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.     

Some Necessary and Sufficient Condition of Multivariate Random Variable Satisfy Normal Distribution

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Imprecise probability models for learning multinomial distributions from data. Applications to learning credal networks

This paper considers the problem of learning multinomial distributions from a sample of independent observations. The Bayesian approach usually assumes a prior Dirichlet distribution about the probabilities of the different possible values. However, there is no consensus on the parameters of this Dirichlet distribution. Here, it will be shown that this is not a simple problem, providing examples in which different selection criteria are reasonable. To solve it the Imprecise Dirichlet Model (IDM) was introduced. But this model has important drawbacks, as the problems associated to learning from indirect observations. As an alternative approach, the Imprecise Sample Size Dirichlet Model (ISSDM) is introduced and its properties are studied. The prior distribution over the parameters of a multinomial distribution is the basis to learn Bayesian networks using Bayesian scores. Here, we will show that the ISSDM can be used to learn imprecise Bayesian networks, also called credal networks when all the distributions share a common graphical structure. Some experiments are reported on the use of the ISSDM to learn the structure of a graphical model and to build supervised classifiers.     

Shock models and the MIFRA property

Marshall and Shaked [6] have shown that some multivariate life distributions obtained from their shock model satisfy the IFRA conditions A and B of Esary and Marshall [5]. Block and Savits [2] have introduced a multivariate IFRA condition which is stronger than Conditions A and B. In this paper it is shown that the multivariate life distributions of Marshall and Shaked actually satisfy the Block-Savits MIFRA condition. As a consequence it follows that the damage processes associated with the Marshall-Shaked shock models are multivariate strongly IFRA in the sense of Block and Savits [3].     

多元秩-序模型中回归系数的估计

通过对多元秩.序模型的研究得到了模型的逆回归性质,基于该性质提出了回归系数的估计方法.当自变量满足线性条件时,不用预先设定扰动项的具体分布便可以得到回归系数方向的估计,并且这个估计与回归系数只相差一个正常数因子.证明了估计是√n相目合的.模拟结果表明估计有良好的大样本性质.     

f-Dissimilarity of Several Distributions in Testing Statistical Hypotheses

Various problems in statistics have been treated by the decision rule, based on the concept of distance between distributions. The aim of this paper is to give an approach for testing statistical hypotheses, using a general class of dissimilarity measures among k 2 distributions. The test statistics are obtained by the replacement, in the expression of the dissimilarity measure, of the unknown parameters by their maximum likelihood estimators. The asymptotic distributions of the resulting test statistics are investigated and the results are applied to multinomial and multivariate normal populations.     

On Neyman's conjecture: A characterization of the multinomials

A complete solution is provided to a problem posed by [5] and reformulated by [1] regarding a characterization of (positive and negative) multinomial distributions based, among other things, on the properties of regression in power series distributions.     

On the Existence and Continuous Dependence on Parameter of Solutions to Some Fractional Dirichlet Problem with Application to Lagrange Optimal Control Problem

In the paper, a Lagrange optimal control problem governed by a fractional Dirichlet problem with the Riemann–Liouville derivative is considered. To begin with, based on some variational method, the existence and continuous dependence of solution to the aforementioned Dirichlet problem is investigated. Then, continuous dependence is applied to show the existence of optimal solution to the Lagrange problem. An important point is that the solution to Dirichlet problem does need to be unique; therefore, the above dependence should be understood as a continuity of some multifunction—the concept of the Kuratowski–Painlevé limit of the sequence of sets is used to formulate this property.     

On solutions with power-law singularities of the homogeneous Dirichlet problem for the Laplace equation in domains with biquadratic boundaries

Properties of the Dirichlet problem for the Laplace equation in a bounded plane domain of a special type are studied in a certain class of solutions with power-law singularities. We prove that if a harmonic function is allowed to have a finite number of poles, then it can satisfy the trivial Dirichlet condition on certain curves of the studied family. The specified curves are selected, and it is shown that the set of those curves is dense (in a certain sense) in the studied family.     

