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两均值之差的序贯贝叶斯估计AI 解读

It is desired to estimate the difference between the means of two independent normal distributions as accurately as possible and in a sequential manner when the total number of observations is fixed. The problem is posed in a Bayesian framework with conjugate prior distributions and squared error loss function. It is shown that the optimal sequential design depends on the ratio of the posterior variances of the two means. There exist constants (dependent on the prior…

伯努利试验中的序贯估计AI 解读

The sequential estimation of $p$, the probability of success in a sequence of Bernoulli trials, is considered for the case where loss is taken to be symmetrized relative squared error of estimation, plus a fixed cost $c$ per observation. Using $s_n/n$ as a terminal estimator of $p$, where $s_n$ is the number of successes in $n$ trials, a heuristic rule is derived and shown to perform well for any fixed $0 < p < 1$ as $c \rightarrow 0$. However for any fixed $c > 0$, this…

尾部概率能否近似贝叶斯因子？AI 解读

Inequalities are given relating the tail area for a nested sharp hypothesis to the Bayes factor based on the event of “significance” considered as data. This Bayes factor based on an insufficient statistic is, in turn, expressed as a weighted average of full-data Bayes factors. Lindley's “statistical paradox” is generalized and other comparisons made in the normal sampling context. A new Bayesian interpretation is given for the traditional two-tailed critical level. An…

非平稳泊松过程强度函数的贝叶斯线性估计AI 解读

A Bayesian approach is taken to the problem of estimating the intensity function of a nonstationary Poisson process. The intensity function, Λ(t), − T < t < T, is a priori a second-order stochastic process. Estimators are found which minimize among, respectively, the class of step functions, the class of moving averages, and a class of linear estimators. An approximation for large T to the last estimator is found. These estimators are compared in a numerical example…

普通最小二乘替代方法的模拟研究AI 解读

Estimated regression coefficients and errors in these estimates are computed for 160 artificial data sets drawn from 160 normal linear models structured according to factorial designs. Ordinary multiple regression (OREG) is compared with 56 alternatives which pull some or all estimated regression coefficients some or all the way to zero. Substantial improvements over OREG are exhibited when collinearity effects are present, noncentrality in the original model is small, and…

摘要暂缺

尾部概率能否作为贝叶斯因子的近似？AI 解读

Inequalities are given relating the tail area for a nested sharp hypothesis to the Bayes factor based on the event of “significance” considered as data. This Bayes factor based on an insufficient statistic is, in turn, expressed as a weighted average of full-data Bayes factors. Lindley's “statistical paradox” is generalized and other comparisons made in the normal sampling context. A new Bayesian interpretation is given for the traditional two-tailed critical level. An…

非平稳泊松过程强度函数的贝叶斯线性估计AI 解读

A Bayesian approach is taken to the problem of estimating the intensity function of a nonstationary Poisson process. The intensity function, Λ(t), − T < t < T, is a priori a second-order stochastic process. Estimators are found which minimize among, respectively, the class of step functions, the class of moving averages, and a class of linear estimators. An approximation for large T to the last estimator is found. These estimators are compared in a numerical example…

摘要暂缺

论统计推断争议的解决AI 解读

Summary The controversy concerning the fundamental principles of statistics still remains unresolved. It is suggested that one key to resolving the conflict lies in recognizing that inferential probability derived from observational data is inherently noncoherent, in the sense that their inferential implications cannot be represented by a single probability distribution on the parameter space (except in the Objective Bayesian case). More precisely, for a parameter space R 1,…

最大后验概率分类AI 解读

The problem of classifying each of $n$ observations to one of two sub-populations is considered. The classification rule examined chooses that classification with maximum posterior probability. Limiting behavior of the rule is given and several examples are presented which show that the rule can lead to classifying all observations to the same subpopulation. Three simulation studies are reported to indicate that this extreme behavior may occur in small samples.

多重决策问题的经验贝叶斯方法AI 解读

In the empirical Bayes approach to multiple decision problems, we obtain theorems and lemmas which can be used to obtain asymptotic optimality and rate results in any multiple decision empirical Bayes problem. Applications of these results to a classification problem, a monotone multiple decision, and a selection problem are given. In addition, a special lemma unique to the monotone multiple decision problem gives improved (exact) rate results in that case.

最简单经验贝叶斯决策问题的完全类定理AI 解读

For the problem of empirical Bayes classification into two known probability distributions on a finite outcome space, an essentially complete class of procedures is determined. This class is proven to be minimal essentially complete if there are only two possible outcomes.

