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多元正态均值的精确置信集AI 解读

A class of confidence sets with constant coverage probability for the mean of a p-variate normal distribution is proposed through a pseudo-empirical-Bayes construction. When the dimension is greater than 2, by combining analytical results with some exact numerical calculations the proposed sets are proved to have a uniformly smaller volume than the usual confidence region. Sufficient conditions for the connectedness of the proposed confidence sets are also derived. In…

约束风险不等式及其在非参数泛函估计中的应用AI 解读

A general constrained minimum risk inequality is derived. Given two densities $f_{\theta}$ and $f_0$ we find a lower bound for the risk at the point $\theta$ given an upper bound for the risk at the point 0. The inequality sheds new light on superefficient estimators in the normal location problem and also on an adaptive estimation problem arising in nonparametric functional estimation.

非参数回归与白噪声模型的渐近等价性AI 解读

The principal result is that, under conditions, to any nonparametric regression problem there corresponds an asymptotically equivalent sequence of white noise with drift problems, and conversely. This asymptotic equivalence is in a global and uniform sense. Any normalized risk function attainable in one problem is asymptotically attainable in the other, with the difference in normalized risks converging to zero uniformly over the entire parameter space. The results are…

最优置信集合、生物等效性与帕斯卡蜗线AI 解读

We begin with a decision-theoretic investigation into confidence sets that minimize expected volume at a given parameter value. Such sets are constructed by inverting a family of uniformly most powerful tests, and hence they also enjoy the optimality property of being uniformly most accurate. In addition, these sets possess Bayesian optimal volume properties and represent the first case (to our knowledge) of a frequentist 1 – α confidence set that possesses a Bayesian…

摘要暂缺

奇异问题中有信息无偏估计量的不存在性AI 解读

In many nonparametric problems, such as density estimation, nonparametric regression and so on, all the existing informative estimators are biased (asymptotic or finite sample). There has long been a suspicion that either informative unbiased estimators do not exist for such problems or they must be quite complicated. In this paper, we clarify the nonexistence of informative unbiased estimators in all singular problems both for fixed sample size and asymptotically (this…

有效性约束下的置信度估计AI 解读

We examine the decision theoretic estimated confidence approach proposed by Kiefer, Robinson and Berger, and focus on results under the frequentist validity constraint previously described by Brown and by Berger. Our main result is that the usual constant coverage probability estimator for the usual confidence set of a linear model is admissible under the frequentist validity constraint. Note that it is inadmissible without the frequentist validity constraint when the…

极小极大风险的信息不等式下界及非参数回归应用AI 解读

This paper compares three methods for producing lower bounds on the minimax risk under quadratic loss. The first uses the bounds from Brown and Gajek. The second method also uses the information inequality and results in bounds which are always at least as good as those from the first method. The third method is the hardest-linear-family method described by Donoho and Liu. These methods are applied in four examples, the last of which relates to a frequently considered…

贝叶斯风险的信息不等式AI 解读

This paper presents lower bounds, derived from the information inequality, for the Bayes risk under scaled quadratic loss. Some numerical results are also presented which give some idea concerning the precision of these bounds. An appendix contains a proof of the information inequality without conditions on the estimator. This result is a direct extension of an earlier result of Fabian and Hannan.

概率密度点值估计风险的下界AI 解读

This article considers estimation of f(0) of a density function satisfying a Lipshitz condition in a neighborhood of 0. A nonstandard use of the Cramer-Rao inequality yields numerical lower bounds on the minimax squared error risk of any estimator. These bounds are then compared with the minimax risk of the asymptotically optimal kernel-type estimator. The asymptotic bounds obtained (as the sample size n → ∞) are not quite as good as those in Donoho and Liu (in press a, b),…

