNA fingerprint mapping via a priori models. J Comput Biol 12: 283–297CrossRefGoogle Scholar
  32. Withers CS, Nadarajah S (2008) Analytic bias reduction for k-sample functionals. Sankhyā A 70: 186–222MathSciNetMATHGoogle Scholar
  33. Withers CS, Nadarajah S (2009) Moments from cumulants and vice versa. Int J Math Educ Sci Technol 40: 842–845CrossRefGoogle Scholar
  34. Withers CS, Nadarajah S (2010) Moment inequalities for positive random variables. Comptes Rendus Math 348: 687–690MathSciNetMATHCrossRefGoogle Scholar
  35. Withers CS, Nadarajah S (2011) Moments in terms of cumulants when cumulants increase factorially. Util Math 84: 339–347MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Applied Mathematics GroupIndustrial Research LimitedLower HuttNew Zealand
  2. 2.School of MathematicsUniversity of ManchesterManchesterUK

Personalised recommendations