Parameter estimation of a bivariate compound Poisson process

In this article, we review the concept of a Lévy copula to describe the dependence structure of a bivariate compound Poisson process. In this first statistical approach we consider a parametric model for the Lévy copula and estimate the parameters of the full dependent model based on a maximum likelihood approach. This approach ensures that the estimated model remains in the class of multivariate compound Poisson processes. A simulation study investigates the small sample behaviour of the MLEs, where we also suggest a new simulation algorithm. Finally, we apply our method to Danish fire insurance data.     

Multivariate reinsurance designs for minimizing an insurer’s capital requirement

This paper investigates optimal reinsurance strategies for an insurer with multiple lines of business under the criterion of minimizing its total capital requirement calculated based on the multivariate lower-orthant Value-at-Risk. The reinsurance is purchased by the insurer for each line of business separately. The premium principles used to compute the reinsurance premiums are allowed to differ from one line of business to another, but they all satisfy three mild conditions: distribution invariance, risk loading and preserving the convex order, which are satisfied by many popular premium principles. Our results show that an optimal strategy for the insurer is to buy a two-layer reinsurance policy for each line of business, and it reduces to be a one-layer reinsurance contract for premium principles satisfying some additional mild conditions, which are met by the expected value principle, standard deviation principle and Wang’s principle among many others. In the end of this paper, some numerical examples are presented to illustrate the effects of marginal distributions, risk dependence structure and reinsurance premium principles on the optimal layer reinsurance.     

A Multivariate CLT for Local Dependence withn log nRate and Applications to Multivariate Graph Related Statistics

This paper concerns the rate of convergence in the central limit theorem for certain local dependence structures. The main goal of the paper is to obtain estimates of the rate in the multidimensional case. Certain one-dimensional results are also improved by using some more flexible characteristics of dependence. Assuming the summands are bounded, we obtain rates close to those for independent variables. As an application we study the rate of the normal approximation of certain graph related statistics which arise in testing equality of several multivariate distributions     

On the asymptotics of numbers of observations in random regions determined by order statistics

In this paper, we consider random variables counting numbers of observations that fall into regions determined by extreme order statistics and Borel sets. We study multivariate asymptotic behavior of these random variables and express their joint limiting law in terms of independent multinomial and negative multinomial laws. First, we give our results for samples with deterministic size; next we explain how to generalize them to the case of randomly indexed samples.     

A Complete Characterization of Multivariate Normal Stable Tweedie Models through a Monge-Ampère Property

Extending normal gamma and normal inverse Gaussian models, multivariate normal stable Tweedie (NST) models are composed by a fixed univariate stable Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are real independent Gaussian variables with the same variance equal to the fixed component. Within the framework of multivariate exponential families, the NST models are recently classified by their covariance matrices V(m) depending on the mean vector m. In this paper, we prove the characterization of all the NST models through their determinants of V(m), also called generalized variance functions, which are power of only one component of m. This result is established under the NST assumptions of Monge-Ampère property and steepness. It completes the two special cases of NST, namely normal Poisson and normal gamma models. As a matter of fact, it provides explicit solutions of particular Monge-Ampère equations in differential geometry.     

A two-stage difference method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped

A novel two-stage difference method is proposed for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped. At the first stage, approximate values of the sum of the pure fourth derivatives of the desired solution are sought on a cubic grid. At the second stage, the system of difference equations approximating the Dirichlet problem is corrected by introducing the quantities determined at the first stage. The difference equations at the first and second stages are formulated using the simplest six-point averaging operator. Under the assumptions that the given boundary values are six times differentiable at the faces of the parallelepiped, those derivatives satisfy the Hölder condition, and the boundary values are continuous at the edges and their second derivatives satisfy a matching condition implied by the Laplace equation, it is proved that the difference solution to the Dirichlet problem converges uniformly as O(h ⁴lnh ^?1), where h is the mesh size.     

