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This paper investigates estimation of the mean vector under invariant quadratic loss for a spherically symmetric location family with a residual vector with density of the form $f(x,u)=\eta ^{(p+n)/2}f(\eta \{\|x-\theta \|^{2}+\|u\|^{2}\})$, where $\eta $ is unknown. We show that the natural estimator $x$ is admissible for $p=1,2$. Also, for $p\geq 3$, we find classes of generalized Bayes estimators that are admissible within the class of equivariant estimators of the form $\{1-\xi (x/\|u\|)\}x$…

置信分布与元分析的统一框架AI 解读

This article develops a unifying framework, as well as robust meta-analysis approaches, for combining studies from independent sources. The device used in this combination is a confidence distribution (CD), which uses a distribution function, instead of a point (point estimator) or an interval (confidence interval), to estimate a parameter of interest. A CD function contains a wealth of information for inferences, and it is a useful device for combining studies from…

实验室间研究中的同时估计与不一致性降低AI 解读

Summary Several procedures that are designed to reduce nonconformity in interlaboratory studies by shrinking data towards a consensus weighted mean are suggested. Some of them are shown to have a smaller quadratic risk than the vector sample means. Shrinkage towards a weighted mean in a random-effects model and a statistic appearing in models which allow for systematic errors are also considered. The results are illustrated by two examples of collaborative studies.

一类新的广义贝叶斯极小极大岭回归估计量AI 解读

Let y=Aβ+ɛ, where y is an N×1 vector of observations, β is a p×1 vector of unknown regression coefficients, A is an N×p design matrix and ɛ is a spherically symmetric error term with unknown scale parameter σ. We consider estimation of β under general quadratic loss functions, and, in particular, extend the work of Strawderman [J. Amer. Statist. Assoc. 73 (1978) 623–627] and Casella [Ann. Statist. 8 (1980) 1036–1056, J. Amer. Statist. Assoc. 80 (1985) 753–758] by finding…

通过置信分布合并独立来源的信息AI 解读

This paper develops new methodology, together with related theories, for combining information from independent studies through confidence distributions. A formal definition of a confidence distribution and its asymptotic counterpart (i.e., asymptotic confidence distribution) are given and illustrated in the context of combining information. Two general combination methods are developed: the first along the lines of combining p-values, with some notable differences in regard…

贝叶斯极小极大估计量的构造AI 解读

Bayes estimation of the mean of a multivariate normal distribution is considered under quadratic loss. We show that, when particular spherical priors are used, the superharmonicity of the square root of the marginal density provides a viable method for constructing (possibly proper) Bayes (and admissible) minimax estimators. Examples illustrate the theory; most notably it is shown that a multivariate Student-$t$ prior yields a proper Bayes minimax estimate.

给定风险的估计量、Riccati 微分方程与 Brown 定理的新证明AI 解读

For the canonical problem of estimating the mean of a multivariate normal distribution with a known covariance matrix using a squared error loss, we give a general method for finding estimates that have risk functions identical to that of a given inadmissible estimate. In the case of more than one dimension, the estimates considered are spherically symmetric, but in one dimension no such assumption is made. Generally speaking, we characterize all estimates which have the…

层次先验的选择：正态均值估计的可容许性AI 解读

In hierarchical Bayesian modeling of normal means, it is common to complete the prior specification by choosing a constant prior density for unmodeled hyperparameters (e.g., variances and highest-level means). This common practice often results in an inadequate overall prior, inadequate in the sense that estimators resulting from its use can be inadmissible under quadratic loss. In this paper, hierarchical priors for normal means are categorized in terms of admissibility and…

改进 James–Stein 正部估计量AI 解读

The purpose of this paper is to give an explicit estimator dominating the positive-part James-Stein rule. The James-Stein estimator improves on the "usual" estimator $X$ of a multivariate normal mean vector $\theta$ if the dimension $p$ of the problem is at least 3. It has been known since at least 1964 that the positive-part version of this estimator improves on the James-Stein estimator. Brown's 1971 results imply that the positive-part version is itself inadmissible…

合并无偏与可能有偏估计量的 James–Stein 型方法AI 解读

We present a method for combining unbiased sample data with possibly biased auxiliary information. The estimator we derive is similar in spirit to the James–Stein estimator. We prove that the estimator dominates the sample mean under quadratic loss. When the auxiliary information is unbiased, our estimator has risk slightly greater than the usual combined estimator. As the bias increases, however, the risk of the usual estimator is unbounded, while the risk of our estimator…

