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贝叶斯序贯估计AI 解读

For fixed $\theta$, let $X_1, X_2, \cdots$ be a sequence of independent identically distributed random variables having density $f_\theta(x)$. Using a sequential Bayes decision theoretic approach we consider the problem of estimating any strictly monotone function $g(\theta)$ when the error incurred by a wrong estimate is measured by squared error loss and the sampling cost is $c$ units per observation. A heuristic stopping rule is suggested. It is shown that the excess risk…

大观测值下后验分布的近似行为AI 解读

Let $X$ be a real valued random variable with a family of possible distributions belonging to a one parameter exponential family with the natural parameter $\theta \in(\underline{\theta}, + \infty)$. Let $g$ be a prior probability density for $\theta$ with unbounded support. Under some additional assumptions it is shown that for large values of $x$ the posterior distribution of $\theta$ given $X = x$ is approximately normally distributed about its mode. If $\delta_g$ denotes…

置信水平、后验概率与Buehler例子AI 解读

A confidence level is sometimes treated as a probability and accordingly given substantial "confidence." And with certain statistical models a confidence level can also be an objective posterior probability (Fraser and MacKay, 1975). An instance involving such a probability with a binary error was discussed at the Symposium on Foundations of Statistical Inference, University of Waterloo, 1970 (Godambe and Sprott, 1971, page 49). An example by Buehler (ibid., page 337) of a…

稳定分布滞后的存在性AI 解读

This paper attempts to determine conditions under which distributed lag analysis is appropriate. Results indicate that lag functions are stable and linear under fairly general (but constant) objective criteria and decision constraints as long as the underlying economic environment is characterized by a stationary Gauss-Markov process, and observed environmental variables are Gaussian perturbations of that process. The results appear particularly useful for specifying the lag…

针对对称备择的正态性检验AI 解读

A tractable location and scale invariant test for normality against all symmetric alternatives is proposed. The test statistic closely approximates a combination of two traditional test statistics, and is shown to have a Bayesian interpretation. A simulation study shows that the test compares favourably with a number of other recently proposed tests.

用于线性回归控制的最优回归设计AI 解读

A Bayesian decision theory approach to the choice of regression design is considered when it is intended to use the regression to help control the dependent variable at a chosen value, the control to be effected by choosing certain of the independent variables and fixing them at selected values.

不变分布与方差分析模型AI 解读

Subjective joint distributions over the outcomes of a set of possible observations, arising from a classical analysis of variance layout, are considered. These will often incorporate invariance properties associated with the symmetries of the layout. Then we show a formal equivalence with Bayesian multistage analysis of variance models. The invariance sheds further light on the analysis of such models. The argument is general, but is illustrated fully for the one-way layout.

利用多项式进行预测的贝叶斯方法AI 解读

SUMMARY We have an unknown function h(x) which we want to estimate within a finite interval. The observed values of h(x) are independent observations of a random variable y whose mean is to be approximated by a polynomial of unknown degree. The problem of estimating h(x) then translates into that of predicting y. We assume that the mean of y is a polynomial of an arbitrarily large degree and derive a prior distribution for its coefficients which expresses the belief that…

临床试验的三阶段序贯模型AI 解读

The sequential version of Colton's decision-theoretic model for comparing two medical treatments is modified by considering both the patients who are in the experiment and also the patients who do not participate in the experiment, while the experiment is in progress. The consideration of the patients who do not participate in the experiment leads to major changes in the restdts. Both the minimax and Bayesian solutions are obtained and are compared with Colton's equivalent…

针对对称备择的正态性检验AI 解读

A tractable location and scale invariant test for normality against all symmetric alternatives is proposed. The test statistic closely approximates a combination of two traditional test statistics, and is shown to have a Bayesian interpretation. A simulation study shows that the test compares favourably with a number of other recently proposed tests.

用于控制的线性回归最优设计AI 解读

A Bayesian decision theory approach to the choice of regression design is considered when it is intended to use the regression to help control the dependent variable at a chosen value, the control to be effected by choosing certain of the independent variables and fixing them at selected values.

不变分布与方差分析模型AI 解读

Subjective joint distributions over the outcomes of a set of possible observations, arising from a classical analysis of variance layout, are considered. These will often incorporate invariance properties associated with the symmetries of the layout. Then we show a formal equivalence with Bayesian multistage analysis of variance models. The invariance sheds further light on the analysis of such models. The argument is general, but is illustrated fully for the one-way layout.