有限参数取值下的近似最优序贯检验AI 解读

Combinations of one-sided sequential probability ratio tests (SPRT's) are shown to be "nearly optimal" for problems involving a finite number of possible underlying distributions. Subject to error probability constraints, expected sample sizes (or weighted averages of them) are minimized to within $o(1)$ asymptotically. For sequential decision problems, simple explicit procedures are proposed which "do exactly what a Bayes solution would do" with probability approaching one…

贝叶斯分析下 Dempster 上下概率的异常AI 解读

Dempster (1967) calculates upper and lower probabilities for samples from a finite univariate population. These are to be used to make inferences about a second sample to be drawn from the same population. Here we show, by example, that these inferences are not compatible with a Bayesian analysis of the problem. Specifically, Example 1 shows that the upper and lower probabilities are not bounds for Bayesian posterior probabilities derived, from ‘no prior information’, and…

谱估计中的若干贝叶斯考虑AI 解读

This paper considers the problem of incorporating prior information about the shape and smoothness of a specrtal density into the formation of a spectral estimate. Two types of finite dimensional parameters are considered, the spectral ordinates at a specified collection of frequencies and the amount of power in each of a set of frequency bands. A method which is conditional on the .asymptotic distribution of periodogram averages is proposed. A formal procedure applies…

联立方程的贝叶斯全信息分析AI 解读

The paper reviews and extends a Bayesian full information analysis of the simultaneous equations model, based upon an extended natural conjugate prior density. The extended prior density belongs to a closed family and is compatible with the independent specification for each equation of a marginal prior density in the multivariate Student form. The paper establishes properties of the resulting posterior density, of some conditional and marginal densities, and of the…

已知变异系数时正态分布均值的估计AI 解读

Estimation of the mean θ of a normal distribution N(θ, aθ2) with known coefficient of variation a 1/2 is treated as a decision problem with squared-error loss. It is shown that all estimators linear in the sample mean and sample standard deviation s, as well as the maximum likelihood estimator (MLE), are dominated in risk by the admissible, minimum risk, scale equivariant estimator . A class of Bayes estimators against inverted-gamma priors is constructed, and shown to…

基于不完整观测的生存曲线非参数贝叶斯估计AI 解读

This article presents a nonparametric Bayesian estimator of a survival curve based on incomplete or arbitrarily right-censored data. This estimator, a Bayes estimator under a squared-error loss function assuming a Dirichlet process prior, is shown to be a Bayesian extension of the usual product limit (Kaplan-Meier) nonparametric estimator.

联立方程的贝叶斯全信息分析AI 解读

Abstract The paper reviews and extends a Bayesian full information analysis of the simultaneous equations model, based upon an extended natural conjugate prior density. The extended prior density belongs to a closed family and is compatible with the independent specification for each equation of a marginal prior density in the multivariate Student form. The paper establishes properties of the resulting posterior density, of some conditional and marginal densities, and of the…

预测分析中的渐近推断AI 解读

An asymptotic approach to inference problems in prediction analysis is discussed. An approximation for the Bayesian predictive distribution is derived and a correspondence with a classical method noted. A comparative example is given for the exponential model.

Consider a modified version of the empirical Bayes decision problem where the component problems in the sequence are not identical in that the sample size may vary. In this case there is not a single Bayes envelope $R(\bullet)$, but rather a sequence of envelopes $R^{m(n)}(\bullet)$ where $m(n)$ is the sample size in the $n$th problem. Let $\mathbf{\theta} = (\theta_1, \theta_2, \cdots)$ be a sequence of i.i.d. $G$ random variables and let the conditional distribution of the…

两状态分量复合决策问题的渐近解：贝叶斯方法与比例参数的弥散先验AI 解读

Gilliland and Hannan (1974, Section 3) consider a general finite state compact risk component and reduce the problem of treating the asymptotic excess compound risk of Bayes compound rules to the question of $L_1$ consistency of certain induced estimators. This present paper considers the two state case and for several classes of diffuse symmetric priors on proportions establishes the $L_1$ consistency with rate. The rate $O(n^{-\frac{1}{2}})$ uniform in state sequences is…

线性模型中的参数估计AI 解读

The first lecture in this series is devoted to a survey of contributions during the last five years to estimation of parameters by linear functions of observations in the Gauss-Markoff model. Some new results are also given. The classes of BLE (Bayes linear estimators) and ALE (admissible linear estimators) are characterized when the loss function is quadratic. It is shown that ALE's are either BLE's or limits of BLE's. Biased estimators like ridge and shrunken estimators…