IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v166y2020ics0167715220301838.html
   My bibliography  Save this article

Online estimation of integrated squared density derivatives

Author

Listed:
  • Mokkadem, Abdelkader
  • Pelletier, Mariane

Abstract

Hall and Marron (1987) introduced kernel estimators of integrals of the squared m-order derivatives of a probability density. Mokkadem and Pelletier (2020) gave recursive versions of their estimators, but the main drawback of these estimators is that their update requires the use of all past data. The aim of this paper is the study of online versions of the estimators introduced by Hall and Marron (1987), that is of estimators which are not only recursive, but which also have the property that their update uses only the last available data. Rates of convergence in mean squared error (MSE) are calculated. Similarly to the estimators of Hall and Marron (1987) and of Mokkadem and Pelletier (2020), our online estimators achieve the parametric rate n−1 when m=0 or when higher order kernels are used. For the case when the parametric rate is not obtained, we also study an online version of the estimator proposed by Jones and Sheather (1991). Finally, we provide recursive estimators of the optimal bandwidth in the framework of density estimation.

Suggested Citation

  • Mokkadem, Abdelkader & Pelletier, Mariane, 2020. "Online estimation of integrated squared density derivatives," Statistics & Probability Letters, Elsevier, vol. 166(C).
    • Handle: RePEc:eee:stapro:v:166:y:2020:i:c:s0167715220301838
    DOI: 10.1016/j.spl.2020.108880
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715220301838
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2020.108880?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Hall, Peter & Marron, J. S., 1987. "Estimation of integrated squared density derivatives," Statistics & Probability Letters, Elsevier, vol. 6(2), pages 109-115, November.
    2. Evarist Giné & David M. Mason, 2008. "Uniform in Bandwidth Estimation of Integral Functionals of the Density Function," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(4), pages 739-761, December.
    3. Jones, M. C. & Sheather, S. J., 1991. "Using non-stochastic terms to advantage in kernel-based estimation of integrated squared density derivatives," Statistics & Probability Letters, Elsevier, vol. 11(6), pages 511-514, June.
    4. Godichon-Baggioni, Antoine, 2016. "Estimating the geometric median in Hilbert spaces with stochastic gradient algorithms: Lp and almost sure rates of convergence," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 209-222.
    5. Anton Schick & Wolfgang Wefelmeyer, 2008. "Root-n consistency in weighted L 1 -spaces for density estimators of invertible linear processes," Statistical Inference for Stochastic Processes, Springer, vol. 11(3), pages 281-310, October.
    6. Mokkadem, Abdelkader & Pelletier, Mariane, 2020. "Recursive estimators of integrated squared density derivatives," Statistics & Probability Letters, Elsevier, vol. 157(C).
    7. Anton Schick & Wolfgang Wefelmeyer, 2004. "Root n consistent and optimal density estimators for moving average processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(1), pages 63-78, March.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. José E. Chacón & Carlos Tenreiro, 2012. "Exact and Asymptotically Optimal Bandwidths for Kernel Estimation of Density Functionals," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 523-548, September.
    2. Tiee-Jian Wu & Chih-Yuan Hsu & Huang-Yu Chen & Hui-Chun Yu, 2014. "Root $$n$$ n estimates of vectors of integrated density partial derivative functionals," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(5), pages 865-895, October.
    3. Hall, Peter & Wolff, Rodney C. L., 1995. "Estimators of integrals of powers of density derivatives," Statistics & Probability Letters, Elsevier, vol. 24(2), pages 105-110, August.
    4. Christopher Partlett & Prakash Patil, 2017. "Measuring asymmetry and testing symmetry," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(2), pages 429-460, April.
    5. Rudolf Grübel, 1994. "Estimation of density functionals," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(1), pages 67-75, March.
    6. Farmen, Mark & Marron, J. S., 1999. "An assessment of finite sample performance of adaptive methods in density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 30(2), pages 143-168, April.
    7. Dimitrios Bagkavos, 2011. "Local linear hazard rate estimation and bandwidth selection," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(5), pages 1019-1046, October.
    8. 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.
    9. Hidehiko Ichimura & Oliver Linton, 2001. "Asymptotic expansions for some semiparametric program evaluation estimators," CeMMAP working papers CWP04/01, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    10. Miguel Reyes & Mario Francisco-Fernández & Ricardo Cao, 2017. "Bandwidth selection in kernel density estimation for interval-grouped data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(3), pages 527-545, September.
    11. 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.
    12. Gonzalez-Manteiga, W. & Sanchez-Sellero, C. & Wand, M. P., 1996. "Accuracy of binned kernel functional approximations," Computational Statistics & Data Analysis, Elsevier, vol. 22(1), pages 1-16, June.
    13. Powell, James L. & Stoker, Thomas M., 1996. "Optimal bandwidth choice for density-weighted averages," Journal of Econometrics, Elsevier, vol. 75(2), pages 291-316, December.
    14. Saavedra, Ángeles & Cao, Ricardo, 1999. "Rate of convergence of a convolution-type estimator of the marginal density of a MA(1) process," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 129-155, April.
    15. Mizushima, Takamasa, 2000. "Multisample tests for scale based on kernel density estimation," Statistics & Probability Letters, Elsevier, vol. 49(1), pages 81-91, August.
    16. Berwin A. TURLACH, "undated". "Bandwidth selection in kernel density estimation: a rewiew," Statistic und Oekonometrie 9307, Humboldt Universitaet Berlin.
    17. Nils-Bastian Heidenreich & Anja Schindler & Stefan Sperlich, 2013. "Bandwidth selection for kernel density estimation: a review of fully automatic selectors," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 97(4), pages 403-433, October.
    18. Catalina Bolance & Montserrat Guillen & David Pitt, 2014. "Non-parametric Models for Univariate Claim Severity Distributions - an approach using R," Working Papers 2014-01, Universitat de Barcelona, UB Riskcenter.
    19. Dabo-Niang, Sophie & Francq, Christian & Zakoïan, Jean-Michel, 2010. "Combining Nonparametric and Optimal Linear Time Series Predictions," Journal of the American Statistical Association, American Statistical Association, vol. 105(492), pages 1554-1565.
    20. Vexler, Albert & Gao, Xinyu & Zhou, Jiaojiao, 2023. "How to implement signed-rank wilcox.test() type procedures when a center of symmetry is unknown," Computational Statistics & Data Analysis, Elsevier, vol. 184(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:166:y:2020:i:c:s0167715220301838. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.