Discrete Associated Kernel Research Articles

Semiparametric and multiplicative bias correction techniques for second-order discrete kernels

In this article, we propose to estimate the probability mass function (pmf) of a discrete supported random variable by a semiparametric bias corrected method using discrete associated kernels. This method consists in applying a two-stage multiplicative bias correction (MBC) approach for the initial parametric model in order to improve the accuracy of the estimator measured in terms of the vanishing bias. Various properties of the resulting semiparametric MBC discrete associated kernel estimator are provided (bias, variance, and mean integrated squared error). The common cross-validation technique and the Kullback–Leibler divergence are adapted for bandwidth selection. Monte Carlo simulations and a real-data application for count data illustrate the performance of the semiparametric-MBC estimator.

On Underdispersed Count Kernels for Smoothing Probability Mass Functions

Only a few count smoothers are available for the widespread use of discrete associated kernel estimators, and their constructions lack systematic approaches. This paper proposes the mean dispersion technique for building count kernels. It is only applicable to count distributions that exhibit the underdispersion property, which ensures the convergence of the corresponding estimators. In addition to the well-known binomial and recent CoM-Poisson kernels, we introduce two new ones such the double Poisson and gamma-count kernels. Despite the challenging problem of obtaining explicit expressions, these kernels effectively smooth densities. Their good performances are pointed out from both numerical and comparative analyses, particularly for small and moderate sample sizes. The optimal tuning parameter is here investigated by integrated squared errors. Also, the added advantage of faster computation times is really very interesting. Thus, the overall accuracy of two newly suggested kernels appears to be between the two old ones. Finally, an application including a tail probability estimation on a real count data and some concluding remarks are given.

Bayesian bandwidth selection with two multiplicative bias correction estimators for discrete kernel

–Two classes of multiplicative bias correction (MBC) methods with discrete associated kernels are used for probability mass function estimation. The MBC estimators reduce the order of magnitude in bias, whereas the order of magnitude in variance remains unchanged. For the choice of the bandwidth, we proposed the Bayes global method against the unbiased cross-validation approach. The performance of both methods (Bayesian and cross validation methods) are evaluated under two criteria: the integrated square error criteria and the integrated square bias through simulation studies. We also present a real data application of the proposed methods. The obtained results show that the Bayes global method performs better than cross-validation one for MBC estimators.

On finite sample properties of nonparametric discrete asymmetric kernel estimators

ABSTRACTThe 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.

On bandwidth parameter choices for discrete nonparametric kernel estimator

This note concentrates on the nonparametric estimation of a probability mass function (p.m.f.) using discrete associated kernels. An expression of the optimal bandwidth minimizing the asymptotic part of the global squared error is given. Some asymptotic expressions of bias and variance of the cross-validation criterion are also presented. At last, the two bandwidth selection procedures are illustrated through some simulations and an application on a real count data set.

Bayesian local bandwidth selector in multivariate associated kernel estimator for joint probability mass functions

ABSTRACTThis work treats non-parametric estimation of multivariate probability mass functions, using multivariate discrete associated kernels. We propose a Bayesian local approach to select the matrix of bandwidths considering the multivariate Dirac Discrete Uniform and the product of binomial kernels, and treating the bandwidths as a diagonal matrix of parameters with some prior distribution. The performances of this approach and the cross-validation method are compared using simulations and real count data sets. The obtained results show that the Bayes local method performs better than cross-validation in terms of integrated squared error.

Nonparametric estimation for probability mass function with Disake: an R package for discrete associated kernel estimators

Kernel smoothing is one of the most widely used nonparametric data smoothing techniques. We introduce a new R package, Disake, for computing discrete associated kernel estimators for probability mass function. When working with a kernel estimator, two choices must be made: the kernel function and the smoothing parameter. The Disake package focuses on discrete associated kernels and also on cross-validation and local Bayesian techniques to select the appropriate bandwidth. Applications on simulated data and real data show that the binomial kernel is appropriate for small or moderate count data while the empirical estimator or the discrete triangular kernel is indicated for large samples.

