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通过共同边际正态变量的依赖性比较统计实验AI 解读

In this note we study comparison of experiments via the positive dependence of normal variables with a common univariate marginal distribution. We show that positive dependence has an adverse effect on the information concerning the common mean $\theta$, and give a partial ordering of the information via a majorization ordering of the correlation matrices. In the special case when the random variables are equally correlated, the main theorem yields a result for the…

分类与多假设检验中的最优划分不等式AI 解读

Optimal-partitioning and minimax risk inequalities are obtained for the classification and multi-hypotheses testing problems. Best possible bounds are derived for the minimax risk for location parameter families, based on the tail concentrations and Levy concentrations of the distributions. Special attention is given to continuous distributions with the maximum likelihood ratio property and to symmetric unimodal continuous distributions. Bounds for general (including…

具有共同边际分布的一类正相依随机变量的不等式AI 解读

This paper concerns a partial ordering of positive dependence of a class of random variables which have a common marginal distribution and are not necessarily exchangeable. The main theorem is obtained by applying a moment inequality via majorization. Inequalities for exchangeable random variables, for random variables whose marginal densities possess the semigroup property and for the multivariate normal distribution are then obtained as special cases.

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Schur 凹随机变量的矩形与椭球概率不等式AI 解读

It is shown that if the density $f(\mathbf{x})$ of $\mathbf{X} = (X_1, \cdots, X_n)$ is Schur-concave, then (1) $P(|X_i| \leq a_i, i = 1, \cdots, n)$ is a Schur-concave function of $(\phi(a_1), \cdots, \phi(a_n))$, and (2) $P\{\Sigma(X_i/a_i)^2 \leq 1\}$ is a Schur-concave function of $(\phi(a^2_1), \cdots, \phi(a^2_n))$, where $\phi; \lbrack 0, \infty) \rightarrow \lbrack 0, \infty)$ is any increasing and convex function. By letting $\phi(a) = a$, (1) implies that $P(|X_i| \leq a_i, i = 1, \cdots, n) \leq P(|X_i| \leq \bar{a}, i = 1, \cdots, n)$…

排序与选择问题的一种自适应解法AI 解读

An adaptive approach is considered as an alternative to the classical indifference-zone formulation of the problems of ranking and selection. With a fixed $\gamma^\ast$, the proposed procedure calls for the termination of sampling when the estimated probability of correct selection exceeds $\gamma^\ast$ for the first time. Asymptotic properties of this procedure are proved as $\gamma^\ast \rightarrow 1$, and Monte Carlo results show that the procedure is well behaved even…

条件独立同分布随机变量的一个序关系定理AI 解读

Let $\mathbf{a}$ and $\mathbf{b}$ be $r$-dimensional real vectors. It is shown that if $\mathbf{a}$ majorizes $\mathbf{b}$, then $E(\Pi^r_{j = 1} X_j^a j) \geqq E(\Pi^r_{j = 1} X_j^b j)$ holds for nonnegative random variables $X_1, \cdots, X_r$ whose joint pdf is permutation symmetric. If in addition the components of $\mathbf{a, b}$ are nonnegative integers, then for every Borel-measurable set $A$, $\Pi^r_{j = 1} P\lbrack\cap^{a_j}_{i = 1} \{Z_i \in A\}\rbrack \geqq \Pi^r_{j = 1} P\lbrack\cap^{b_j}_{i = 1} \{Z_i \in A\}\rbrack$…

逆线性回归中观测数量的最优分配AI 解读

Consider the problem of estimating $x$ under the inverse linear regression model $Y_i = \alpha + \beta x_i + \varepsilon_i,\quad Z_j = \alpha + \beta x + \varepsilon_j'$ for $i = 1,\cdots, n,\cdots, j = 1,\cdots, m,\cdots,$ where $\{\varepsilon_i\}, \{\varepsilon_j'\}$ are two sequences of i.i.d. rv's with 0 means and finite variances, $\{x_i\}$ is a sequence of known constants and $\alpha, \beta, x$ are unknown parameters. For fixed $T = m + n$, this paper considers a…

通过分组估计正态总体的百分位数AI 解读

In this article we apply the group-testing idea to the estimation of a percentile of a normal population when both the mean and the variance are unknown. The structure is such that we observe the sum of the normal, independent measurements of m units in one single observation (or test). Under the assumption of a fixed total cost T, we have obtained the optimal value of the group size m for both the small-sample and the large-sample (T → ∞) cases. Comparison with the…

通过分组测量估计正态总体分位点AI 解读

In this article we apply the group-testing idea to the estimation of a percentile of a normal population when both the mean and the variance are unknown. The structure is such that we observe the sum of the normal, independent measurements of m units in one single observation (or test). Under the assumption of a fixed total cost T, we have obtained the optimal value of the group size m for both the small-sample and the large-sample (T → ∞) cases. Comparison with the…