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Kernel Smoothed Probability Mass Functions for Ordered Datatypes

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

Abstract

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
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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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