临床试验的三阶段序贯模型AI 解读

The sequential version of Colton's decision-theoretic model for comparing two medical treatments is modified by considering both the patients who are in the experiment and also the patients who do not participate in the experiment, while the experiment is in progress. The consideration of the patients who do not participate in the experiment leads to major changes in the restdts. Both the minimax and Bayesian solutions are obtained and are compared with Colton's equivalent…

使用多项式进行预测的贝叶斯方法AI 解读

We have an unknown function h(x) which we want to estimate within a finite interval. The observed values of h(x) are independent observations of a random variable y whose mean is to be approximated by a polynomial of unknown degree. The problem of estimating h(x) then translates into that of predicting y. We assume that the mean of y is a polynomial of an arbitrarily large degree and derive a prior distribution for its coefficients which expresses the belief that these…

保守统计推断AI 解读

Summary It is argued that statistical problems can usefully be divided into two classes: one-person problems for which Bayesian techniques are appropriate, and many-person problems, for which Bayesian techniques are not appropriate. Two conditions are presented which it is felt that statistical procedures for many-person problems should satisfy. Some of the consequences of these conditions are then investigated and discussed.

一致的非参数回归AI 解读

Let $(X, Y)$ be a pair of random variables such that $X$ is $\mathbb{R}^d$-valued and $Y$ is $\mathbb{R}^{d'}$-valued. Given a random sample $(X_1, Y_1), \cdots, (X_n, Y_n)$ from the distribution of $(X, Y)$, the conditional distribution $P^Y(\bullet \mid X)$ of $Y$ given $X$ can be estimated nonparametrically by $\hat{P}_n^Y(A \mid X) = \sum^n_1 W_{ni}(X)I_A(Y_i)$, where the weight function $W_n$ is of the form $W_{ni}(X) = W_{ni}(X, X_1, \cdots, X_n), 1 \leqq i \leqq n$.…

泊松过程的上、下概率推断AI 解读

A general system of inference which leads to upper and lower posterior distributions has been proposed by Dempster (1966, 1968a, 1968b). This general theory of inference is applied to various estimation problems arising in Poisson processes. In particular, the emphasis is on the estimation of the parameter of the process and on the prediction of the number of future events, given the observed data. In this article, all upper and lower probability inferences are generated by…

估计多条平稳时间序列均值函数的经验贝叶斯方法AI 解读

An empirical Bayes procedure is considered for estimating a mean value function imbedded within a collection of N stationary time series. Two cases are considered: a mean value function that is a fixed function of time and one that is a stationary normal process. Estimators for the spectral densities are derived from prior data using an extension to time series of the usual one-way analysis of variance. These estimators are inserted into approximations for the classical…

论干扰参数的消除AI 解读

Eliminating nuisance parameters from a model is universally recognized as a major problem of statistics. A surprisingly large number of elimination methods have been proposed by various writers on the topic. In this article we propose to critically review two such elimination methods. We shall be concerned with some particular cases of the marginalizing and the conditioning methods. The origin of these methods may be traced to the work of Sir Ronald A. Fisher. The contents…

内在统计推断AI 解读

For any linear model there is, according to an invariance argument, a uniquely defined prior representing ignorance, which will be called the inner prior. It is shown that the corresponding posterior probabilities are weighted averages of conditional confidence levels. This frequency interpretation compares favorably with the usual interpretation of confidence levels as unconditional probabilities of coverage. Of particular interest to users of statistical methods is that…

泊松过程的上、下概率推断AI 解读

A general system of inference which leads to upper and lower posterior distributions has been proposed by Dempster (1966, 1968a, 1968b). This general theory of inference is applied to various estimation problems arising in Poisson processes. In particular, the emphasis is on the estimation of the parameter of the process and on the prediction of the number of future events, given the observed data. In this article, all upper and lower probability inferences are generated by…

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单调似然比族中经验贝叶斯单调多重决策问题的改进收敛率AI 解读

In the context of the two-action, linear loss, exponential family multiple decision problem, Van Houwelingen (1973), (1976) has shown that faster rates of convergence are deducible for monotone empirical Bayes procedures than result from application of the bound established for general empirical Bayes procedures by Johns and Van Ryzin (1972). This note generalizes a (1973) Van Houwelingen bound to arbitrary $k$-action, monotone loss, MLR family multiple decision problems. An…

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…