IDEAS home Printed from
   My bibliography  Save this paper

Kernel Smoothed Probability Mass Functions for Ordered Datatypes


  • Jeffrey S. Racine
  • Qi Li
  • Karen X. Yan


We propose a kernel function for ordered categorical data that overcomes certain limitations present in ordered kernel functions that have appeared in the literature on the estimation of probability mass functions for multinomial ordered data. Some of these limitations arise from assumptions made about the support of the random variable that may be at odds with the data at hand. Furthermore, many existing ordered kernel functions lack a particularly appealing property, namely the ability to deliver discrete uniform probability estimates for some value of the smoothing parameter. To overcome these limitations, we propose an asymmetric empirical support kernel function that adapts to the data at hand and possesses certain desirable features. In particular, there are no difficulties arising from zero counts caused by gaps in the data while it encompasses both the empirical proportions and the discrete uniform probabilities at the lower and upper boundaries of the smoothing parameter. We propose using likelihood and least squares cross-validation for smoothing parameter selection, and study the asymptotic behaviour of these data-driven methods. We use Monte Carlo simulations to examine the finite sample performance of the proposed estimator and we also provide a simple empirical example to illustrate the usefulness of the proposed estimator in applied settings.

Suggested Citation

  • Jeffrey S. Racine & Qi Li & Karen X. Yan, 2017. "Kernel Smoothed Probability Mass Functions for Ordered Datatypes," Department of Economics Working Papers 2017-14, McMaster University.
  • Handle: RePEc:mcm:deptwp:2017-14

    Download full text from publisher

    File URL:
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    1. Peter Hall & Qi Li & Jeffrey S. Racine, 2007. "Nonparametric Estimation of Regression Functions in the Presence of Irrelevant Regressors," The Review of Economics and Statistics, MIT Press, vol. 89(4), pages 784-789, November.
    2. Hausman, Jerry & Hall, Bronwyn H & Griliches, Zvi, 1984. "Econometric Models for Count Data with an Application to the Patents-R&D Relationship," Econometrica, Econometric Society, vol. 52(4), pages 909-938, July.
    3. Peter Hall & Jeff Racine & Qi Li, 2004. "Cross-Validation and the Estimation of Conditional Probability Densities," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 1015-1026, December.
    4. Chu, Chi-Yang & Henderson, Daniel J. & Parmeter, Christopher F., 2017. "On discrete Epanechnikov kernel functions," Computational Statistics & Data Analysis, Elsevier, vol. 116(C), pages 79-105.
    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. Qi Li & Juan Lin & Jeffrey S. Racine, 2013. "Optimal Bandwidth Selection for Nonparametric Conditional Distribution and Quantile Functions," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(1), pages 57-65, January.
    2. Michael S. Delgado & Daniel J. Henderson & Christopher F. Parmeter, 2014. "Does Education Matter for Economic Growth?," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 76(3), pages 334-359, June.
    3. Jean Pierre Huiban & Camille Mastromarco & Antonio Musolesi & Michel Simioni, 2016. "The impact of pollution abatement investments on production technology: new insights from frontier analysis," Working Papers hal-01512154, HAL.
    4. Nolwenn Roudaut & Anne Vanhems, 2012. "Explaining firms efficiency in the Ivorian manufacturing sector: a robust nonparametric approach," Journal of Productivity Analysis, Springer, vol. 37(2), pages 155-169, April.
    5. Chen, Xirong & Li, Degui & Li, Qi & Li, Zheng, 2019. "Nonparametric estimation of conditional quantile functions in the presence of irrelevant covariates," Journal of Econometrics, Elsevier, vol. 212(2), pages 433-450.
    6. Phillip Heiler & Jana Mareckova, 2019. "Shrinkage for Categorical Regressors," Papers 1901.01898,
    7. Hsiao, Cheng & Li, Qi & Racine, Jeffrey S., 2007. "A consistent model specification test with mixed discrete and continuous data," Journal of Econometrics, Elsevier, vol. 140(2), pages 802-826, October.
    8. Haupt, Harry & Schnurbus, Joachim & Semmler, Willi, 2018. "Estimation of grouped, time-varying convergence in economic growth," Econometrics and Statistics, Elsevier, vol. 8(C), pages 141-158.
    9. Racine, Jeffrey S. & Li, Kevin, 2017. "Nonparametric conditional quantile estimation: A locally weighted quantile kernel approach," Journal of Econometrics, Elsevier, vol. 201(1), pages 72-94.
    10. Tingting Cheng & Jiti Gao & Xibin Zhang, 2019. "Nonparametric localized bandwidth selection for Kernel density estimation," Econometric Reviews, Taylor & Francis Journals, vol. 38(7), pages 733-762, August.
    11. Jean Pierre Huiban & Camilla Mastromarco & Antonio Musolesi & Michel Simioni, 2018. "Reconciling the Porter hypothesis with the traditional paradigm about environmental regulation: a nonparametric approach," Journal of Productivity Analysis, Springer, vol. 50(3), pages 85-100, December.
    12. Chen, Xirong & Gao, Wenzheng & Li, Zheng, 2018. "A data-driven bandwidth selection method for the smoothed maximum score estimator," Economics Letters, Elsevier, vol. 170(C), pages 24-26.
    13. repec:wyi:journl:002112 is not listed on IDEAS
    14. Zonglin He & Jean D. Opsomer, 2015. "Local polynomial regression with an ordinal covariate," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(4), pages 516-531, December.
    15. Simar, Leopold & Zelenyuk, Valentin, 2011. "To Smooth or Not to Smooth? The Case of Discrete Variables in Nonparametric Regressions," LIDAM Discussion Papers ISBA 2011042, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    16. Li, Qi & Maasoumi, Esfandiar & Racine, Jeffrey S., 2009. "A nonparametric test for equality of distributions with mixed categorical and continuous data," Journal of Econometrics, Elsevier, vol. 148(2), pages 186-200, February.
    17. Zongwu Cai & Qi Li, 2013. "Some Recent Develop- ments on Nonparametric Econometrics," Working Papers 2013-10-14, Wang Yanan Institute for Studies in Economics (WISE), Xiamen University.
    18. Jean Pierre Huiban & Camilla Mastromarco & Antonio Musolesi & Michel Simioni, 2018. "The impact of pollution abatement investments on production technology: a nonparametric approach," SEEDS Working Papers 0918, SEEDS, Sustainability Environmental Economics and Dynamics Studies, revised Sep 2018.
    19. Persson, Emma & Häggström, Jenny & Waernbaum, Ingeborg & de Luna, Xavier, 2017. "Data-driven algorithms for dimension reduction in causal inference," Computational Statistics & Data Analysis, Elsevier, vol. 105(C), pages 280-292.
    20. Hayfield, Tristen & Racine, Jeffrey S., 2008. "Nonparametric Econometrics: The np Package," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 27(i05).

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    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:mcm:deptwp:2017-14. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (). General contact details of provider: .

    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 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.

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.