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Kernel smoothed probability mass functions for ordered datatypes

Author

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

Abstract

We propose a kernel function for ordered categorical data that overcomes limitations present in ordered kernel functions appearing in the literature on the estimation of probability mass functions for multinomial ordered data. Some limitations arise from assumptions made about the support of the underlying random variable. 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. We propose an asymmetric empirical support kernel function that adapts to the data at hand and possesses certain desirable features. 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 likelihood and least-squares cross-validation for smoothing parameter selection and study their asymptotic and finite-sample behaviour.

Suggested Citation

  • Jeffrey S. Racine & Qi Li & Karen X. Yan, 2020. "Kernel smoothed probability mass functions for ordered datatypes," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 32(3), pages 563-586, July.
    • Handle: RePEc:taf:gnstxx:v:32:y:2020:i:3:p:563-586
    DOI: 10.1080/10485252.2020.1759595
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    References listed on IDEAS

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    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.
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    Cited by:

    1. Ben O’Neill, 2022. "Smallest covering regions and highest density regions for discrete distributions," Computational Statistics, Springer, vol. 37(3), pages 1229-1254, July.
    2. Senga Kiessé, Tristan & Durrieu, Gilles, 2024. "On a discrete symmetric optimal associated kernel for estimating count data distributions," Statistics & Probability Letters, Elsevier, vol. 208(C).
    3. Pang Du & Christopher F. Parmeter & Jeffrey S. Racine, 2021. "Shape Constrained Kernel PDF and PMF Estimation," Department of Economics Working Papers 2021-05, McMaster University.

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