多元线性回归中的辅助统计量悖论AI 解读

Consider a multiple linear regression in which $Y_i, i = 1, \cdots, n$, are independent normal variables with variance $\sigma^2$ and $E(Y_i) = \alpha + V'_i\beta$, where $V_i \in \mathbb{R}^r$ and $\beta \in \mathbb{R}^r.$ Let $\hat{\alpha}$ denote the usual least squares estimator of $\alpha$. Suppose that $V_i$ are themselves observations of independent multivariate normal random variables with mean 0 and known, nonsingular covariance matrix $\theta$. Then $\hat{\alpha}$ i…

普遍支配与随机支配：最小二乘估计量的 U 可容许性与不可容许性AI 解读

Assume the standard linear model $X_{n \times 1} = A_{n \times p} \theta_{p \times 1} + \varepsilon_{n \times 1},$ where $\varepsilon$ has an $n$-variate normal distribution with zero mean vector and identity covariance matrix. The least squares estimator for the coefficient $\theta$ is $\hat{\theta} \equiv (A'A)^{-1}A'X$. It is well known that $\hat{\theta}$ is dominated by James-Stein type estimators under the sum of squared error loss $|\theta - \hat{\theta}|^2$ when $p \geq 3$…

离散、连续不变非参数估计及多项分布对应问题的可容许性AI 解读

Discrete and multinomial analogs are defined for classical (continuous) invariant nonparametric problems of estimating the sample cumulative distribution function (sample c.d.f.) and the sample median. Admissibility of classical estimators and their analogs is investigated. In discrete (including multinomial) settings the sample c.d.f. is shown to be an admissible estimator of the population c.d.f. under the invariant weighted Cramer-von Mises loss function $L_1(F, \hat{F}) = \int \big\lbrack (F(t) - \hat{F}(t))^2/(F(t)(1 - F(t))) \big\rbrack dF(t).$…

摘要暂缺

含无差别区间问题中 Student t 检验的替代方法AI 解读

Consider a sample from a normal population with mean, $\mu$, and variance unknown. Suppose it is desired to test $H_0:\mu \leq \mu_0$ versus $H_1:\mu \geq \mu_1$, with the region $H^I_1:\mu_0 < \mu < \mu_1$ being a (nonempty) indifference zone. It is shown that the usual Student's $t$-test is inadmissible for this problem. An alternative test is proposed. The two sided problem with indifference region is also discussed. By contrast with the above result, the usual Student's $t$…

Jack Kiefer 在实验设计之外的研究AI 解读

This article is a survey of Jack Kiefer's published works in areas other than design of experiments. The approximately 50 articles can be divided into categories by subject matter. These categories will be discussed individually in the following review. At times Jack worked in several areas simultaneously and ideas from one area sometimes influenced work in another. Some of these influences are noted in the following survey, as the categories are discussed in roughly the…

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…

判断估计量可容许性的启发式方法及其应用AI 解读

Questions of admissibility of statistical estimators are reduced to considerations involving differential inequalities. The coefficients of these inequalities involve moments of the underlying distributions; and so are, in principle, not difficult to derive. The methods are "heuristic" because it is necessary to verify on an ad-hoc basis that error terms are small. Some conditions on the structure of the problem are given which we believe will guarantee that these error…

损失函数未完全指定时的多个位置参数估计AI 解读

If p ≥ 3 separate normal means are to be estimated, the usual, invariant estimator is “inadmissible” if the p separate loss functions can be pooled into a single loss function for the ensemble-of-means problem. Here, we formulate a version of the problem where the losses cannot be pooled in this way, and find a surprisingly weak necessary and sufficient condition under which the usual estimator is inadmissible in a sense appropriate to this formulation. We also examine the…

损失函数未完全指定时的估计：多个位置参数情形AI 解读

If p ≥ 3 separate normal means are to be estimated, the usual, invariant estimator is “inadmissible” if the p separate loss functions can be pooled into a single loss function for the ensemble-of-means problem. Here, we formulate a version of the problem where the losses cannot be pooled in this way, and find a surprisingly weak necessary and sufficient condition under which the usual estimator is inadmissible in a sense appropriate to this formulation. We also examine the…