带ARMA(1,1)条件异方差相关的随机波动模型的MCMC算法

我们首先提出了一个带ARMA(1,1)条件异方差相关的随机波动模型,它是基本的随机波动模型的一个自然的推广.进一步,对于这一新模型,我们给出了一个马尔可夫链蒙特卡罗(M CM C)算法.最后,利用该模型的模拟数据,展示了M CM C算法在这种模型中的应用.     

基于MCMC方法的金融贝叶斯半参数随机波动模型研究

现有随机波动(SV)模型依赖于参数条件分布形式假设,无法充分描述金融资产收益的偏态厚尾等典型特点,而非参数分布能够更全面地刻画这些特性。本文将SV模型和非参数分布相结合,构建一类半参数SV模型;同时在贝叶斯框架内,发展有效MCMC抽样解决模型的参数估计难问题,并利用对数预测尾部得分(LPTS)法分析模型的极端风险预测能力;最后以我国美元/人民币汇率市场为例,对半参数SV模型在收益特性刻画以及极端风险预测方面的实际效果进行了检验。     

基于MCMC方法的ARFIMA模型贝叶斯分析及实证研究

在经济领域中,时间序列具有序列相关和长记忆等特征,用考虑了时间序列短记忆性和长记忆的ARFIMA来模型分析研究经济时间序列有利于提高拟合及预测的精度。近几十年来对ARFIMA模型参数估计和分数差分算子阶数d的研究越来越多,该模型的应用也越来越广泛。基于贝叶斯方法在参数估计中的优越性,本文结合众多应用此方法的文献所得到的后验分布特点,提出了合理的先验分布,考虑到计算难度,采用MCMC方法对模型的参数进行估计,最后应用我国过去几十年的GDP数据进行实证分析,得到了ARFIMA模型参数的后验分布图、均值、方差及95%的置信区间。     

带有缺失数据的结构方程模型中的模型选择问题

结构方程模型在社会学、教育学、医学、市场营销学和行为学中有很广泛的应用。在这些领域中,缺失数据比较常见,很多学者提出了带有缺失数据的结构方程模型,并对此模型进行过很多研究。在这一类模型的应用中,模型选择非常重要,本文将一个基于贝叶斯准则的统计量,称为L_v测度,应用到此类模型中进行模型选择。最后,本文通过一个模拟研究及实例分析来说明L_v测度的有效性及应用,并在实例分析中给出了根据贝叶斯因子进行模型选择的结果,以此来进一步说明该测度的有效性。     

基于MCMC抽样的金融贝叶斯半参数GARCH模型研究

GARCH模型是研究金融资产收益的重要模型,然而现有参数GARCH模型依然不能有效刻画金融资产收益偏态厚尾特性且存在模型设定风险。本文在非参数分布和GARCH模型基础上,建立半参数GARCH模型以提高模型的有效性;同时在贝叶斯框架内发展有效MCMC抽样解决模型的参数估计难问题,并利用DIC4研究模型比较问题;最后通过模拟研究和实证研究考察MCMC抽样的有效性,检验半参数GARCH模型在刻画金融资产收益特性和风险价值预测方面的实际效果。     

小样本下两阶段MCMC参数估计方法——基于信用风险强度模型的研究

应用我国金融市场数据估计信用风险强度模型参数时,常遇到由小样本而导致的偏差问题,对此本文提出了两阶段MCMC参数估计方法:第一阶段用Lee和Mykland的跳辨识方法估计跳跃项参数;第二阶段用MC-MC方法估计扩散和漂移项参数。误差分析的结果表明两阶段MCMC方法小样本下信用风险模型参数估计的效果要明显好于单纯的MCMC方法。作为应用,采用我国第一支个人住房抵押贷款支持证券"建元2005-1"的违约和提前还款数据,估计了信用风险强度模型的参数。     

基于MCMC模拟的贝叶斯厚尾金融随机波动模型分析

针对现有金融时间序列模型建模方法难以刻画模型参数的渐变性问题,利用贝叶斯分析方法构建贝叶斯厚尾SV模型。首先对反映波动性特征的厚尾金融随机波动模型(SV-T)进行贝叶斯分析,构造了基于Gibbs抽样的MCMC数值计算过程进行仿真分析,并利用DIC准则对SV-N模型和SV-T模型进行优劣比较。研究结果表明:在模拟我国股市的波动性方面,SV-T模型比SV-N模型更优,更能反应我国股市的尖峰厚尾的特性,并且证明了我国股市具有很强的波动持续性。     

