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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…

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,…

多项分布函数和的中心极限定理AI 解读

Let $(N_1, \cdots, N_k)$ be a multinomial vector with $n = \sum N_i$ and with parameter $(p_1, \cdots, p_k), \sum p_i = 1$. Let $f_1, \cdots, f_k$ be real-valued functions defined on the integers $\{0, 1, \cdots, n\}$, and let $S_k = \sum^k_{i=1} f_i(N_i)$. Suppose $k \rightarrow \infty$, and as $k \rightarrow \infty$, that $n \rightarrow \infty$ and $\max_{1\leqq i \leqq k}p_i \rightarrow 0$. Conditions on the $\{f_i\}$ are given which guarantee that $S_k$, suitably…

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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