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倒数矩的界及其在 Stein 估计与后分层中的应用AI 解读

SUMMARY Bounds on the kth reciprocal moment of a distribution are given in terms of the mean, variance and left extremity of the distribution. A methodology for obtaining distribution-specific bounds, with improved accuracy, is outlined. Some natural approximations are noted. Applications are given in Stein estimation, and in classical and Bayesian post-stratification.

检验产出与通胀权衡中的卢卡斯假说AI 解读

A Bayesian highest-posterior interval test that accounts for heteroscedasticity and for two regimes of autocorrelation is derived to test the Lucas variance hypothesis on output-inflation trade-offs. Japanese and U.S. data show that heteroscedasticity and autocorrelation tend to influence the test results. A Bayesian procedure and Akaike's information criterion are used to determine the order or autoregressive schemes, and a join point to partition the sample is estimated by…

抽样检验中的经验贝叶斯估计AI 解读

This paper considers an empirical Bayes approach to estimating the number of defectives in a hypergeometric distribution. It involves application of an empirical Bayes identity which has proved useful in providing simple Bayes estimates for various other models. Data on the reliability of coding the answers to questions from a household survey illustrate how an empirical Bayes estimate can be computed without knowledge of the prior distribution of the number of defectives.

含噪离散时间阶梯函数的贝叶斯与经验贝叶斯估计AI 解读

Consider the problem of estimating, in a Bayesian framework and in the presence of additive Gaussian noise, a signal which is a step function. Best linear estimates and Bayes estimates are derived, evaluated and compared. A characterization of the Bayes estimates is presented. This characterization has an intuitive interpretation and also provides a way to compute the Bayes estimates with a number of operations of the order of $T^3$ where $T$ is the fixed time span. An…

高斯过程贝叶斯估计量的稳定性AI 解读

We consider the Bayes estimator $\delta_0$ for a Gaussian signal process observed in the presence of additive Gaussian noise under contamination of the signal or noise by QN-laws, introduced by Gualtierotti (1979). Upper bounds on the increase in the mean square error of $\delta_0$ over the minimum possible mean square error under contaminated noise or contaminated signal are given. It is shown that the performance of $\delta_0$ is relatively close to optimal for small…

位置—尺度回归模型的近似推断AI 解读

It is shown that χ2 and multivariate t distributions provide good approximations to posterior distributions arising in location-scale regression models. In the case of a linear regression model, Bayesian inferences based on an improper prior have conditional frequency, as well as fiducial/structural interpretation; so the results are also applicable in a non-Bayesian context. Numerical support for the approximations is presented for the case of a Weibull regression model and…

位置尺度回归模型的近似推断AI 解读

Abstract It is shown that χ2 and multivariate t distributions provide good approximations to posterior distributions arising in location-scale regression models. In the case of a linear regression model, Bayesian inferences based on an improper prior have conditional frequency, as well as fiducial/structural interpretation; so the results are also applicable in a non-Bayesian context. Numerical support for the approximations is presented for the case of a Weibull regression…

区分正态线性模型的标准贝叶斯程序之替代方法AI 解读

We consider the standard Bayesian procedure for discrimination, focusing on its tendency to give low posterior probabilities to some correct models, even in conditions of moderate to large sampling sizes. Furthermore, in some situations that could well occur in practice, the standard procedure is even asymptotically inconsistent. We claim that the explanation is that the standard procedure inflates the posterior probability of the model that has the smallest expected…

回归模型贝叶斯分析中的最小二乘近似AI 解读

SUMMARY Least-Squares approximation of the parameters in a regression model are worked out in a Bayesian framework and shown to provide, at a low computational cost, a setup to handle non-normal models and/or non-natural conjugate prior distributions. This allows one to balance an approximate solution to a reasonable model against an exact solution to a simplified model.

序贯检验维纳过程漂移符号的贝叶斯风险界AI 解读

Let $x(t)$ be a Wiener process with drift $\mu$ and variance 1 per unit time. The following problem is treated; test $H:\mu \leq 0$ vs. $A:\mu > 0$ with the loss function $|\mu|$ if the wrong decision is made and 0 otherwise, and with $c =$ cost of observation per unit time, where $\mu$ has a prior distribution which is normal with mean 0 and variance $\sigma^2_0$. An idea of Bickel and Yahav is followed to obtain a lower bound for the Bayes risk which is strict as $\sigma_0 \rightarrow \infty$…

凸集的估计AI 解读

Given independent observations $x_1, \cdots, x_n$ drawn uniformly from an unknown compact convex set $D$ in $\mathbb{R}^p$ ($p$ known) it is desired to estimate $D$ from the observations. This problem was first considered, for $p = 2$, by Ripley and Rasson (1977). We consider a decision-theoretic approach where the loss function is $L(D, \hat{D}) = m(D \Delta \hat{D})$. We prove the completeness of the Bayes estimation rules. A form for the nonrandomized Bayes estimation…

子集选择渐近一致性的一般理论及应用AI 解读

The problem of selecting a random nonempty subset from $k$ populatinos, characterized by $\theta_1, \cdots, \theta_k$ with possible nuisance parameters $\sigma$, is considered using a decision-theoretic approach. The concept of asymptotic consistency is defined as the property that the risk of a procedure at $(\theta, \sigma)$ tends to the minimum loss at $(\theta, \sigma)$. Necessary and sufficient conditions for both pointwise and uniform (on compact sets) consistency for…

贝叶斯两阶段抽样估计程序AI 解读

Bayes double sample procedures are given for estimating the mean of exponential family distributions. The distribution can be multiparameter and so the case of normal mean with unknown variance is included. Linear combination loss functions are usually assumed. Stein's double sample procedure, and other intuitive double sample procedures for estimating a binomial parameter, are studied. Recommendations are given, including guidelines for the size of the initial samples.

