Partition-Weighted Monte Carlo Estimation

Although various efficient and sophisticated Markov chain Monte Carlo sampling methods have been developed during the last decade, the sample mean is still a dominant in computing Bayesian posterior quantities. The sample mean is simple, but may not be efficient. The weighted sample mean is a natural generalization of the sample mean. In this paper, a new weighted sample mean is proposed by partitioning the support of posterior distribution, so that the same weight is assigned to observations that belong to the same subset in the partition. A novel application of this new weighted sample mean in computing ratios of normalizing constants and necessary theory are provided. Illustrative examples are given to demonstrate the methodology.     

Estimation of the statistical error of the direct simulation Monte Carlo method

The statistical error of the direct simulation Monte Carlo method for numerical solution of the rarefied gas dynamics problems is investigated. Based on the central limit theorem for Markov processes, asymptotic confidence intervals for the errors connected with the number of time steps are obtained for estimates of the three main macroparameters of the flow (density, velocity, and temperature). For the quantities involved in the expressions for the confidence intervals, practical recommendations are given concerning their numerical evaluation simultaneously with the calculation of the flow macroparameters. The proposed approaches to constructing the confidence intervals are illustrated using the classical problem of heat transfer between two infinite parallel plates as an example.     

Study of weighted Monte Carlo algorithms with branching

Various weighted algorithms for numerical statistical simulation are formulated and studied. The trajectory of an algorithm branches when the current weighting factor exceeds unity. As a result, the weight of an individual branch does not exceed unity and the variance of the estimate for the computed functional is finite. The unbiasedness and finiteness of the variance of estimates are analyzed using the recurrence “partial“ averaging method formulated in this study. The estimation of the particle reproduction factor and solutions to the Helmholtz equation are considered as applications. The comparative complexity of the algorithms is examined using a test problem. The variances of weighted algorithms with branching as applied to integral equations with power nonlinearity are analyzed.     

基于蒙特卡罗模拟的供应链风险估计研究

以市场需求波动风险为例,基于蒙特卡罗模拟研究了供应链风险估计问题.首先,对市场需求波动风险及其损失度量进行理论分析,利用市场需求波动风险情境下的供应链系统库存成本损失来度量市场需求波动风险的损失.其次,选择供应链末端需求为蒙特卡罗方法待模拟的随机变量,基于需求建立了市场需求波动风险概率测度模型和风险损失度量模型,确定了市场需求波动风险概率和风险损失为需求的相关量.然后,通过实例的仿真求解验证了模型.最后,给出了利用本模型方法进行供应链风险估计时需要注意的问题及进一步研究的问题.研究表明:蒙特卡罗方法对供应链风险估计具有较强的鲁棒性.     

General collision branching processes with two parameters

A new class of branching models, the general collision branching processes with two parameters, is considered in this paper. For such models, it is necessary to evaluate the absorbing probabilities and mean extinction times for both absorbing states. Regularity and uniqueness criteria are firstly established. Explicit expressions are then obtained for the extinction probability vector, the mean extinction times and the conditional mean extinction times. The explosion behavior of these models is investigated and an explicit expression for mean explosion time is established. The mean global holding time is also obtained. It is revealed that these properties are substantially different between the super-explosive and sub-explosive cases. This work was partially supported by National Natural Science Foundation of China (Grant No. 10771216), Research Grants Council of Hong Kong (Grant No. HKU 7010/06P) and Scientific Research Foundation for Returned Overseas Chinese Scholars, State Education Ministry of China (Grant No. [2007]1108)     

Benchmark Estimation for Markov chain Monte Carlo Samples

While studying various features of the posterior distribution of a vector-valued parameter using an MCMC sample, a subsample is often all that is available for analysis. The goal of benchmark estimation is to use the best available information, that is, the full MCMC sample, to improve future estimates made on the basis of the subsample. We discuss a simple approach to do this and provide a theoretical basis for the method. The methodology and benefits of benchmark estimation are illustrated using a well-known example from the literature. We obtain nearly a 90% reduction in MSE with the technique based on a 1-in-10 subsample and show that greater benefits accrue with the thinner subsamples that are often used in practice.     

A Monte Carlo Estimation of the Entropy for Markov Chains

We introduce an estimate of the entropy of the marginal density p ^t of a (eventually inhomogeneous) Markov chain at time t≥1. This estimate is based on a double Monte Carlo integration over simulated i.i.d. copies of the Markov chain, whose transition density kernel is supposed to be known. The technique is extended to compute the external entropy , where the p ₁ ^t s are the successive marginal densities of another Markov process at time t. We prove, under mild conditions, weak consistency and asymptotic normality of both estimators. The strong consistency is also obtained under stronger assumptions. These estimators can be used to study by simulation the convergence of p ^t to its stationary distribution. Potential applications for this work are presented: (1) a diagnostic by simulation of the stability property of a Markovian dynamical system with respect to various initial conditions; (2) a study of the rate in the Central Limit Theorem for i.i.d. random variables. Simulated examples are provided as illustration.      

Unbiased estimates of the Neumann series sum by the Monte Carlo method

The conditions of unbiasedness are studied for linear estimates constructed in solving linear integral equations of the second kind by the Monte Carlo method. Earlier results obtained in this field are extended, and the cases are considered in which the well-known sufficient conditions of unbiasedness are found to be also necessary conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 43, pp. 155–168, 1974.     

Smoothed Monte Carlo estimators for the time-in-the-red in risk processes

We consider a modified version of the de Finetti model in insurance risk theory in which, when surpluses become negative the company has the possibility of borrowing, and thus continue its operation. For this model we examine the problem of estimating the time-in-the red over a finite horizon via simulation. We propose a smoothed estimator based on a conditioning argument which is very simple to implement as well as particularly efficient, especially when the claim distribution is heavy tailed. We establish unbiasedness for this estimator and show that its variance is lower than the naïve estimator based on counts. Finally we present a number of simulation results showing that the smoothed estimator has variance which is often significantly lower than that of the naïve Monte-Carlo estimator.     

Discrete stochastic consistent estimators of the Monte Carlo method

The efficiency of discrete stochastic consistent estimators (the weighted uniform sampling and estimator with a correcting multiplier) of the Monte Carlo method is investigated. Confidence intervals and upper bounds on the variances are obtained, and the computational cost of the corresponding discrete stochastic numerical scheme is estimated.