Using a Markov Chain to Construct a Tractable Approximation of an Intractable Probability Distribution

Abstract.  Let π denote an intractable probability distribution that we would like to explore. Suppose that we have a positive recurrent, irreducible Markov chain that satisfies a minorization condition and has π as its invariant measure. We provide a method of using simulations from the Markov chain to construct a statistical estimate of π from which it is straightforward to sample. We show that this estimate is 'strongly consistent' in the sense that the total variation distance between the estimate and π converges to 0 almost surely as the number of simulations grows. Moreover, we use some recently developed asymptotic results to provide guidance as to how much simulation is necessary. Draws from the estimate can be used to approximate features of π or as intelligent starting values for the original Markov chain. We illustrate our methods with two examples.     

论《战国策》的"地理势能"

《战国策》作为战国典籍,在思维上具有发散特征,主张注重对形势的顺应和改造,言辞上磅礴恣肆,较之战国时代其他典籍,有着鲜明的个性特色。作者由《战国策》4种“势”的情况中的地势,引出了“地理势能”这个新概念,认为地理的分裂造成了这种潜在的蓄势力量,从而形成了战国中的外交合和力与张力,对策士的心理和行为产生了影响,并因此造就了《战国策》中的一系列特征。同时还分析了地理势能与4种“势”的关系,其中地理势能与时势、事势是并向的,与人势是对向的。     

试论尔朱氏集团的兴亡

在北魏迁都洛阳之后,代北地区的尔朱氏部族迅速发展。到孝明帝时期,代北爆发起义,尔朱荣以其训练有素的部民为核心组建了强大的军队。这支军队是北魏末年,唯一一支纪律严明,训练有素,指挥有力的军队。尔朱荣凭借其军队迅速控制了北魏朝廷,平息了反抗势力。但是,尔朱荣没有意识到洛阳官僚、汉豪族的敌意。而且,在尔朱氏集团内部形成了以贺拔岳为核心的武川集团和以高欢为核心的晋州集团。在尔朱荣被洛阳官僚刺杀后,尔朱氏家族发生内讧,被高欢一一消灭。     

论股权分置与股票市场系统风险

本文以股票市场系统风险的基本理论为基础,对我国股权分置的现状进行了分析,论述了股权分置与股票市场系统风险之间的关系,并提出了相应的对策。     

The ubiquitous angle

Summary.  Previously we used the geometry of n -dimensional space to derive the paired samples t -test and its p -value. In the present paper we describe the 'ubiquitous' application of these results to single degree of freedom linear model hypothesis tests. As examples, we derive the p - and t -values for the independent samples t -test, for testing a contrast in an analysis of variance and for testing the slope in a simple linear regression analysis. An angle θ in n -dimensional space is again pivotal in the development of the ideas. The relationships between p , t , θ , F and the correlation coefficient are also described by using a 'statistical triangle'.     

Fisher Information in Type II Progressive Censored Samples

Recently, progressively Type II censored samples have attracted attention in the study and analysis of life-testing data. Here we propose an indirect approach for computing the Fisher information (FI) in progressively Type II censored samples that simplifies the calculations. Some recurrence relations for the FI in progressively Type II censored samples are derived that facilitate the FI computation using the proposed decomposition. This paper presents a standard recurrence relation that simplifies computation of the FI in progressively Type II censored samples to a sum; FI in collections order statistics (OS). We compute the FI in a collections of progressively Type II censored samples for some known distributions.     

Bayesian credibility limits for small finite population proportions:stratified and cluster sampling

Bayes credibility limits for small proportions from stratified and fixed size cluster samples are discussed. Ericson’s (JRSS B (1969)) Beta Binomial and Dirichlet-Multinomial priors are used. Approximate limits that are appropriate for large samples and small proportions are derived in both cases. These allow asymptotic comparisons of the efficacy of stratified and cluster sampling relative to simple random sampling for estimating small proportions. Procedures for the selection of hyper parameters are also presented.     

Estimation of the location and scale parameters of the extreme value distmbution based on multiply type-II censored samples

In this paper, we consider the problem of estimating the location and scale parameters of an extreme value distribution based on multiply Type-II censored samples. We first describe the best linear unbiased estimators and the maximum likelihood estimators of these parameters. After observing that the best linear unbiased estimators need the construction of some tables for its coefficients and that the maximum likelihood estimators do not exist in an explicit algebraic form and hence need to be found by numerical methods, we develop approximate maximum likelihood estimators by appropriately approximating the likelihood equations. In addition to being simple explicit estimators, these estimators turn out to be nearly as efficient as the best linear unbiased estimators and the maximum likelihood estimators. Next, we derive the asymptotic variances and covariance of these estimators in terms of the first two single moments and the product moments of order statistics from the standard extreme value distribution. Finally, we present an example in order to illustrate all the methods of estimation of parameters discussed in this paper.     

One-Sided Estimating and Testing Problems for Scale Models from Grouped Samples

Abstract

This article mainly analyzes estimating and testing problems for scale models from grouped samples. Suppose the support region of a density function, which does not depend on parameters, is divided into some disjoint intervals, grouped samples are the number of observations falling in each intervals respectively. The studying of grouped samples may be dated back to the beginning of the century, in which only one sample location and/or scale models is considered. (Shi, N.-Z., Gao, W., Zhang, B.-X. (2001 Shi, N.-Z., Gao, W. and Zhang, B.-X. 2001. One-sided estimating and testing problems for location models from grouped samples. Comm. Statist.—Simul. Comput, 30(4): 895–898.  [Google Scholar]). One-sided estimating and testing problems for location models from grouped samples. Comm. Statist.—Simul. Comput. 30(4)) had investigated one-sided problems for location models, this article discusses one-sided estimating and testing problems for scale models. Some algorithms for obtaining the maximum likelihood estimates of the parameters subject to order restrictions are proposed.