IDEAS home Printed from https://ideas.repec.org/a/spr/sankha/v83y2021i1d10.1007_s13171-019-00171-6.html
   My bibliography  Save this article

Data Dependent Asymmetric Kernels for Estimating the Density Function

Author

Listed:
  • R. N. Rattihalli

    (Central University of Rajastan)

  • S. B. Patil

    (Shivaji University Kolhapur)

Abstract

In the nonparametric setup, smooth density estimators are obtained by using a kernel function and it is used to account for contribution of each observation to the estimator. Generally, these kernel functions are symmetric about zero. We propose an estimator by using data dependent asymmetric kernels being generated by an unimodal symmetric kernel. To account for the contribution of the ith order statistic to the estimator, a suitable asymmetric kernel is used. Asymptotic Bias, Mean Square Error (MSE) and Mean Integrated Square Error (MISE) of the estimator are obtained with higher accuracies by using some properties of the densities of order statistics. The convergence rates depend on both n, the sample size and hn, the band width. Expressions for the optimum band width minimizing the asymptotic MISE and the rate of convergence of the optimum MISE are obtained. Simulation study and data analysis indicate that the proposed estimator not only reduces bias outside the support significantly but also it is closer to the true density function as compared to that of the usual estimator based on a common symmetric kernel. Scope for the multivariate generalization of the method is given. Some methods of generating multivariate skew symmetric kernels are described. As an illustration one of them has been used in the simulation study and estimation of density in bivariate case.

Suggested Citation

  • R. N. Rattihalli & S. B. Patil, 2021. "Data Dependent Asymmetric Kernels for Estimating the Density Function," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 155-186, February.
  • Handle: RePEc:spr:sankha:v:83:y:2021:i:1:d:10.1007_s13171-019-00171-6
    DOI: 10.1007/s13171-019-00171-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13171-019-00171-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s13171-019-00171-6?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. Mnatsakanov, Robert & Sarkisian, Khachatur, 2012. "Varying kernel density estimation on R+," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1337-1345.
    2. Jin Zhang & Xueren Wang, 2009. "Robust normal reference bandwidth for kernel density estimation," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 63(1), pages 13-23, February.
    3. Chen, Song Xi, 1999. "Beta kernel estimators for density functions," Computational Statistics & Data Analysis, Elsevier, vol. 31(2), pages 131-145, August.
    4. A. Azzalini & A.W. Bowman, 1990. "A Look at Some Data on the Old Faithful Geyser," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 39(3), pages 357-365, November.
    5. Duong, Tarn, 2007. "ks: Kernel Density Estimation and Kernel Discriminant Analysis for Multivariate Data in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 21(i07).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Mahdi Salehi & Andriette Bekker & Mohammad Arashi, 2023. "A Semi-parametric Density Estimation with Application in Clustering," Journal of Classification, Springer;The Classification Society, vol. 40(1), pages 52-78, April.

    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. Juxia Xiao & Xu Li & Jianhong Shi, 2019. "Local linear smoothers using inverse Gaussian regression," Statistical Papers, Springer, vol. 60(4), pages 1225-1253, August.
    2. Hirukawa, Masayuki, 2010. "Nonparametric multiplicative bias correction for kernel-type density estimation on the unit interval," Computational Statistics & Data Analysis, Elsevier, vol. 54(2), pages 473-495, February.
    3. Ouimet, Frédéric & Tolosana-Delgado, Raimon, 2022. "Asymptotic properties of Dirichlet kernel density estimators," Journal of Multivariate Analysis, Elsevier, vol. 187(C).
    4. Gaku Igarashi, 2018. "Multivariate Density Estimation Using a Multivariate Weighted Log-Normal Kernel," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(2), pages 247-266, August.
    5. Xu Li & Juxia Xiao & Weixing Song & Jianhong Shi, 2019. "Local linear regression with reciprocal inverse Gaussian kernel," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(6), pages 733-758, August.
    6. N. Balakrishna & Hira L. Koul, 2017. "Varying kernel marginal density estimator for a positive time series," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(3), pages 531-552, July.
    7. Senga Kiessé, Tristan & Corson, Michael S. & Eugène, Maguy, 2022. "The potential of kernel density estimation for modelling relations among dairy farm characteristics," Agricultural Systems, Elsevier, vol. 199(C).
    8. Mark S. Handcock & Adrian E. Raftery & Jeremy M. Tantrum, 2007. "Model‐based clustering for social networks," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 170(2), pages 301-354, March.
    9. Bouezmarni, Taoufik & Van Bellegem, Sébastien, 2009. "Nonparametric Beta Kernel Estimator for Long Memory Time Series," IDEI Working Papers 633, Institut d'Économie Industrielle (IDEI), Toulouse.
    10. Hagmann, M. & Scaillet, O., 2007. "Local multiplicative bias correction for asymmetric kernel density estimators," Journal of Econometrics, Elsevier, vol. 141(1), pages 213-249, November.
    11. Fernandes, Marcelo & Grammig, Joachim, 2005. "Nonparametric specification tests for conditional duration models," Journal of Econometrics, Elsevier, vol. 127(1), pages 35-68, July.
    12. Nikolay Gospodinov & Masayuki Hirukawa, 2008. "Time Series Nonparametric Regression Using Asymmetric Kernels with an Application to Estimation of Scalar Diffusion Processes," CIRJE F-Series CIRJE-F-573, CIRJE, Faculty of Economics, University of Tokyo.
    13. Salvatore D. Tomarchio & Antonio Punzo, 2019. "Modelling the loss given default distribution via a family of zero‐and‐one inflated mixture models," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 182(4), pages 1247-1266, October.
    14. Grammig, Joachim G. & Peter, Franziska J., 2008. "International price discovery in the presence of market microstructure effects," CFR Working Papers 08-10, University of Cologne, Centre for Financial Research (CFR).
    15. Fred Huffer & Cheolyong Park, 2000. "A test for multivariate structure," Journal of Applied Statistics, Taylor & Francis Journals, vol. 27(5), pages 633-650.
    16. Suqin Ge & João Macieira, 2024. "Unobserved Worker Quality and Inter‐Industry Wage Differentials," Journal of Industrial Economics, Wiley Blackwell, vol. 72(1), pages 459-515, March.
    17. Adriano Z. Zambom & Ronaldo Dias, 2013. "A Review of Kernel Density Estimation with Applications to Econometrics," International Econometric Review (IER), Econometric Research Association, vol. 5(1), pages 20-42, April.
    18. Song Li & Mervyn J. Silvapulle & Param Silvapulle & Xibin Zhang, 2015. "Bayesian Approaches to Nonparametric Estimation of Densities on the Unit Interval," Econometric Reviews, Taylor & Francis Journals, vol. 34(3), pages 394-412, March.
    19. Bauwens, Luc & Giot, Pierre & Grammig, Joachim & Veredas, David, 2004. "A comparison of financial duration models via density forecasts," International Journal of Forecasting, Elsevier, vol. 20(4), pages 589-609.
    20. Funke, Benedikt & Hirukawa, Masayuki, 2019. "Nonparametric estimation and testing on discontinuity of positive supported densities: a kernel truncation approach," Econometrics and Statistics, Elsevier, vol. 9(C), pages 156-170.

    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:spr:sankha:v:83:y:2021:i:1:d:10.1007_s13171-019-00171-6. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.