Kernel Smoothed Probability Mass Functions for Ordered Datatypes

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