Discrete triangular distributions and non-parametric estimation for probability mass function

Abstract

Discrete triangular distributions are introduced, in order to serve as kernels in the non-parametric estimation for probability mass function. They are locally symmetric around every point of estimation. Their variances depend on the smoothing bandwidth and establish a bridge between Dirac and discrete uniform distributions. The boundary bias related to the discrete triangular kernel estimator is solved through a modification of the kernel near the boundary. The mean integrated squared errors and then the optimal bandwidth are investigated. We also study the adequate bandwidth for excess zeros. The performance of the discrete triangular kernel estimator is illustrated using simulated count data. An application to count data from football is described and compared with a binomial kernel estimator.

Adaptive smoothing in associated kernel discrete functions estimation using Bayesian approach

“…, x n be an independent random sample drawn from f . The main goal is to derive the variable bandwidths for the adaptive discrete associated kernel estimator of f (·) given by Equation (2). Our approach consists of associating the variable bandwidth h i for each observation x i and treating h i as a random quantity with a prior distribution π(·).…”

“…There is also the other one family of discrete kernel, called discrete *Corresponding author. Email: nabilzougab@yahoo.fr triangular distributions which are proposed by Kokonendji et al [2] and improved recently by Kokonendji and Zocchi [3]. Similar to the kernel density estimation, the performance of discrete kernel method depends crucially on the bandwidth.…”

Bayesian bandwidth selection in discrete multivariate associated kernel estimators for probability mass functions

“…But this proposed discrete kernel is appropriate for categorical data and finite count distributions; see also Racine and Li (2004) for rf estimation with both categorical and continuous data. Therefore, recently the so-called discrete associated kernel method is largely developed by several authors; see, e.g., Kokonendji and Senga Kiessé (2011), Kokonendji, Senga Kiessé, and Balakrishnan (2009), Kokonendji et al (2007), Wansouwé, Kokonendji, and Kolyang (2015), Zougab, Adjabi, and Kokonendji (2012) and Zougab, Adjabi, and Kokonendji (2013a). Note that the semi-parametric estimation of pmf and count rf (crf) is also investigated, see for example Abdous, Kokonendji, and Senga Kiessé (2012) and Senga Kiessé, Zougab, and Kokonendji (2015).…”

“…Nonparametric estimation of probability density and probability mass functions (pdf and pmf) by kernel method is one of the most investigated topics in statistical inference; see, e.g., Aitchison and Aitken (1976), Kokonendji and Senga Kiessé (2011), Kokonendji, Senga Kiessé, and Zocchi (2007), Silverman (1986), Simonoff (1996), Wand and Jones (1995) and Wang and Ryzin (1981); see also Somé and Kokonendji (2015) for nonparametric multivariate regression function (rf) estimation by using the associated continuous and discrete kernel method. The discrete kernel estimation has been far less investigated in comparison with continuous kernel estimation.…”

Discrete triangular associated kernel and bandwidth choices in semiparametric estimation for count data

“…This paper pursues the works on estimatorf n in Equation (2) using the famous example of discrete symmetric triangular associated kernels introduced by Kokonendji et al [4]. Under some assumptions, a mathematical result on pointwise consistency off n is formulated followed by a proposition on global consistency off n using discrete triangular kernels.…”