基于MCMC的随机库存系统中提前期需求分布的计算

在需求和提前期均是随机的库存系统中,提前期需求的分布是由提前期分布与需求分布复合而成的,这个复合分布的计算通常是困难的。本文采用基于Gibbs抽样的马尔科夫链蒙特卡洛(MCMC,Markov chain Monte Carlo)方法,抽取条件分布样本作为提前期需求分布的样本,通过样本来计算提前期需求分布密度、服务水平和损失函数。这种方法避免了直接求解复杂积分计算上的困难,也克服了近似分布拟合偏差过大的问题,有效地解决了随机需求与随机提前期的复杂库存系统中提前期需求确定问题。理论与数值分析结果表明:与现有方法相比较,基于MCMC的方法具有计算简便、拟合精度高、通用性好等特点。     

基于MCMC算法的时变Copula-GARCH-M-t模型及组合风险预测

为了准确地量化资产之间的时变相依结构和预测组合风险,本文考虑到投资者对资产风险偏好的差异,假设资产收益率序列的新息服从标准t分布,提出时变Copula-GARCH-M-t模型,推导了模型参数的两步MCMC估计方法,还得到了组合风险(VaR和CVaR)的一步预测方法。最后选取上证综合指数和标准普尔500指数,验证了所提模型及方法的可行性和优越性,同时该模型较为准确地量化了两指数在次贷危机后的时变相依结构特征。     

基于MCMC算法的贝叶斯变结构分位自回归模型研究

为了更全面细致的刻画时间序列变结构性的特征及其相依性,提出了一类马尔可夫变结构分位自回归模型。利用非对称Laplace分布构建了模型的似然函数,证明了当回归系数的先验分布选择为扩散先验分布时,参数的各阶后验矩都是存在的,并给出了能确定变点位置和性质的隐含变量的后验完全条件分布。仿真分析结果发现马尔可夫变结构分位自回归模型可以全面有效地实现对时间序列数据变结构性的刻画。并应用贝叶斯Markov分位自回归方法分析了中国证券市场的变结构性,结果发现中国证券市场在不同阶段尾部表现出不同的相依性。     

Structured Markov Chain Monte Carlo

Abstract

This article introduces a general method for Bayesian computing in richly parameterized models, structured Markov chain Monte Carlo (SMCMC), that is based on a blocked hybrid of the Gibbs sampling and Metropolis—Hastings algorithms. SMCMC speeds algorithm convergence by using the structure that is present in the problem to suggest an appropriate Metropolis—Hastings candidate distribution. Although the approach is easiest to describe for hierarchical normal linear models, we show that its extension to both nonnormal and nonlinear cases is straightforward. After describing the method in detail we compare its performance (in terms of run time and autocorrelation in the samples) to other existing methods, including the single-site updating Gibbs sampler available in the popular BUGS software package. Our results suggest significant improvements in convergence for many problems using SMCMC, as well as broad applicability of the method, including previously intractable hierarchical nonlinear model settings.     

Evolutionary Monte Carlo Methods for Clustering

The problem of clustering a group of observations according to some objective function (e.g., K-means clustering, variable selection) or a density (e.g., posterior from a Dirichlet process mixture model prior) can be cast in the framework of Monte Carlo sampling for cluster indicators. We propose a new method called the evolutionary Monte Carlo clustering (EMCC) algorithm, in which three new “crossover moves,” based on swapping and reshuffling sub cluster intersections, are proposed. We apply the EMCC algorithm to several clustering problems including Bernoulli clustering, biological sequence motif clustering, BIC based variable selection, and mixture of normals clustering. We compare EMCC's performance both as a sampler and as a stochastic optimizer with Gibbs sampling, “split-merge” Metropolis–Hastings algorithms, K-means clustering, and the MCLUST algorithm.     

Population Monte Carlo

Importance sampling methods can be iterated like MCMC algorithms, while being more robust against dependence and starting values. The population Monte Carlo principle consists of iterated generations of importance samples, with importance functions depending on the previously generated importance samples. The advantage over MCMC algorithms is that the scheme is unbiased at any iteration and can thus be stopped at any time, while iterations improve the performances of the importance function, thus leading to an adaptive importance sampling. We illustrate this method on a mixture example with multiscale importance functions. A second example reanalyzes the ion channel model using an importance sampling scheme based on a hidden Markov representation, and compares population Monte Carlo with a corresponding MCMC algorithm.     

基于贝叶斯MCMC算法的美式期权定价

鉴于美式期权的定价具有后向迭代搜索特征,本文结合Longstaff和Schwartz提出的美式期权定价的最小二乘模拟方法,研究基于马尔科夫链蒙特卡洛算法对回归方程系数的估计,实现对美式期权的双重模拟定价.通过对无红利美式看跌股票期权定价进行大量实证模拟,从期权价值定价误差等方面同著名的最小二乘蒙特卡洛模拟方法进行对比分析,结果表明基于MCMC回归算法给出的美式期权定价具有更高的精确度.模拟实证结果表明本文提出的对美式期权定价方法具有较好的可行性、有效性与广泛的适用性.该方法的不足之处就是类似于一般的蒙特卡洛方法,会使得求解的计算量有所加大.     

