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新观测属于两个多元正态总体之一的概率可信区间AI 解读

Summary The moments of the logarithm of the likelihood ratio of a new observation, drawn from one of two multivariate normal populations, are obtained from the posterior moment generating function. The distribution of the logarithm of the likelihood ratio is approximated using Pearson curves. The distribution of the probability p that a new observation belongs to one of the two populations can be obtained by transformation. The relationship between the expected value of this…

不完整数据问题中的近似后验分布AI 解读

Summary We consider the problem of developing a simple approximation to a posterior distribution arising from incomplete data sampling. We compare approximations based on the normal distribution and upon conjugate distributions, theoretically where feasible and in some numerical examples. Two general conclusions can be made on the basis of our numerical work. First, when there is missing data, approximations which fail to take loss of information into account give overly…

有限总体抽样中的平衡样本与稳健贝叶斯推断AI 解读

Bayesian inference in finite populations uses probability models at two stages: (i) to describe relationships among population units and (ii) to express uncertainty concerning the values of parameters appearing at stage (i). Here we consider the Bayes posterior distribution of the population total when a multivariate nornxal regression model is used at stage (i), with a diffuse prior distribution on the regression coefficients. We study the situation where the stage (i)…

不连续决策过程与门限自回归时间序列建模AI 解读

Setting the problem of approximating an underlying nonlinear time series model within the framework of Bayesian decision theory, we demonstrate how the general analysis of discontinuous decision processes developed by Smith, Harrison & Zeeman (1981) leads naturally to a threshold autoregression.

贝叶斯经验贝叶斯AI 解读

A Bayesian approach is given for various kinds of empirical Bayes problems. In particular it is shown that empirical Bayes procedures are really non-Bayesian, asymptotically optimal, classical procedures for mixtures. In some situations these procedures are Bayes with respect to some prior and in other situations, there is no prior for which they are Bayes. Several examples of these concepts are given as well as a general theory showing the difference between an empirical…

删失生存数据下两个指数分布的序贯比较AI 解读

A sequential stopping procedure is proposed for a clinical trial which compares two exponential distributions using censored survival data. The procedure extends an approach first developed by Anscombe (1963) for comparing two treatments with normally distributed and instantaneously observed responses. Patients are assumed to arrive at random and are assigned alternately to one of the two treatments. An optimal stopping region for the ratio of failure rates of the two…

非参数判别的渐近误分类概率AI 解读

Let $(X, Y), (X_1, Y_1), \cdots, (X_n, Y_n)$ be independent identically distributed random vectors from $R^d \times \{0, 1\}$, and let $\hat{Y}$ be the $k$-nearest neighbor estimate of $Y$ from $X$ and the $(X_i, Y_i)$'s. We show that for all distributions of $(X, Y)$, the limit of $L_n = P(\hat{Y} \neq Y)$ exists and satisfies $\lim_{n\rightarrow\infty} L_n \leq (1 + a_k)R^\ast, a_k = \frac{\alpha \sqrt k}{k - 3.25}\big(1 + \frac{\beta}{\sqrt{k-3}}\big), k \text{odd}, k \geq 5,$…

非参数回归函数估计的几乎处处收敛AI 解读

Let $(X, Y), (X_1, Y_1), \cdots, (X_n, Y_n)$ be independent identically distributed random vectors from $R^d \times R$, and let $E(|Y|^p) < \infty$ for some $p \geq 1$. We wish to estimate the regression function $m(x) = E(Y \mid X = x)$ by $m_n(x)$, a function of $x$ and $(X_1, Y_1), \cdots, (X_n, Y_n)$. For large classes of kernel estimates and nearest neighbor estimates, sufficient conditions are given for $E\{|m_n(x) - m(x)|^p\} \rightarrow 0$ as $n \rightarrow \infty$,…

有限样本空间统计问题的完全类定理AI 解读

This paper contains a complete class theorem (Theorem 3.2) which applies to most statistical estimation problems having a finite sample space. This theorem also applies to many other statistical problems with finite sample spaces. The description of this complete class involves a stepwise algorithm. At each step of the process it is necessary to construct the Bayes procedures in a suitably modified version of the original problem. The complete class is a minimal complete…

论贝叶斯稳态预测模型AI 解读

Summary A recent criterion of Smith (1979) seeks to generalize the steady forecasting model of Harrison and Stevens (1976) to take account of non-normality in the observation and system variables. In this note, predictive distributions and forecasts corresponding to quadratic and step loss functions are examined when the observation variable has a distribution belonging to the exponential family. It is shown that Smith's generalized model does not always yield forecasts…

生存时间数据的经验贝叶斯分析AI 解读

Summary An empirical Bayes analysis of life-time data is described. The paper examines and extends the model discussed by Kalbfleisch (1978) in which the cumulative hazard function is treated as a gamma process. It is shown that the results obtained from the model by using grouped data (the usual case in practice) are more satisfactory than those obtained using continuous data. The paper also shows that a parameter representing the variability of the underlying gamma process…

比较两个正态总体均值的渐近贝叶斯序贯检验AI 解读

Summary We consider sequential tests to decide whether the difference between the means of two normal populations is negative or positive. A Bayesian formulation is adopted, with costs of sampling which differ for the two populations and depend on the difference between the means. The asymptotic “shape”, in the sense of Schwarz (1962), of the Bayes continuation region is found, and this is used to derive an asymptotically Bayes test. A number of asymptotic optimality…

最大似然估计与贝叶斯估计的一致性AI 解读

It is proved in the paper that every possible set of conditions which implies consistency of the maximum likelihood method also implies consistency of Bayes estimates for a large class of prior distributions. The method of the proof is based on a necessary and sufficient condition for the consistency of maximum likelihood estimates which is due to Perlman (1972). Perlman's result is improved as far as certain questions of measurability are concerned.

