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标准误的非参数估计：刀切法、自助法及其他方法AI 解读

We discuss several nonparametric methods for attaching a standard error to a point estimate: the jackknife, the bootstrap, half-sampling, subsampling, balanced repeated replications, the infinitesimal jackknife, influence function techniques and the delta method. The last three methods are shown to be identical. All the methods derive from the same basic idea, which is also the idea underlying the common parametric methods. Extended numerical comparisons are made for the…

最大似然估计精度的评估：观测与期望Fisher信息AI 解读

This paper concerns normal approximations to the distribution of the maximum likelihood estimator in one-parameter families. The traditional variance approximation is 1/§, where θ is the maximum likelihood estimator and § is the expected total Fisher information. Many writers, including R. A. Fisher, have argued in favour of the variance estimate 1/I(x), where I(x) is the observed information, i.e. minus the second derivative of the log likelihood function at θ given data x.…

零一数据的回归与方差分析：残差变异的度量AI 解读

We consider regression situations for which the response variable is dichotomous. The most common analysis fits successively richer linear logistic models and measures the residual variation from the model by minus twice the maximized log likelihood. General measures of residual variation are considered here, including ordinary squared error and prediction error as well as the log likelihood. All of these are shown to be satisfactory in a certain primitive sense, unlike…

最大似然估计精度评估：观测与期望 Fisher 信息AI 解读

This paper concerns normal approximations to the distribution of the maximum likelihood estimator in one-parameter families. The traditional variance approximation is 1/§, where θ is the maximum likelihood estimator and § is the expected total Fisher information. Many writers, including R. A. Fisher, have argued in favour of the variance estimate 1/I(x), where I(x) is the observed information, i.e. minus the second derivative of the log likelihood function at θ given data x.…

Cox 删失数据似然函数的效率AI 解读

D.R. Cox has suggested a simple method for the regression analysis of censored data. We carry out an information calculation which shows that Cox's method has full asymptotic efficiency under conditions which are likely to be satisfied in many realistic situations. The connection of Cox's method with the Kaplan-Meier estimator of a survival curve is made explicit.

多元正态均值的极小极大估计量族AI 解读

Ever since Stein's result, that the sample mean vector $\mathbf{X}$ of a $k \geqq 3$ dimensional normal distribution is an inadmissible estimator of its expectation $\mathbf{\theta}$, statisticians have searched for uniformly better (minimax) estimators. Unbiased estimators are derived here of the risk of arbitrary orthogonally-invariant and scale-invariant estimators of $\mathbf{\theta}$ when the dispersion matrix $\sum$ of $\mathbf{X}$ is unknown and must be estimated.…

多元经验贝叶斯与协方差矩阵估计AI 解读

The problem of estimating several normal mean vectors in an empirical Bayes situation is considered. In this case, it reduces to the problem of estimating the inverse of a covariance matrix in the standard multivariate normal situation using a particular loss function. Estimators which dominate any constant multiple of the inverse sample covariance matrix are presented. These estimators work by shrinking the sample eigenvalues toward a central value, in much the same way as…

Logistic 回归与正态判别分析的效率比较AI 解读

A random vector x arises from one of two multivariate normal distributions differing in mean but not covariance. A training set x 1, x 2, ··· x n of previous cases, along with their correct assignments, is known. These can be used to estimate Fisher's discriminant by maximum likelihood and then to assign x on the basis of the estimated discriminant, a method known as the normal discrimination procedure. Logistic regression does the same thing but with the estimation of…

使用 Stein 估计量及其推广进行数据分析AI 解读

In 1961, James and Stein exhibited an estimator of the mean of a multivariate normal distribution having uniformly lower mean squared error than the sample mean. This estimator is reviewed briefly in an empirical Bayes context. Stein's rule and its generalizations are then applied to predict baseball averages, to estimate toxomosis prevalence rates, and to estimate the exact size of Pearson's chi-square test with results from a computer simulation. In each of these examples,…

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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限制贝叶斯与经验贝叶斯估计量的风险：第二部分，经验贝叶斯情形AI 解读

We discuss compromises between Stein's estimator and the MLE which limit the risk to individual components of the estimation problem while sacrificing only a small fraction of the savings in total squared error loss given by Stein's rule. The compromise estimators “limit translation” away from the MLE. The calculations are pursued in an empirical Bayesian manner by considering their performance against an entire family of prior distributions on the unknown parameters.

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向量观测的经验贝叶斯估计：Stein 方法的扩展AI 解读

The statistician is considering several independent normal linear models with identical structures, and desires to estimate the vector of unknown parameters in each of them. An estimator is constructed which dominates the usual Gauss-Markov estimator in terms of total squared error loss. This estimator is shown to have good efficiency in the Bayesian situation where the parameter vectors themselves have a normal prior distribution. A practical example is given.

限制贝叶斯与经验贝叶斯估计量的风险：第一部分，贝叶斯情形AI 解读

The first part of this article considers the Bayesian problem of estimating the mean, θ, of a normal distribution when the mean itself has a normal prior. The usual Bayes estimator for this situation has high risk if θ is far from the mean of the prior distribution. We suggest rules which do not have this bad property and still perform well against the normal prior. These rules are compromises between the Bayes rule and the MLE. Similar rules are suggested for the empirical…

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使序贯实验保持分组平衡AI 解读

Subjects arrive sequentially at an experimental site and must be assigned immediately to treatment or control groups. In order to avoid biasing the results of the experiment it is customary to make the assignments by independent flips of a fair coin, but in small-sized experiments this may result in a severe imbalance between the numbers of treatments and controls. This paper discusses a new method of assigning the subjects which tends to balance the experiment, but at the…

对称性条件下的 Student t 检验AI 解读

The size and power of Student's t-test are discussed under weaker than normal conditions. It is shown that assuming only a symmetry condition for the null hypothesis leads to effective bounds on the dispersion of the t-statistic. (The symmetry condition is weak enough to include all cases of independent but not necessarily identically distributed observations, each symmetric about the origin.) The connection between Student's test and the usual non-parametric tests is…

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