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曲线型多元回归中用于预测的变量选择AI 解读

A Bayesian formulation of the problem of analysing data from a curvilinear regression of $y$ on $x_1, x_2, \cdots, x_r$ in order to predict a future value of $y$ is considered. The problem is to obtain a criterion to decide which is the best subset of $x_1, x_2, \cdots, x_r$ to perform this prediction. Under very strict assumptions the criterion obtained is shown to use the same statistic as the orthodox (least squares) approach.

期望一致性与广义贝叶斯推断AI 解读

In a previous paper, the authors introduced a new criterion of expectation consistency between probability distributions of discrete data given discrete parameter values and arbitrary posterior probability distributions for the parameter. It is here shown, under very weak assumptions, that expectation consistency implies that the posterior distributions are generalized Bayes. However when the posterior distributions are generalized Bayes, the implied prior distribution need…

参数随机分散的线性回归AI 解读

Consider a collection of individual linear regression models, in which each individual parameter vector is independently drawn from a common multivariate normal distribution and is fixed over successive observations on that individual. Maximum likelihood estimators of the mean and dispersion of the parameters and of the disturbance variance are derived. These estimators yield empirical Bayes estimators of the individual parameter veotors. The properties of the estimators are…

有先后顺序变量的多维列联表分析：修正路径分析方法AI 解读

Models and methods for analyzing the relations among a set of polytomous variables, when some of the variables are posterior to others, are presented. The techniques proposed here yield ‘path diagrams’ that are somewhat analogous to those used in path analysis. Earlier path analysis models are not suited to the case where the variables are polytomous, or dichotomous, but the models proposed herein are. For each of our models, methods are presented for (a) testing whether the…

具有先后关系的多维列联表分析：修正路径分析方法AI 解读

Models and methods for analyzing the relations among a set of polytomous variables, when some of the variables are posterior to others, are presented. The techniques proposed here yield ‘path diagrams’ that are somewhat analogous to those used in path analysis. Earlier path analysis models are not suited to the case where the variables are polytomous, or dichotomous, but the models proposed herein are. For each of our models, methods are presented for (a) testing whether the…

参数随机分散的线性回归AI 解读

Consider a collection of individual linear regression models, in which each individual parameter vector is independently drawn from a common multivariate normal distribution and is fixed over successive observations on that individual. Maximum likelihood estimators of the mean and dispersion of the parameters and of the disturbance variance are derived. These estimators yield empirical Bayes estimators of the individual parameter veotors. The properties of the estimators are…

频率学派推断理论中的若干困难AI 解读

A study of relationships between confidence regions being Bayesian, and the existence of some generalizations of Fisher's notion of relevant subsets. For a betting scheme introduced by Buehler, and for finite parameter space, it is shown that non-Bayesian procedures allow a winning strategy for a statistician's adversary. It is further shown, for finite parameter space, non-Bayesian procedures must admit conditional confidence levels bounded away from the unconditional…

指数族中参数非正与为正的贝叶斯序贯检验的渐近形状AI 解读

We show that, in a weak sense, as the cost of observation tends to zero, the shape of the continuation region of the Bayes solution for the exponential family problem given above is approximated by that of a corresponding problem for the Wiener process with drift. The approach is an extension of that used in Bickel and Yahav, Proc. Sixth Berkeley Symp. Math. Statist. Prob. (1971).

若干非参数问题的贝叶斯分析AI 解读

The Bayesian approach to statistical problems, though fruitful in many ways, has been rather unsuccessful in treating nonparametric problems. This is due primarily to the difficulty in finding workable prior distributions on the parameter space, which in nonparametric ploblems is taken to be a set of probability distributions on a given sample space. There are two desirable properties of a prior distribution for nonparametric problems. (I) The support of the prior…

摘要暂缺

十种成对多重比较程序的蒙特卡洛评价AI 解读

Computer simulation techniques were used to study the Type I and Type III error rates and the correct decision rates for ten pairwise multiple comparison procedures. Results indicated that Scheffé's test, Tukey's test, and the Student-Newman-Keuls test are less appropriate than either the least significant difference with the restriction that the analysis of variance F value be significant at α = .05, two Bayesian modifications of the least significant difference, or…