位置参数后验分布的决策论推导AI 解读

In this paper we develop a decision theoretic formulation for the problem of deriving posterior distributions for a parameter $\theta$, when the prior information is vague. Let $\pi(d\theta)$ be the true but unknown prior, $Q_\pi(d\theta\mid X)$ the corresponding posterior and $\delta(d\theta\mid X)$ an estimate of the posterior based on an observation $X$. The loss function is specified as a measure of distance between $Q_\pi(\cdot\mid X)$ and $\delta(\cdot\mid X)$, and the…

受限参数空间中的最优固定大小置信程序AI 解读

Optimal fixed size confidence procedures are derived for the mean of a normal random variable with known variance, when the mean is restricted to a compact interval. These confidence procedures are, in turn, based on the solution of a related minimax decision problem which is characterized by a zero-one loss function and a compact interval parameter space. The minimax rules obtained are nonrandomized, admissible, Bayes procedures. The decision-theoretic results are extended…

控制问题中估计量不可容许性的尖锐充要条件AI 解读

Suppose $\mathbf{x} = (x_1, \cdots, x_m)^t$ is an $m$-variate normal random variable with mean vector $\mathbf{\theta} = (\theta_1, \cdots, \theta_m)^t$ and identity dispersion matrix. We consider the control problem which, in canonical form, is the problem of estimating $\mathbf{\theta}$ with respect to the loss $L(\theta, \delta) = (1 - \theta^t\delta)^2,$ where $\delta(x) = (\delta_1(x), \cdots, \delta_m(x))^t$. A necessary and sufficient condition for the admissibility…

先验随阶段变化的经验贝叶斯方法AI 解读

We consider modified empirical Bayes problems in which the prior distribution of $\Theta$ at stage $n + 1$ is $G^{(n+1)}(\theta)$. The Bayes optimality criterion is now given by the sequence of functionals $R(G^{(n+1)}$. The observations $X_1, \cdots, X_n$ are no longer i.i.d so decision procedures are constructed based on modified empirical density estimates for $f_G^{(n+1)}(x)$. Asymptotic optimality together with asymptotic convergence rates is established for two action…

追踪1982年西班牙社会党胜选：贝叶斯分析AI 解读

An estimate of how people intended to vote was obtained four weeks before the Spanish general elections of October 1982. The evolution of opinion was followed during the campaign. Using a small sample of polling stations, the victory of the Spanish Socialist Party was predicted with great accuracy only two hours after the polls closed. A Bayesian hierarchical model was used.

追踪1982年西班牙社会党胜选：贝叶斯分析AI 解读

An estimate of how people intended to vote was obtained four weeks before the Spanish general elections of October 1982. The evolution of opinion was followed during the campaign. Using a small sample of polling stations, the victory of the Spanish Socialist Party was predicted with great accuracy only two hours after the polls closed. A Bayesian hierarchical model was used.

非遍历模型的渐近最优推断AI 解读

0. An Over-view.- 1. Introduction.- 2. The Classical Fisher-Rao Model for Asymptotic Inference.- 3. Generalisation of the Fisher-Rao Model to Non-ergodic Type Processes.- 4. Mixture Experiments and Conditional Inference.- 5. Non-local Results.- 1. A General Model and Its Local Approximation.- 1. Introduction.- 2. LAMN Families.- 3. Consequences of the LAMN Condition.- 4. Sufficient Conditions for the LAMN Property.- 5. Asymptotic Sufficiency.- 6. An Example (Galton-Watson…

先验信息极少时的模型选择AI 解读

A NUMBER OF AUTHORS argue that a Bayesian posterior odds criterion is appropriate for model selection.2 This paper considers how to derive this criterion when there is minimal prior information. We propose minimizing measures of prior information relative to the models in question rather than relative to the parameters of the particular models. In so doing, we obtain an expression for the odds that is invariant to the parameterization of the particular models and overcomes…

岭估计与 Stein 估计的另一种经验贝叶斯解释AI 解读

SUMMARY Ridge and Stein-like estimators have been motivated as being empirical Bayes estimates with respect to the prior β ~ N(0, τ2 V- 1) for the vector of regression coefficients β in the usual model y = Xβ + ϵ, where the hyperparameter τ2 is estimated from the data. V = I results in a ridge pattern of shrinkage while V = Xt X gives a Stein-like pattern. In this article an alternative formulation is considered, according to which the data represent a sample from a…

随机效应模型的一种新贝叶斯分析AI 解读

SUMMARY For a typical random effects model in the one way analysis of variance a new Bayesian analysis is offered. An exchangeable prior distribution is defined over the parameters. It is suggested that rather than use the usual “F” test one should consider this as an estimation problem. The posterior distribution of a function of the parameters which measures the amount of variation present is found.

贝叶斯线性回归中的异常值模型与先验分布AI 解读

SUMMARY Bayesian inference in regression models is considered using heavy-tailed error distributions to accommodate outliers. The particular class of distributions that can be constructed as scale mixtures of normal distributions are examined and use is made of them as both error models and prior distributions in Bayesian linear modelling, including simple regression and more complex hierarchical models with structured priors depending on unknown hyperprior parameters.