线性回归模型的Bayes统计推断

Bayes方法虽融合了样本信息和先验信息，但利用的先验信息都是有历史经验和专家估计所得，因此可靠度不高。该文研究了正态线性回归模型：Y=Xβ＋e，e—N（0，σ^2。L），其中σ^2已知，β为未知参数向量，对传统的Bayes方法进行了改进，即把Bayes方法中的后验信息作为改进Bayes的无验信息并融合样本信息进行统计推断，在二次损失函数下得到了β的改进的Bayes估计。由于改进的Bayes方法的先验信息中有样本信息，因此其准确度比传统的Bayes方法准确度更高。     

正态分布在不同先验下的Bayes估计及风险比较

本文讨论在均值未知,方差已知的正态分布情况下通过在共轭先验以及Jeffreys先验二种先验下的Bayes估计问题,在平方损失函数下和线性损失函数下Bayes风险的比较.数据计算可以看出,在Jeffreys先验下的Bayes风险要比在共轭先验下的Bayes风险要大,但是当样本量增大时,两者的后验风险越来越靠近.     

对数正态分布场合的BAYES分析和大样本的后验分布

本文绘出了对数正态分布场合中的两参数μ，σ2均未知时的Bayes分析，并给出相应的例子．讨论了大样本的后验分布，推广了[2]中的结果．     

限制参数空间上的Fiducial推断

给出了在限制参数空间上,利用Fiducial方法求参数的区间估计的一般方法,并且讨论了一些常见的典型问题,结果表明所得的区间估计是合理的．另外,本文还证明了在限制参数空间上,刻度族和位置族中参数的条件Fiducial分布与无信息先验的Bayes 后验分布一致,推广了Lindely的结论．     

不同先验分布下艾拉姆咖分布参数的Bayes估计

首先在定数截尾场合下,分别取共轭先验、Jeffreys先验和无信息先验,给出了艾拉姆咖分布参数的Bayes点估计和区间估计;其次用极大似然法得到超参数的估计值;然后通过随机模拟得到参数估计的均值和均方误差;最后由一个实例给出了不同截尾样本下参数的三种点估计和区间估计,并把它们进行了比较.     

位置参数分布族中Fiducial分布与后验分布的关系

本文考虑本质位置参数分布族中,参数的Fiducial分布与后验分布的等同问题.首先讨论了如何给出Fiducial分布,分析结果表明以分布函数形式给出Fiducial分布要比密度函数形式合理,同时,证明了所给的Fiducial分布具有频率性质.然后,研究在参数受到单侧限制时,Fiducial分布与后验分布等同的问题,给出的充要条件是分布族为指数分布族,此时,先验分布是一个广义先验分布,它不能被Lebesgue测度控制.最后,证明了在参数限制在一个有限区间内时,Fiducial分布与任何先验(包括广义先验分布)下的后验分布不等同.     

位置参数分布族中Fiducial分布与后验分布的关系

本文考虑本质位置参数分布族中,参数的Fiducial分布与后验分布的等同问题．首先讨论了如何给出Fiducial分布,分析结果表明以分布函数形式给出Fiducial分布要比密度函数形式合理,同时,证明了所给的Fiducial分布具有频率性质．然后,研究在参数受到单侧限制时,Fiducial分布与后验分布等同的问题,给出的充要条件是分布族为指数分布族,此时,先验分布是一个广义先验分布,它不能被Lebesgue测度控制．最后,证明了在参数限制在一个有限区间内时,Fiducial分布与任何先验(包括广义先验分布)下的后验分布不等同．     

定数截尾试验下两参数Weibull分布尺度参数的最优稳健Bayes估计

以Г-后验期望损失作为标准,研究了定数截尾试验下两参数W e ibu ll分布尺度参数θ的最优稳健Bayes估计问题.假设尺度参数θ的先验分布在分布族Г上变化,形状参数β已知时,在0-1损失下,得到了θ的最优稳健区间估计,在均方损失下得到θ的最优稳健点估计及区间估计;β未知时,得到了θ的最优稳健点估计及区间估计.最后给出了数值例子,说明了方法的有效性.     

扩散先验分布下Bayes多总体分类判别方法的构造

本文研究了各总体服从多元正态分布 ,其未知参数的先验分布均为扩散先验分布时 ,如何利用待判样品的预报密度函数、构造后验概率比并据此对样品进行分类与判别 ;此方法并不需要假设各总体分布的协方差相同 ,而且在预试样本容量较小时仍然可行。     

正态分布的共轭分布及贝叶斯估计

本文介绍含有一个或两个未知参数的正态分布Ｎ（μ，）的共轭分布，以及对正态总体的未知参数进行估计的贝叶斯方法。     

位置刻度分布族中参数的两步置信区间

寻找一些分布中的参数的具有预先给定宽度和预先给定覆盖概率的置信区间是令人感兴趣的，对于位置刻度分布族中位置参数和刻度参数，这种类型的置信区间的存在性问题已在文献中被解决，本文利用两步抽样，具体地构造出这样的固定宽度置信区间，此外，对于Cauchy分布，第一阶段的最优抽样量和一些统计量的分位点也被计算出，所得到的结果具有应用价值。     

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or the variance parameter of the normal distribution with a normal-inverse-gamma prior, we analytically calculate the Bayes posterior estimator with respect to a conjugate normal-inverse-gamma prior distribution under Stein's loss function. This estimator minimizes the Posterior Expected Stein's Loss (PESL). We also analytically calculate the Bayes posterior estimator and the PESL under the squared error loss function. The numerical simulations exemplify our theoretical studies that the PESLs do not depend on the sample, and that the Bayes posterior estimator and the PESL under the squared error loss function are unanimously larger than those under Stein's loss function. Finally, we calculate the Bayes posterior estimators and the PESLs of the monthly simple returns of the SSE Composite Index.     

