Mode estimation for discrete distributions

The problem of estimating the mode of a discrete distribution is considered. New characterizations of discrete unimodal and multi-modal distributions are obtained. The proposed mode estimator is essentially the sample mode, modulo appropriate modifications when the sample mode is not well defined. In the case of i.i.d. observations coming from a unimodal discrete distribution, our proposed mode estimator is shown to possess a number of strong asymptotic properties. Many of these results extend to the case of multi-modal discrete distributions as well. Our method also applies — and we have similar asymptotic results — to the problem of mode estimation based on finitely many observations on a Markov chain whose equilibrium distribution is the underlying unimodal distribution. For unimodal discrete distributions, we also propose a consistent large sample test of mode based on the proposed statistic. Applications of mode estimation problem in Monte-Carlo optimization problem using the Hastings Metropolis chain and in prediction problem using binary response variable, specially in the context of dose-response experiments, are also illustrated.     

A bias reducing technique in kernel distribution function estimation

In this paper we suggest a bias reducing technique in kerneldistribution function estimation. In fact, it uses a convex combination of three kernel estimators, and it turned out that the bias has been reduced to the fourth power of the bandwidth, while the bias of the kernel distribution function estimator has the second power of the bandwidth. Also, the variance of the proposed estimator remains at the same order as the kernel distribution function estimator. Numerical results based on simulation studies show this phenomenon, too.     

The boundary value problem for discrete analytic functions

This paper is on further development of discrete complex analysis introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal.     

Parameter estimation accuracy in three randomized discrete distributions

Translated from Chislennye Metody Resheniya Obratnykh Zadach Matematicheskoi Fiziki, pp. 63–68, 1988.     

Multiparameter estimation for some multivariate discrete distributions with possibly dependent components

Summary In multiparameter estimation for multivariate discrete distributions with infinite support, inadmissibility problems in situations where the multivariate probability distribution function isnot a product of the one-dimensional marginal probability distribution functions have previously been unexplored. This paper examines the inadmissibility problem in some of these situations. Special attention is given to estimating the mean of a negative multinomial distribution. In estimating the mean vector, certain Clevenson-Zidek type estimators are shown to be uniformly better than the usual estimator under a large class of generally scaled squared loss functions. Some of the results are generalized to other multivariate discrete distributions and to situations where several independent negative multinomial distributions are considered.     

The bias and risk functions of some Stein-rules in elliptically contoured distributions

In this paper, we derive the bias and risk functions of a class of shrinkage estimators of several mean parameter matrices of matrix-variate elliptically contoured distributions. More specifically, we generalize some recent findings in three ways. First, the class of distributions under consideration is more general than the Gaussian distribution case, which is often studied in literature. Second, the uncertain subspace candidate is more general than that considered in literature. Finally, we generalize some recent identities, which are useful in establishing the risk and the bias of matrix shrinkage estimators.     

Fully discrete approximation methods for the estimation of parabolic systems and boundary parameters

A spatially and temporally discrete numerical approximation scheme is developed for the identification of a class of semilinear parabolic systems with unknown boundary parameters. The identification problem is formulated as a least squares fit to data subject to an equivalent representation for the dynamics in the form of an abstract evolution equation. Finite-dimensional difference equation state approximations are constructed using a cubic spline-based, Galerkin method and the Padé rational function approximations to the exponential. A sequence of approximating identification problems result, the solutions of which are shown to exist and, in a certain sense, approximate solutions to the original identification problem. Numerical results for two examples, one involving the modeling of biological mixing in deep sea sediment cores, and the other, the estimation of transport parameters for indoor mixing, are discussed. In both examples, the identification is based upon actual experimental data.Parts of the research were carried out while the authors were visitors at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, Virginia, which is operated under NASA Contracts No. NAS1-17070 and No. NAS1-17130.Research supported in part by NSF Grant MCS-8205355, AFOSR Contract 81-0198 and ARO Contract ARO-DAAG-29-K-0029.     

Minimax designs for linear regression models with bias in a reproducing kernel Hilbert space in a discrete set

Consider the design problem for estimation and extrapolation in approximately linear regression models with possible misspecification. The design space is a discrete set consisting of finitely many points, and the model bias comes from a reproducing kernel Hilbert space. Two different design criteria are proposed by applying the minimax approach for estimating the parameters of the regression response and extrapolating the regression response to points outside of the design space. A simulated annealing algorithm is applied to construct the minimax designs.These minimax designs are compared with the classical D-optimal designs and all-bias extrapolation designs. Numerical results indicate that the simulated annealing algorithm is feasible and the minimax designs are robust against bias caused by model misspecification.     

Approach regions for the square root of the Poisson kernel and boundary functions in certain Orlicz spaces

If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of L ^p and weak L ^p boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces L _Φ having the property L ^∞ ? L _Φ L ^p , 1 ? p > ∞. The second contains spaces L _Φ that resemble L ^p spaces.     

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

This work takes advantage of semiparametric modelling which improves significantly in many situations the estimation accuracy of the purely nonparametric approach. Herein for semiparametric estimations of probability mass function (pmf) of count data, and an unknown count regression function (crf), the kernel used is a binomial one and the bandiwdth selection is investigated by developing Bayesian approaches. About the latter, Bayes local and global bandwidth approaches are used to establish data-driven selection procedures in semiparametric framework. From conjugate beta prior distributions of the smoothing parameter and under the squared errors loss function, Bayes estimate for pmf is obtained in closed form. This is not available for the crf which is computed by the Markov Chain Monte Carlo technique. Simulation studies demonstrate that both proposed methods perform better than the classical cross-validation procedures, in particular the smoothing quality and execution times are optimized. All applications are made on real data sets.     

Approximation of distributions of integer-valued additive functions by discrete charges. II

V. Kapsukas Vilnius State University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 1, pp. 179–201, January–March, 1989.     

On a new system of discrete distributions and characterizations of several discrete distributions by equality of distributions

This paper deals with a new system of discrete distributions. It also gives several characterizations of the Waring (and hence the Yule) distribution (and its truncated versions), the super-Poisson, the discrete uniform and other discrete distributions by using this system and other such systems existing in the literature, and linear regression. Continuous analogues of the above results are also briefly discussed.     

Extensions of group-valued set functions

The problem of common extension ofcharges (finitely additive measures) is generalised to include group-valued functions defined on a system of sets (u-systems). To eachu-systemU an Abelian groupH(U) is attached. Every Abelian group is isomorphic to one of the formH(U). The groupH(U) is an indicator for extendability of charges fromU to the Boolean algebra generated byU. AllG-valued measures extend if and only if Ext(H(U),G)=0, for instance. Supported as van Vleck visiting professor at Wesleyan University, Connecticut in 1993. Partially supported by the Graduierten KollogTheoretische und experimentelle Methoden der reinen Mathematik of Essen University, a project No. G-0294-081.06/93 of the German-Israeli Foundation for Scientific Research & Development and by the German Academic Exchange, DAAD 1994.     

Parameter estimation by Hellinger type distance for multivariate distributions based upon probability generating functions

Maximum likelihood (ML) estimation is a popular method for parameter estimation when modeling discrete or count observations but unfortunately it may be sensitive to outliers. Alternative robust methods like minimum Hellinger distance (MHD) have been proposed for estimation. However, in the multivariate case, the MHD method leads to computer intensive estimation especially when the joint probability density function is complicated. In this paper, a Hellinger type distance measure based on the probability generating function is proposed as a tool for quick and robust parameter estimation. The proposed method yields consistent estimators, performs well for simulated and real data, and can be computationally much faster than ML or MHD estimation.