球对称分布下 James–Stein 估计量的推广AI 解读

This paper is primarily concerned with extending the results of Stein to spherically symmetric distributions. Specifically, when $X \sim f(\|X - \theta\|^2)$, we investigate conditions under which estimators of the form $X + ag(X)$ dominate $X$ for loss functions $\|\delta - \theta\|^2$ and loss functions which are concave in $\|\delta - \theta\|^2$. Additionally, if the scale is unknown we investigate estimators of the location parameter of the form $X + aVg(X)$ in two…

二次损失下多元正态均值的两阶段序贯估计AI 解读

In estimating a multivariate normal mean under quadratic loss, this paper looks into the existence of two-stage sequential estimators that are better both in risk (mean square error) and sample size than the usual estimator of a given fixed sample size. In other words, given any sample size $n$, we are looking for two-stage sequential estimators truncated at $n$, with a positive probability of stopping earlier and risk lower than that of the sample mean based on $n$ observati…

指数分布分位数的估计AI 解读

We consider the problem of estimating a quantile ξ + bσ of an exponential distribution on the basis of a random sample of size n ≥ 2. Here ξ and σ are unknown location and scale parameters and b is a given constant. For quadratic loss, it is established that the best equivariant estimator is inadmissible if 0 ≤ b < n-1 or b > 1 + n 1. For b > 1 + n 1 the estimator elsewhere, provides a noticeable improvement over δ0.

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…

预测变量范围受限时的线性回归置信带AI 解读

The classic procedure for constructing a confidence band around a regression function is that of Scheffé (1959), which provides a band of infinite length. This band can be unnecessarily wide if the region of interest to the experimenter is finite. A class of sets is defined that restricts the range of the predictor variables. Confidence bands for a regression function over are constructed, which, for fixed α, are narrower than the Scheffé bands. Also, a procedure is…

摘要暂缺

凹损失下球对称分布位置参数的极小极大估计AI 解读

For $p \geqslant 4$ and one observation $X$ on a $p$-dimensional spherically symmetric distribution, minimax estimators of $\theta$ whose risks are smaller than the risk of $X$ (the best invariant estimator) are found when the loss is a nondecreasing concave function of quadratic loss. For $n$ observations $X_1, X_2, \cdots, X_n$, we have classes of minimax estimators which are better than the usual procedures, such as the best invariant estimator, $\bar{X}$, or a maximum…

极小极大自适应广义岭回归估计量AI 解读

We consider the problem of estimating the vector of regression coefficients of a linear model using generalized ridge regression estimators where the ridge constant is chosen on the basis of the data. For general quadratic loss we produce such estimators whose risk function dominates that of the least squares procedure provided the number of regressors is at least three. We study the problem by the usual reduction to estimating the mean vector of a multivariate normal…

球对称单峰分布位置参数在二次损失下的极小极大估计AI 解读

Families of minimax estimators are found for the location parameter of a $p$-variate $(p \geqq 3)$ spherically symmetric unimodal distribution with respect to general quadratic loss. The estimators of James and Stein, Baranchik, Bock and Strawderman are all considered for this general problem. Specifically, when the loss is general quadratic loss given by $L(\delta, \theta) = (\delta - \theta)'D(\delta - \theta)$ where $D$ is a known $p \times p$ positive definite matrix,…

The problem of estimating powers of the variance of a normal distribution is considered when loss is essentially squared error. A class of minimax estimators is found by extending the techniques of Stein. It is shown, at least for estimating the variance, that a subclass of the above consists of generalized Bayes estimators.

共同未知方差下多元正态均值的正规贝叶斯极小极大估计AI 解读

We investigate the problem of estimating the mean vector $\mathbf{\theta}$ of a multivariate normal distribution with covariance matrix equal to $\sigma^2\mathbf{I}_p, \sigma^2$ unknown, and loss $\|\delta - \mathbf{\theta}\|^2/\sigma^2$. We first find a class of minimax estimators for this problem which enlarges a class given by Baranchik. This result is then used to show that for sufficiently large sample sizes (which never need exceed 4) proper Bayes minimax estimators…

平移或尺度参数的容许置信区间与点估计AI 解读

Sufficient conditions for admissibility of the best invariant confidence interval for a translation or scale parameter are given, for a very wide class of loss functions. These conditions result by adapting a theorem of L. D. Brown [2]. Simpler sufficient conditions are found for a subclass of loss functions of special interest. The subclass of losses involves three components. One concerned with coverage of the true value, another concerned with the distance from the…