Quasi-Conjugate Bayes Estimates for GPD Parameters and Application to Heavy Tails Modelling

We present a quasi-conjugate Bayes approach for estimating Generalized Pareto Distribution (GPD) parameters, distribution tails and extreme quantiles within the Peaks-Over-Threshold framework. Damsleth conjugate Bayes structure on Gamma distributions is transfered to GPD. Posterior estimates are then computed by Gibbs samplers with Hastings-Metropolis steps. Accurate Bayes credibility intervals are also defined, they provide assessment of the quality of the extreme events estimates. An empirical Bayesian method is used in this work, but the suggested approach could incorporate prior information. It is shown that the obtained quasi-conjugate Bayes estimators compare well with the GPD standard estimators when simulated and real data sets are studied. AMS 2000 Subject Classification Primary—62G32, 62F15, 62G09     

Reliability analysis for Weibull distribution with homogeneous heavily censored data based on Bayesian and least-squares methods

The reliability for Weibull distribution with homogeneous heavily censored data is analyzed in this study. The universal model of heavily censored data and existing methods, including maximum likelihood, least-squares, E-Bayesian estimation, and hierarchical Bayesian methods, are introduced. An improved method is proposed based on Bayesian inference and least-squares method. In this method, the Bayes estimations of failure probabilities are focused on for all the samples. The conjugate prior distribution of failure probability is set, and an optimization model is developed by maximizing the information entropy of prior distribution to determine the hyper-parameters. By integrating the likelihood function, the posterior distribution of failure probability is then derived to yield the Bayes estimation of failure probability. The estimations of reliability parameters are obtained by fitting distribution curve using least-squares method. The four existing methods are compared with the proposed method in terms of applicability, precision, efficiency, robustness, and simplicity. Specifically, the closed form expressions concerning E-Bayesian estimation and hierarchical Bayesian methods are derived and used. The comparisons demonstrate that the improved method is superior. Finally, three illustrative examples are presented to show the application of the proposed method.     

One-armed bandit process with a covariate

We generalize the bandit process with a covariate introduced by Woodroofe in several significant directions: a linear regression model characterizing the unknown arm, an unknown variance for regression residuals and general discounting sequence for a non-stationary model. With the Bayesian regression approach, we assume a normal-gamma conjugate prior distribution of the unknown parameters. It is shown that the optimal strategy is determined by a sequence of index values which are monotonic and determined by the observed value of the covariate and updated posterior distributions. We further show that the myopic strategy is not optimal in general. Such structural properties help to understand the tradeoff between information gathering and immediate expected payoff and may provide certain insight for covariate adjusted response adaptive design of clinical trials.     

Extensions of the conjugate prior through the Kullback-Leibler separators

The conjugate prior for the exponential family, referred to also as the natural conjugate prior, is represented in terms of the Kullback-Leibler separator. This representation permits us to extend the conjugate prior to that for a general family of sampling distributions. Further, by replacing the Kullback-Leibler separator with its dual form, we define another form of a prior, which will be called the mean conjugate prior. Various results on duality between the two conjugate priors are shown. Implications of this approach include richer families of prior distributions induced by a sampling distribution and the empirical Bayes estimation of a high-dimensional mean parameter.     

工序能力Bayes推断

本文从Ｂａｙｅｓ观点研究工序能力，对无信息先验和共轭先验，给出了Ｃｐ的后验分布、条件期望估计和最大后验估计、Ｂａｙｅｓ置信下限和判断工序是否有能力的临界值，适于对相似工序作统计推断。     

随机估计和VDR检验

众所周知统计推断有三种理论:普遍承认的Neyman理论(频率学派),Bayes推断和信仰推断(Fiducial)。Bayes推断基于后验分布,由先验分布和样本分布求得。信仰推断是基于信仰分布(Confidence Distribution,简称CD),直接利用样本求得。两者推断方式一致,都是用分布函数作推断,称为分布推断。从分析传统的参数估计、假设检验特性来看,经典统计推断也可以视为分布推断。通常将置信上限看做置信度的函数。其反函数,即置信度是置信上界的函数,恰是分布函数,该分布恰是近年来引起许多学者兴趣的CD。在本文中,基于随机化估计(其分布是一CD)的概率密度函数,提出VDR检验。常见正态分布期望或方差的检验,多元正态分布期望的Hoteling检验等是其特例。VDR(vertical density representation)检验适合于多元分布参数检验,实现了非正态的多元线性变换分布族的参数检验。VDR构造的参数的置信域有最小Lebesgue测度。