Stein 估计规则及其竞争方法：经验贝叶斯视角AI 解读

Stein's estimator for k normal means is known to dominate the MLE if k ≥ 3. In this article we ask if Stein's estimator is any good in its own right. Our answer is yes: the positive part version of Stein's estimator is one member of a class of “good” rules that have Bayesian properties and also dominate the MLE. Other members of this class are also useful in various situations. Our approach is by means of empirical Bayes ideas. In the later sections we discuss rules for more…

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摘要暂缺

节点数量未知时的贝叶斯样条回归AI 解读

Summary The problem of selecting the model and coefficients for a spline regression curve with normal errors is considered. The locations of all possible knots are assumed to be known; which subset of these is the set of actual knots is unknown. Each subset forms a model. Priors of the form of a marginal distribution on the index of the model and a conditional distribution for each model on the coefficients and variance are allowed. The posterior distribution is explicitly…

Box–Jenkins 传递函数—噪声模型的贝叶斯估计AI 解读

Summary The problem of estimating the coefficients of the transfer function-noise models proposed by Box and Jenkins is considered. A Bayesian approach is taken and the exact posterior density function of the coefficients is derived. Because of the complexity of this function, inference about individual parameters is virtually impossible except in the most simple cases. Accordingly, it is desirable to consider certain approximations. In particular, it is shown that inference…

贝叶斯推断与结构推断中的边际化悖论AI 解读

Summary We describe a range of routine statistical problems in which marginal posterior distributions derived from improper prior measures are found to have an unBayesian property—one that could not occur if proper prior measures were employed. This paradoxical possibility is shown to have several facets that can be successfully analysed in the framework of a general group structure. The results cast a shadow on the uncritical use of improper prior measures. A separate…

协方差矩阵未知时多元正态均值的广义贝叶斯极小极大估计AI 解读

Let $\mathbf{X}$ be a $p$-variate $(p \geqq 3)$ vector normally distributed with mean $\mathbf{\theta}$ and covariance matrix $\Sigma$, positive definite but unknown. Let $A$ be a $p \times p$ Wishart matrix with parameters $(n, \Sigma)$, independent of $\mathbf{X}$. To estimate $\mathbf{\theta}$ relative to quadratic loss function $(\hat{\mathbf{\theta}} - \mathbf{\theta})'\Sigma^{-1}(\hat{\mathbf{\theta}} - \mathbf{\theta})$, we obtain a family of minimax estimators $\mathbf{\delta}(\mathbf{X}, \mathbf{A})$…

多元线性回归的经验贝叶斯方法AI 解读

We consider estimation (subject to quadratic loss) of the vector of coefficients of a multiple linear regression model in which the error vector is assumed to have 0 mean and covariance matrix $\sigma^2 I$ but is not assumed to take on a specific parametric form, e.g., Normal. The vector of coefficients is taken to be randomly distributed according to some unknown prior. Restricted minimax solutions are exhibited relative to equivalence classes on the space of all prior…

维度约束下估计量的收敛AI 解读

Consider independent identically distributed observations whose distribution depends on a parameter $\theta$. Measure the distance between two parameter points $\theta_1, \theta_2$ by the Hellinger distance $h(\theta_1, \theta_2)$. Suppose that for $n$ observations there is a good but not perfect test of $\theta_0$ against $\theta_n$. Then $n^{\frac{1}{2}}h(\theta_0, \theta_n)$ stays away from zero and infinity. The usual parametric examples, regular or irregular, also have…

基于贝叶斯方法的随机差分方程多期预测AI 解读

Given n observations on a system of linear stochastic difference equations with appropriate initial conditions, and given a prior density (possibly diffuse) of its parameters, this paper obtains the predictor of the time series k periods into the future with minimum mean squared error. Completely analytical solution is given for predictions from the first-order univariate system, and, in the general higher-order multivariate case, for k up to 5. econometric equations to…

大观测值下的后验期望AI 解读

For a single observation X = x from a distribution having unknown location parameter Θ we investigate the asymptotic behaviour of the posterior distribution of Θ, as x tends to infinity. Conditions are given for this posterior distribution to approach the prior distribution, and dual conditions under which the posterior distribution is ‘ asymptotically fiducial’. The former conditions apply when the prior distribution is normal and the error has a Student distribution, the…