Célestin C. Kokonendji

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.

On Strict Arcsine Distribution

Abstract It is shown that the strict arcsine distribution is overdispersed, skewed to the right, and leptokurtic. Comparing to the negative binomial and Poisson-inverse Gaussian distributions, the strict arcsine can be seen as their alternative. Moreover, we discuss about unimodality and bimodality conditions, and Poisson mixture distribution. Application of the distribution to automobile claim frequency data is attempted and its performance is demonstrated for some data set.

Bayesian estimation of adaptive bandwidth matrices in multivariate kernel density estimation

Bandwidth selection in multivariate kernel density estimation has received considerable attention. In addition to classical methods of bandwidth selection, such as plug-in and cross-validation methods, Bayesian approaches have also been previously investigated. Bayesian estimation of adaptive bandwidth matrices in multivariate kernel density estimation is investigated, when the quadratic and entropy loss functions are used. Under the quadratic loss function, the proposed method is evaluated through a simulation study and two real data sets, which were already discussed in the literature. For these real-data applications, very interesting advantages of the proposed method are pointed out.

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.

The Lindsay transform of natural exponential families

Let gL be an infinitely divisible positive measure on R. If the measure p, is such that x-2[ p,(dx)p,({0})o0(dx)] is the L6vy measure associated with gi and is infinitely divisible, we consider for all positive reals a and [ the measure T(,p(L) which is the convolution of g~C and p . For example, if gi is the inverse Gaussian law, then p, is a gamma law with paramter 2. Then Tap(g ) is an extension of the Lindsay transform of the first order, restricted to the distributions which are infinitely divisible. The main aim of this paper is to point out that it is possible to apply this transformation to all natural exponential families (NEF) with strictly cubic variance functions P. We then obtain NEF with variance functions of the form VAiP(WA), where A is an affine function of the mean of the NEF. Some of these latter types appear scattered in the literature.

Comparing UMVU and ML estimators of the generalized variance for natural exponential families

In this paper we compare the uniformly minimum variance unbiased (UMVU) estimator and maximum likelihood (ML) estimator of the generalized variance in the context of natural exponential families (NEFs) on , d>1. We conjecture that for irreducible NEFs the proportionality holds if and only if the generalized variance has a specific form. In particular, we show that the estimators are proportional in the simple and homogeneous quadratic NEFs and prove that the UMVU estimator is preferable in terms of mean squared error except for the case of multinomial family.

Implicit Distributions and Estimation

Abstract In this article, we introduce a notion of implicit distribution for a parameter in a dominated statistical model. It is a conditional distribution of the parameter given the data. We show that the implicit distribution exists for the Jørgensen parameter in an exponential dispersion model. A method of point estimation is provided for some illustrative examples.

Estimateurs de la variance généralisée pour des familles exponentielles non gaussiennes

Within the framework of the non-Gaussian natural exponential families, we construct two UMVU estimators of the generalized variance according to whether the distribution is infinitely divisible or not. This result improves Kokonendji and Seshadri [5] and we can calculate their variance for the simple quadratic families. The multinomial and Poisson-Gaussian cases are studied.

A Bayesian Approach to Bandwidth Selection in Univariate Associate Kernel Estimation

The fundamental problem in the associate kernel estimation of density or probability mass function (pmf) is the choice of the bandwidth. In this paper, we use a Bayesian approach based upon likelihood cross-validation and a Monte Carlo Markov chain (MCMC) method for deriving the global optimal bandwidth. A comparative simulation study of the MCMC method and the classical methods that adopt the asymptotic mean integrated square error (AMISE) as criterion and the cross validation is presented for data generated from known densities and pmf, using standard AMISE and the practical integrated squared error. The simulation results show the superiority of the MCMC method over the classical methods.

Adaptive smoothing in associated kernel discrete functions estimation using Bayesian approach

This paper demonstrates that cross-validation (CV) and Bayesian adaptive bandwidth selection can be applied in the estimation of associated kernel discrete functions. This idea is originally proposed by Brewer [A Bayesian model for local smoothing in kernel density estimation, Stat. Comput. 10 (2000), pp. 299–309] to derive variable bandwidths in adaptive kernel density estimation. Our approach considers the adaptive binomial kernel estimator and treats the variable bandwidths as parameters with beta prior distribution. The best variable bandwidth selector is estimated by the posterior mean in the Bayesian sense under squared error loss. Monte Carlo simulations are conducted to examine the performance of the proposed Bayesian adaptive approach in comparison with the performance of the Asymptotic mean integrated squared error estimator and CV technique for selecting a global (fixed) bandwidth proposed in Kokonendji and Senga Kiessé [Discrete associated kernels method and extensions, Stat. Methodol. 8 (2011), pp. 497–516]. The Bayesian adaptive bandwidth estimator performs better than the global bandwidth, in particular for small and moderate sample sizes.