Maximum‐norm a posteriori error bounds for a collocation method applied to a singularly perturbed reaction–diffusion problem in three dimensions

Collocation with triquadratic C¹‐splines for a singularly perturbed reaction–diffusion problem in three dimension is studied. A posteriori error bound in the maximum norm is derived for the collocation method on arbitrary tensor‐product meshes which is robust in the perturbation parameter. Numerical results are presented that support our theoretical estimate.     

A Penalized Likelihood Method for Classification With Matrix-Valued Predictors

We propose a penalized likelihood method to fit the linear discriminant analysis model when the predictor is matrix valued. We simultaneously estimate the means and the precision matrix, which we assume has a Kronecker product decomposition. Our penalties encourage pairs of response category mean matrix estimators to have equal entries and also encourage zeros in the precision matrix estimator. To compute our estimators, we use a blockwise coordinate descent algorithm. To update the optimization variables corresponding to response category mean matrices, we use an alternating minimization algorithm that takes advantage of the Kronecker structure of the precision matrix. We show that our method can outperform relevant competitors in classification, even when our modeling assumptions are violated. We analyze three real datasets to demonstrate our method’s applicability. Supplementary materials, including an R package implementing our method, are available online.     

The maximum surplus in a finite‐time interval for a discrete‐time risk model with exchangeable,dependent claim occurrences

This paper investigates a discrete‐time risk model that involves exchangeable dependent loss generating claim occurrences and compound binomially distributed aggregate loss amounts. First, a general framework is presented to derive the distribution of a surplus sequence using the model. This framework is then applied to obtain the distribution of any function of a surplus sequence in a finite‐time interval. Specifically, the distribution of the maximum surplus is obtained under nonruin conditions. Based on this distribution, the computation of the minimum surplus distribution is given. Asset and risk management–oriented implications are discussed for the obtained distributions based on numerical evaluations. In addition, comparisons are made involving the corresponding results of the classical discrete‐time compound binomial risk model, for which claim occurrences are independent and identically distributed.     

Log‐symmetric regression models: information criteria and application to movie business and industry data with economic implications

This work deals with log‐symmetric regression models, which are particularly useful when the response variable is continuous, strictly positive, and following an asymmetric distribution, with the possibility of modeling atypical observations by means of robust estimation. In these regression models, the distribution of the random errors is a member of the log‐symmetric family, which is composed by the log‐contaminated‐normal, log‐hyperbolic, log‐normal, log‐power‐exponential, log‐slash and log‐Student‐t distributions, among others. One way to select the best family member in log‐symmetric regression models is using information criteria. In this paper, we formulate log‐symmetric regression models and conduct a Monte Carlo simulation study to investigate the accuracy of popular information criteria, as Akaike, Bayesian, and Hannan‐Quinn, and their respective corrected versions to choose adequate log‐symmetric regressions models. As a business application, a movie data set assembled by authors is analyzed to compare and obtain the best possible log‐symmetric regression model for box offices. The results provide relevant information for model selection criteria in log‐symmetric regressions and for the movie industry. Economic implications of our study are discussed after the numerical illustrations.     

Statistical inference for coherent systems with Weibull distributed component lifetimes under complete and incomplete information

Point estimators for the parameters of the component lifetime distribution in coherent systems are evolved assuming to be independently and identically Weibull distributed component lifetimes. We study both complete and incomplete information under continuous monitoring of the essential component lifetimes. First, we prove that the maximum likelihood estimator (MLE) under complete information based on progressively Type‐II censored system lifetimes uniquely exists and we present two approaches to compute the estimates. Furthermore, we consider an ad hoc estimator, a max‐probability plan estimator and the MLE for the parameters under incomplete information. In order to compute the MLEs, we consider a direct maximization of the likelihood and an EM‐algorithm–type approach, respectively. In all cases, we illustrate the results by simulations of the five‐component bridge system and the 10‐component parallel system, respectively.     

Numerical Analysis of the Allen-Cahn Equation with Coarse Meshes

In this paper, we consider the finite difference semi-discretization of the Allen-Cahn equation with the diffuse interface parameter $\varepsilon$. While it is natural to make the mesh size parameter $h$ smaller than $\varepsilon$, it is desirable that $h$ is as big as possible in view of computational costs. In fact, when $h$ is bigger than $\varepsilon$ (i.e., the mesh is relatively coarse), it is observed that the numerical solution does not move at all. The purpose of this paper is to clarify the mechanism of this phenomenon. We will prove that the numerical solution converges to that of the ordinary equation without the diffusion term if $h$ is bigger than $\varepsilon$. Numerical examples are presented to support the result.     

面板数据公共变点的非参数最大似然估计

Gao Pengli;Xia Zhiming(School of Mathematics,Northwest University,Xi'an 710127,China)     

关于Tsallis型非对称熵的若干研究

基于Tsallis熵和非对称熵,本文提出了Tsallis型非对称熵,该熵推广了Tsallis熵和非对称熵,证明了最大的Tsallis型非对称熵原理,并且从该原理中可以获得比Tsallis熵及非对称熵原理更多的分布,从而说明该原理的有用性.     

离散随机变量序列的强偏差定理

利用分析方法建立了用不等式表示的用对数似然比刻划的任意相依离散随机变量序列的强偏差定理,作为推论得到了更一般的离散随机变量序列加权和的强大数定律.     

缺失数据下含几何分布的对数线性模型的EM算法

本文研究缺失数据下对数线性模型参数的极大似然估计问题.通过Monte-Carlo EM算法去拟合所提出的模型.其中,在期望步中利用Metropolis-Hastings算法产生一个缺失数据的样本,在最大化步中利用Newton-Raphson迭代使似然函数最大化.最后,利用观测数据的Fisher信息得到参数极大似然估计的渐近方差和标准误差.