We demonstrate the admissible Bayes character of two residual error criteria for subset selection of independent variables in normal multivariate regression models. In particular, suppose a linear model includes the independent variable list $X$, and suppose all additional independent variable subsets of size $s$ (fixed) from list $Z$ are under consideration for inclusion in the model. Let $\mathbf{\Sigma}$ be the regression error covariance matrix and $\hat{\mathbf{\Sigma}}(\hat{\mathbf{\Sigma}}_\mathscr{J})$…

通过潜变量估计实现多指标多原因模型的 EM 方法AI 解读

This article considers both likelihood and Bayesian estimation procedures for a model with multiple indicators and multiple causes of a single unobservable latent variable. The model is complicated by its reduced form, which displays a mixture of econometric and psychometric themes. We avoid this complexity via the estimation of the unobservable latent variable through direct use of the original structural form. This approach not only provides maximum likelihood estimates…

整群抽样数据的回归分析AI 解读

For the case of different regression relationships in different subgroups of a finite population, only part of which are sampled, the extended least squares estimator of any weighted average of the distinct coefficients is derived, under assumptions relating only to the first two moments of the distribution of the coefficients. Under these assumptions, the estimator is shown to be the best linear ξ-unbiased estimator, while under further distributional assumptions, it is…

线性纯死亡过程的初始规模估计AI 解读

Consider a linear pure death process in which the common death rate and the initial size n of the population are both unknown. An estimate of n is required, based on the observed times of the first j deaths, i.e. with censored sampling. It is shown that the only reliable non-Bayesian method is to use interval estimates, for all the standard point estimation techniques are liable to fail. Appropriate interval estimators are derived and discussed. The corresponding truncated…

利用秩计算变点的后验概率AI 解读

Simple approximate methods of analysis are given for change-point models using ranked observations. The models can be applied to continuous observations, providing a robust analysis in this case. For ranked observations, the models provide analyses for association. Numerical examples, involving continuous and ranked observations, are given.

平均寿命贝叶斯估计量的预后验分析AI 解读

In considering life test experiments from a behaviouristic Bayes point of view, one may speculate on the effect of model misspecification on the posterior mean under alternative stopping rulers. Of coures, under the life distribution model, we expect the mean of the posterior to equal the mean of the prior. However, if we think the model may have been misspecified, we would no longer expect this equality. It is shown that for Bayes estimators of mean life under the…

有界正态均值的估计AI 解读

The problem of estimating a normal mean has received much attention in recent years. If one assumes, however, that the true mean lies in a bounded interval, the problem changes drastically. In this paper we show that if the interval is small (approximately two standard deviations wide) then the Bayes rule against a two point prior is the unique minimax estimator under squared error loss. For somewhat wider intervals we also derive sufficient conditions for minimaxity of the…

二元剂量响应数据下狄利克雷过程的后验分布AI 解读

The posterior distribution of a Dirichlet process from $N$ quantal responses at $r$ dosage levels has been recognized by Antoniak as a mixture of Dirichlet processes. The purpose of this paper is to develop a systematic procedure for computing finite-dimensional distributions of such mixtures which can be equivalently expressed as multivariate beta distributions with random parameter vectors. It is shown that the sequence of random parameter vectors of the updated beta…

内在统计推断（二）AI 解读

According to an invariance principle, for some models having a certain group structure, there is a uniquely defined prior representing ignorance, which is called the inner prior. It is shown that the corresponding posterior probability of a likelihood region has a simple frequency interpretation as a mean conditional confidence level. The central multivariate normal model is considered as an example.

协方差分量模型中的估计AI 解读

Estimation techniques for linear covariance components models are developed and illustrated with special emphasis on explaining computational processes. The estimation of fixed and random effects when the variances and covariances are known is presented in Bayesian terms, Point estimates of the unknown variances and covariances are computed using the EM algorithm for maximum likelihood estimation from incomplete data. The techniques are illustrated with data on law schools,…

准确且简约的正态线性回归模型的可容许选择AI 解读

Let $M_0$ be a normal linear regression model and let $M_1,\cdots, M_K$ be distinct proper linear submodels of $M_0$. Let $\hat k \in \{0,\cdots, K\}$ be a model selection rule based on observed data from the true model. Given $\hat k$, let the unknown parameters of the selected model $M_{\hat k}$ be fitted by the maximum likelihood method. A loss function is introduced which depends additively on two parts: (i) a measure of the difference between the fitted model $M_{\hat k}$…