Annals of Statistics

Asymptotics and optimal bandwidth selection for highest density region estimation

R. J. Samworth and M. P. Wand

Full-text: Open access

Enhanced PDF (824 KB)

Abstract

We study kernel estimation of highest-density regions (HDR). Our main contributions are two-fold. First, we derive a uniform-in-bandwidth asymptotic approximation to a risk that is appropriate for HDR estimation. This approximation is then used to derive a bandwidth selection rule for HDR estimation possessing attractive asymptotic properties. We also present the results of numerical studies that illustrate the benefits of our theory and methodology.

Article information

Source
Ann. Statist., Volume 38, Number 3 (2010), 1767-1792.

Dates
First available in Project Euclid: 24 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aos/1269452654

Digital Object Identifier
doi:10.1214/09-AOS766

Mathematical Reviews number (MathSciNet)
MR2662359

Zentralblatt MATH identifier
1189.62061

Subjects
Primary: 62G07: Density estimation 62G20: Asymptotic properties

Keywords
Density contour density level set kernel density estimator plug-in bandwidth selection

Citation

Samworth, R. J.; Wand, M. P. Asymptotics and optimal bandwidth selection for highest density region estimation. Ann. Statist. 38 (2010), no. 3, 1767--1792. doi:10.1214/09-AOS766. https://projecteuclid.org/euclid.aos/1269452654


Export citation

References

  • Baíllo, A. (2003). Total error in a plug-in estimator of level sets. Statist. Probab. Lett. 65 411–417.
    Zentralblatt MATH: 1116.62338
    Digital Object Identifier: doi:10.1016/j.spl.2003.08.007
  • Baíllo, A., Cuesta-Albertos, J. A. and Cuevas, A. (2001). Convergence rates in nonparametric estimation of level sets. Statist. Probab. Lett. 53 27–35.
    Zentralblatt MATH: 0980.62022
    Digital Object Identifier: doi:10.1016/S0167-7152(01)00006-2
  • Bowman, A. W. (1984). An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71 353–360.
    Mathematical Reviews (MathSciNet): MR767163
    Digital Object Identifier: doi:10.1093/biomet/71.2.353
    JSTOR: links.jstor.org
  • Burkill, J. C. and Burkill, H. (2002). A Second Course in Mathematical Analysis. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR1962361
    Zentralblatt MATH: 1021.00001
  • Cadre, B. (2006). Kernel estimation of density level sets. J. Multivariate Anal. 97 999–1023.
    Mathematical Reviews (MathSciNet): MR2256570
    Zentralblatt MATH: 1085.62039
    Digital Object Identifier: doi:10.1016/j.jmva.2005.05.004
  • Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics 63. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR1720712
    Zentralblatt MATH: 0951.60033
  • Giné, E. and Guillou, A. (2002). Rates of strong uniform consistency for multivariate kernel density estimators. Ann. Inst. H. Poincaré Probab. Statist. 38 907–921.
    Mathematical Reviews (MathSciNet): MR1955344
    Zentralblatt MATH: 1011.62034
    Digital Object Identifier: doi:10.1016/S0246-0203(02)01128-7
  • González-Manteiga, W., Sanchéz-Sellero, C. and Wand, M. P. (1996). Accuracy of binned kernel functional approximations. Comput. Statist. Data Anal. 22 1–16.
    Zentralblatt MATH: 0900.62195
    Digital Object Identifier: doi:10.1016/0167-9473(96)88030-3
  • Hartigan, J. A. (1987). Estimation of a convex density contour in two dimensions. J. Amer. Statist. Assoc. 82 267–270.
    Mathematical Reviews (MathSciNet): MR883354
    Zentralblatt MATH: 0607.62045
    Digital Object Identifier: doi:10.2307/2289162
    JSTOR: links.jstor.org
  • Hyndman, R. J. (1996). Computing and graphing highest density regions. Amer. Statist. 50 120–126.
  • Hyndman, R. J. (2009). hdrcde 2.12. Highest density regions and conditional density estimation. R package. Available at http://cran.r-project.org.
  • Jang, W. (2006). Nonparametric density estimation and clustering in astronomical sky surveys. Comput. Statist. Data Anal. 50 760–774.
    Mathematical Reviews (MathSciNet): MR2207006
    Zentralblatt MATH: 05381600
  • Marron, J. S. and Wand, M. P. (1992). Exact mean integrated squared error. Ann. Statist. 20 712–736.
    Mathematical Reviews (MathSciNet): MR1165589
    Zentralblatt MATH: 0746.62040
    Digital Object Identifier: doi:10.1214/aos/1176348653
    Project Euclid: euclid.aos/1176348653
  • Mason, D. M. and Polonik, W. (2009). Asymptotic normality of plug-in level set estimates. Ann. Appl. Probab. 19 1108–1142.
    Mathematical Reviews (MathSciNet): MR2537201
    Zentralblatt MATH: 1180.62048
    Digital Object Identifier: doi:10.1214/08-AAP569
    Project Euclid: euclid.aoap/1245071021
  • Müller, D. W. and Sawitzki, G. (1991). Excess mass estimates and tests for multimodality. J. Amer. Statist. Assoc. 86 738–746.
    Zentralblatt MATH: 0733.62040
  • Park, B. U. and Marron, J. S. (1990). Comparison of data-driven bandwidth selectors. J. Amer. Statist. Assoc. 85 66–72.
  • Polonik, W. (1995). Measuring mass concentrations and estimating density contour clusters—an excess mass approach. Ann. Statist. 23 855–881.
    Mathematical Reviews (MathSciNet): MR1345204
    Zentralblatt MATH: 0841.62045
    Digital Object Identifier: doi:10.1214/aos/1176324626
    Project Euclid: euclid.aos/1176324626
  • R Development Core Team (2008). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Available at http://www.R-project.org.
  • Rigollet, P. and Vert, R. (2009). Optimal rates for plug-in estimators of density level sets. Bernoulli 15 1154–1178.
    Zentralblatt MATH: 1200.62034
    Digital Object Identifier: doi:10.3150/09-BEJ184
    Project Euclid: euclid.bj/1262962230
  • Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators. Scand. J. Statist. 9 65–78.
    Mathematical Reviews (MathSciNet): MR668683
    Zentralblatt MATH: 0501.62028
  • Sheather, S. J. and Jones, M. C. (1991). A reliable data-based bandwidth selection method for kernel density estimation. J. Roy. Statist. Soc. Ser. B 53 683–690.
    Mathematical Reviews (MathSciNet): MR1125725
    JSTOR: links.jstor.org
    Zentralblatt MATH: 0800.62219
  • Tsybakov, A. B. (1997). On nonparametric estimation of density level sets. Ann. Statist. 25 948–969.
    Mathematical Reviews (MathSciNet): MR1447735
    Zentralblatt MATH: 0881.62039
    Digital Object Identifier: doi:10.1214/aos/1069362732
    Project Euclid: euclid.aos/1069362732
  • Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Monographs on Statistics and Applied Probability 60. Chapman and Hall, London.
    Mathematical Reviews (MathSciNet): MR1319818

The Institute of Mathematical Statistics

Annals of Statistics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more