On finite sample properties of nonparametric discrete asymmetric kernel estimators

Abstract

The discrete kernel method was developed to estimate count data distributions, distinguishing discrete associated kernels based on their asymptotic behaviour. This study investigates the class of discrete asymmetric kernels and their resulting non-consistent estimators, but this theoretical drawback of the estimators is balanced by some interesting features in small/medium samples. The role of modal probability and variance of discrete asymmetric kernels is highlighted to help better understand the performance of these kernels, in particular how the binomial kernel outperforms other asymmetric kernels. The performance of discrete asymmetric kernel estimators of probability mass functions is illustrated using simulations, in addition to applications to real data sets.

Trivariate Kernel Density Estimation of Spatiotemporal Crime Events with Case Study for Lithuania

“…Choice of the kernel function. A kernel function can affect the quality of the bandwidth estimation [2]. Appropriate PMF kernels, e.g., negative binomial kernels, should be used for discrete data types.…”

“…The first class includes Dirac-type symmetric kernels such as discrete triangular [24], Aitchison-Aitken [10], Wang-van Ryzin [11], and discrete Epanechnikov [25] kernel functions. The other class of kernels contains discrete asymmetric non-Dirac type kernels constructed from PDFs such as Poisson, binomial, and negative binomial [2]. A kernel estimator for count data is generally estimated as…”

“…The classical symmetric PDF estimators (e.g., Epanechnikov and Gaussian kernels) presume that the underlying data are naturally continuous, which is frequently not the case. Thus, a symmetric kernel may be not suitable for discrete bounded datasets and, instead, other types of kernels should be used [2,7,9].…”

“…For example, Aitchison and Aitken's kernel [10] can be used for unordered discrete or categorical variables; Wang and van Ryzin's kernel [11] can be used for ordered discrete variables [7]. Discrete asymmetric kernel methods have been extensively investigated by Kiessé, Kokonendji, and Somé [2,8]; they have constructed discrete kernels from known discrete PMFs such as Poisson, binomial, and negative binomial [2,12].…”

“…Kernel smoothing with continuous PDF is a well-established approach for the estimation of density and intensity of spatial surfaces based on a sampled dataset [1]. Discrete kernel estimations of PMF have been far less investigated, especially for multivariate spatiotemporal estimations [2].…”