Linear Model Selection Based on Risk Estimation

The problem of selecting one model from a family of linear models to describe a normally distributed observed data vector is considered. The notion of the model of given dimension nearest to the observation vector is introduced and methods of estimating the risk associated with such a nearest model are discussed. This leads to new model selection criteria one of which, called the "partial bootstrap", seems particularly promising. The methods are illustrated by specializing to the problem of estimating the non-zero components of a parameter vector on which noisy observations are available.     

Toward Automatic Model Comparison: An Adaptive Sequential Monte Carlo Approach

Model comparison for the purposes of selection, averaging, and validation is a problem found throughout statistics. Within the Bayesian paradigm, these problems all require the calculation of the posterior probabilities of models within a particular class. Substantial progress has been made in recent years, but difficulties remain in the implementation of existing schemes. This article presents adaptive sequential Monte Carlo (SMC) sampling strategies to characterize the posterior distribution of a collection of models, as well as the parameters of those models. Both a simple product estimator and a combination of SMC and a path sampling estimator are considered and existing theoretical results are extended to include the path sampling variant. A novel approach to the automatic specification of distributions within SMC algorithms is presented and shown to outperform the state of the art in this area. The performance of the proposed strategies is demonstrated via an extensive empirical study. Comparisons with state-of-the-art algorithms show that the proposed algorithms are always competitive, and often substantially superior to alternative techniques, at equal computational cost and considerably less application-specific implementation effort. Supplementary materials for this article are available online.     

On Markov chain Monte Carlo Algorithms for Computing Conditional Expectations Based on Sufficient Statistics

Much work has focused on developing exact tests for the analysis of discrete data using log linear or logistic regression models. A parametric model is tested for a dataset by conditioning on the value of a sufficient statistic and determining the probability of obtaining another dataset as extreme or more extreme relative to the general model, where extremeness is determined by the value of a test statistic such as the chi-square or the log-likelihood ratio. Exact determination of these probabilities can be infeasible for high dimensional problems, and asymptotic approximations to them are often inaccurate when there are small data entries and/or there are many nuisance parameters. In these cases Monte Carlo methods can be used to estimate exact probabilities by randomly generating datasets (tables) that match the sufficient statistic of the original table. However, naive Monte Carlo methods produce tables that are usually far from matching the sufficient statistic. The Markov chain Monte Carlo method used in this work (the regression/attraction approach) uses attraction to concentrate the distribution around the set of tables that match the sufficient statistic, and uses regression to take advantage of information in tables that “almost” match. It is also more general than others in that it does not require the sufficient statistic to be linear, and it can be adapted to problems involving continuous variables. The method is applied to several high dimensional settings including four-way tables with a model of no four-way interaction, and a table of continuous data based on beta distributions. It is powerful enough to deal with the difficult problem of four-way tables and flexible enough to handle continuous data with a nonlinear sufficient statistic.     

Implementations of the Monte Carlo EM Algorithm

The Monte Carlo EM (MCEM) algorithm is a modification of the EM algorithm where the expectation in the E-step is computed numerically through Monte Carlo simulations. The most exible and generally applicable approach to obtaining a Monte Carlo sample in each iteration of an MCEM algorithm is through Markov chain Monte Carlo (MCMC) routines such as the Gibbs and Metropolis–Hastings samplers. Although MCMC estimation presents a tractable solution to problems where the E-step is not available in closed form, two issues arise when implementing this MCEM routine: (1) how do we minimize the computational cost in obtaining an MCMC sample? and (2) how do we choose the Monte Carlo sample size? We address the first question through an application of importance sampling whereby samples drawn during previous EM iterations are recycled rather than running an MCMC sampler each MCEM iteration. The second question is addressed through an application of regenerative simulation. We obtain approximate independent and identical samples by subsampling the generated MCMC sample during different renewal periods. Standard central limit theorems may thus be used to gauge Monte Carlo error. In particular, we apply an automated rule for increasing the Monte Carlo sample size when the Monte Carlo error overwhelms the EM estimate at any given iteration. We illustrate our MCEM algorithm through analyses of two datasets fit by generalized linear mixed models. As a part of these applications, we demonstrate the improvement in computational cost and efficiency of our routine over alternative MCEM strategies.     

基于Matlab的Poisson分布随机数的Monte carlo模拟

给出了三种随机模拟:Poisson分布随机数的方法,用Matlab7.0软件实现了对Poisson分布随机数的随机模拟,并用随机模拟的方法解决了商品销售、母鸡下蛋等实际应用问题.