THE PLATEAU-B(E)ZIER PROBLEM WITH WEAK-AREA FUNCTIONAL

In this paper, we present a new method to solve the Plateau-Bézier problem. A new energy functional called weak-area functional is proposed as the objective functional to obtain the approximate minimal Bézier surface from given boundaries. This functional is constructed based on Dirichlet energy and weak isothermal parameterization condition. Experimental comparisons of the weak-area functional method with existing Dirichlet, quasi-harmonic, the strain energy-minimizing, harmonic and biharmonic masks are performed which show that the weak-area functional method are among the best by choosing appropriate parameters.     

Reducibility for Schr¨odinger Operator with Finite Smooth and Time-Quasi-periodic Potential

In this paper, the author establishes a reduction theorem for linear Schr¨odinger equation with finite smooth and time-quasi-periodic potential subject to Dirichlet boundary condition by means of KAM(Kolmogorov-Arnold-Moser) technique. Moreover, it is proved that the corresponding Schr¨odinger operator possesses the property of pure point spectra and zero Lyapunov exponent.     

A new class of multivariate counting processes and its characterization

In this paper, we suggest a new class of multivariate counting processes which generalizes and extends the multivariate generalized Polya process recently studied in Cha and Giorgio [On a class of multivariate counting processes, Adv. Appl. Probab. 48 (2016), pp. 443–462]. Initially, we define this multivariate counting process by means of mixing. For further characterization of it, we suggest an alternative definition, which facilitates convenient characterization of the proposed process. We also discuss the dependence structure of the proposed multivariate counting process and other stochastic properties such as the joint distributions of the number of events in an arbitrary interval or disjoint intervals and the conditional joint distribution of the arrival times of different types of events given the number of events. The corresponding marginal processes are also characterized.     

Identifiability of mixtures of power-series distributions and related characterizations

The concept of the identifiability of mixtures of distributions is discussed and a sufficient condition for the identifiability of the mixture of a large class of discrete distributions, namely that of the power-series distributions, is given. Specifically, by using probabilistic arguments, an elementary and shorter proof of the Lüxmann-Ellinghaus's (1987,Statist. Probab. Lett.,5, 375–378) result is obtained. Moreover, it is shown that this result is a special case of a stronger result connected with the Stieltjes moment problem. Some recent observations due to Singh and Vasudeva (1984,J. Indian Statist. Assoc.,22, 93–96) and Johnson and Kotz (1989,Ann. Inst. Statist. Math.,41, 13–17) concerning characterizations based on conditional distributions are also revealed as special cases of this latter result. Exploiting the notion of the identifiability of power-series mixtures, characterizations based on regression functions (posterior expectations) are obtained. Finally, multivariate generalizations of the preceding results have also been addressed.     

Semi‐analytical solution of MHD flow through boundary integrals on the pipe wall

A mathematical model is given for the magnetohydrodynamic (MHD) pipe flow as an inner Dirichlet problem in a 2D circular cross section of the pipe, coupled with an outer Dirichlet or Neumann magnetic problem. Inner Dirichlet problem is given as the coupled convection‐diffusion equations for the velocity and the induced current of the fluid coupling also to the outer problem, which is defined with the Laplace equation for the induced magnetic field of the exterior region with either Dirichlet or Neumann boundary condition. Unique solution of inner Dirichlet problem is obtained theoretically reducing it into two boundary integral equations defined on the boundary by using the corresponding fundamental solutions. Exterior solution is also given theoretically on the pipe wall with Poisson integral, and it is unique with Dirichlet boundary condition but exists with an additive constant obtained through coupled boundary and solvability conditions in Neumann wall condition. The collocation method is used to discretize these boundary integrals on the pipe wall. Thus, the proposed procedure is an improved theoretical analysis for combining the solution methods for the interior and exterior regions, which are consolidated numerically showing the flow behavior. The solution is simulated for several values of problem parameters, and the well‐known MHD characteristics are observed inside the pipe for increasing values of Hartmann number maintaining the continuity of induced currents on the pipe wall.