Extensions of discrete triangular distributions and boundary bias in kernel estimation for discrete functions

Abstract

Asymmetric discrete triangular distributions are introduced in order to extend the symmetric ones serving for discrete associated-kernels in the nonparametric estimation for discrete functions. The extension from one to two orders around the mode provides a large family of discrete distributions having a finite support. Establishing a bridge between Dirac and discrete uniform distributions, some different shapes are also obtained and their properties are investigated. In particular, the mean and variance are pointed out. Applications to discrete kernel estimators are given with solution to a boundary bias problem.

Generating discrete analogues of continuous probability distributions-A survey of methods and constructions

“…For more on general continuous triangular and two-sided power distributions see Zocchi and Kokonendji (2013) and for application of discrete triangular distribution in kernel estimation for discrete functions see Kokonendji and Zocchi (2010).…”

Bayesian estimation of bandwidth in semiparametric kernel estimation of unknown probability mass and regression functions of count data

“…These conditions are gathered to obtain good asymptotic behaviours (similar to the Dirac or naive discrete kernel) of the corresponding estimators, as developped in Kokonendji and Zocchi (2010) for the family of discrete triangular kernels. However, for small and moderate sample sizes, the class of the so-called standard discrete kernels, satisfying only (A1) and (A4)-(A5) below, works also very well:…”

On General Continuous Triangular and Two-sided Power Distributions

“…Recently, Kokonendji and Zocchi (2010) introduced the general discrete triangular (GDT) distributions which are asymmetric extensions of the symmetric family of discrete triangular distributions (Kokonendji et al, 2007). Actually, they are mainly used as discrete associated kernels in nonparametric estimation for discrete functions and they have very interesting properties.…”