Performance of discrete associated kernel estimators through the total variation distance

We prove asymptotic results and concentration inequalities for a large class of discrete associated kernel estimators, under the total variation distance. We also propose a data driven bandwidth selection procedure aiming to minimize the total variation. Simulations are conducted.

Discrete Nonparametric Kernel and Parametric Methods for the Modeling of Pavement Deterioration

This article is concerned with one discrete nonparametric kernel and two parametric regression approaches for providing the evolution law of pavement deterioration. The first parametric approach is a survival data analysis method; and the second is a nonlinear mixed-effects model. The nonparametric approach consists of a regression estimator using the discrete associated kernels. Some asymptotic properties of the discrete nonparametric kernel estimator are shown as, in particular, its almost sure consistency. Moreover, two data-driven bandwidth selection methods are also given, with a new theoretical explicit expression of optimal bandwidth provided for this nonparametric estimator. A comparative simulation study is realized with an application of bootstrap methods to a measure of statistical accuracy.

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

This work deals with semiparametric kernel estimator of probability mass functions which are assumed to be modified Poisson distributions. This semiparametric approach is based on discrete associated kernel method appropriated for modelling count data; in particular, the famous discrete symmetric triangular kernels are used. Two data-driven bandwidth selection procedures are investigated and an explicit expression of optimal bandwidth not available until now is provided. Moreover, some asymptotic properties of the cross-validation criterion adapted for discrete semiparametric kernel estimation are studied. Finally, to measure the performance of semiparametric estimator according to each type of bandwidth parameter, some applications are realized on three real count data-sets from sociology and biology.

Binomial kernel and Bayes local bandwidth in discrete function estimation

The Bayesian approach to bandwidth selection in discrete associated kernel estimation of probability mass function is a very good alternative to the classical popular methods such as the methods which adopt the asymptotic mean integrated squared error as a criterion and the cross-validation technique. In this paper, we propose a Bayesian local approach to bandwidth selection considering the binomial kernel estimator and locally treating the bandwidth h as a random quantity with a prior distribution. The local bandwidth is estimated by the posterior mean of h. The performance of this proposed approach and that of the classical methods are compared using simulations of data generated from known discrete functions. The new method is then applied to a real count data set. The smoothing quality of the Bayes estimator is very satisfactory.

On semiparametric regression for count explanatory variables

This work deals with a semiparametric estimation of a count regression function m that can be represented as a product of an unknown discrete parametric function r and an unknown discrete “smooth” function ω. We propose an estimation procedure in two steps: first, we construct an approximation r^ of r, then we use a discrete associated kernel method to estimate nonparametrically the multiplicative correction factor ω=m/r^. The asymptotic and small-sample properties of the proposed estimator are investigated. Its comparison with the classical Nadaraya–Watson type count regression estimator shows that an improvement in terms of bias is achieved.

Discrete associated kernels method and extensions

Discrete kernel estimation of a probability mass function (p.m.f.), often mentioned in the literature, has been far less investigated in comparison with continuous kernel estimation of a probability density function (p.d.f.). In this paper, we are concerned with a general methodology of discrete kernels for smoothing a p.m.f. f . We give a basic of mathematical tools for further investigations. First, we point out a generalizable notion of discrete associated kernel which is defined at each point of the support of f and built from any parametric discrete probability distribution. Then, some properties of the corresponding estimators are shown, in particular pointwise and global (asymptotical) properties. Other discrete kernels are constructed from usual discrete probability distributions such as Poisson, binomial and negative binomial. For small samples sizes, underdispersed discrete kernel estimators are more interesting than the empirical estimator; thus, an importance of discrete kernels is illustrated. The choice of smoothing bandwidth is classically investigated according to cross-validation and, novelly, to excess of zeros methods. Finally, a unification way of this method concerning the general probability function is discussed.

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

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 a solution to a boundary bias problem.