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查尔斯·斯坦与不变性：从 Hunt–Stein 定理谈起AI 解读

When statistical decision theory was emerging as a promising new paradigm, Charles Stein was to play a major role in the development of minimax theory for invariant statistical problems. In some of his earliest work with Gil Hunt, he set out to prove that, in problems where invariant procedures have constant risk, any best invariant test would be minimax among all tests. Although finding it not quite true in general, this led to the legendary Hunt–Stein theorem, which…

通过马尔可夫链论证评价形式后验分布AI 解读

We consider evaluation of proper posterior distributions obtained from improper prior distributions. Our context is estimating a bounded function φ of a parameter when the loss is quadratic. If the posterior mean of φ is admissible for all bounded φ, the posterior is strongly admissible. We give sufficient conditions for strong admissibility. These conditions involve the recurrence of a Markov chain associated with the estimation problem. We develop general sufficient…

统计双联篇：可容许推断与对称马尔可夫链的常返性AI 解读

Given a parametric model and an improper prior distribution, the formal posterior distribution induces decision rules in any decision problem. The results here provide conditions under which this formal Bayes method produces admissible decision rules for all quadratically regular decision problems. The conditions derived are shown to be equivalent to the recurrence of a natural symmetric Markov chain (on the parameter space) generated by the model and the improper prior. The…

Wielandt不等式及其在随机对称矩阵特征值渐近分布中的应用AI 解读

A relatively obscure eigenvalue due to Wielandt is used to give a simple derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix. The asymptotic distributions are obtained under a fairly general setting. An application of the general theory to the bootstrap distribution of the eigenvalues of the sample covariance matrix is given.

Cholesky 分解的最优等变估计量AI 解读

Every positive definite matrix $\Sigma$ has a unique Cholesky decomposition $\Sigma = \theta\theta'$, where $\theta$ is lower triangular with positive diagonal elements. Suppose that $S$ has a Wishart distribution with mean $n\Sigma$ and that $S$ has the Cholesky decomposition $S = XX'$. We show, for a variety of loss functions, that $XD$, where $D$ is diagonal, is a best equivariant estimator of $\theta$. Explicit expressions for $D$ are provided.

摘要暂缺

球对称性的稳健检验AI 解读

Let $R^n$ be Euclidean $n$-space and let $O(n)$ be the group of $n \times n$ orthogonal matrices. Consider $\mathscr{F}_0 = \{f\mid f$ is a density on $R^n, f(x) = f(gx), x \in R^n, g \in O(n)\}$, and let $Q = \{q\mid q: \lbrack 0, \infty) \rightarrow \lbrack 0, \infty), q$ is nonincreasing, $\int_{R^n} q(\| x \|^2) dx = 1\}$. If $\Sigma$ is an $n \times n$ positive definite matrix, set $\mathscr{F}_1(\Sigma) = \{f|f(x) = \mid \Sigma|^{-\frac{1}{2}}q(x'\Sigma^{-1}x), q \in Q\}$…

The main result of the current research describes a monotonicity property of certain invariant tests for the multivariate analysis of variance problem. Suppose $X: r \times p$ has a normal distribution, $EX = \Theta$ and the rows of $X$ are independent, each with unknown covariance matrix $\Sigma: p \times p$. Let $S = p \times p$ have a Wishart distribution $W(\Sigma, p, n), S$ independent of $X$. If $K$ is the acceptance region of an invariant test for the null hypothesis $\Theta = 0$…

有界随机变量线性组合的概率不等式AI 解读

Let $Y_1, \cdots, Y_n$ be independent random variables with mean zero such that $|Y_i| \leqq i, i = 1, \cdots, n$, and let $\theta_1,\cdots, \theta_n$ be real numbers satisfying $\sum^n_1 \theta_i^2 = 1$. Set $S_n(\theta) = \sum^n_1 \theta_i Y_i$ and let $\varphi(x) = (2\pi)^{-\frac{1}{2}} \exp \lbrack - \frac{1}{2} x^2\rbrack$. THEOREM. For $\alpha > 0$, and for all $\theta_1,\cdots, \theta_n$, \begin{\align*}P\{|S_n(\theta)| \geqq \alpha\} &\leqq 2\inf_{0\leqq u \leqq…

摘要暂缺

广义样本协方差矩阵的非奇异性AI 解读

Let $X = (X_1, \cdots, X_n)$ where the $X_i: p \times 1$ are independent random vectors, and let $A: n \times n$ be positive semi-definite symmetric. This paper establishes necessary and sufficient conditions that the random matrix $XAX'$ be positive definite w.p.1. The results are applied to cases where $A$ has a particular form or $X_1, \cdots, X_n$ are i.i.d. In particular, it is shown that in the i.i.d. case, the sample covariance matrix $\sigma(X_i - \bar{X})(X_i - \bar{X})'$…

关于 Scheffé 对 Behrens–Fisher 问题解法的若干讨论AI 解读

The problem considered here is a multivariate generalization of the classical Behrens-Fisher problem. Based on samples of size ni , i = 1, …, k, from k multivariate normal populations (of perhaps different dimension), a test statistic is proposed to test null hypotheses about linear relationships between components of the mean vectors of the k populations. Under the null hypothesis, the test statistic has a central F distribution and under alternatives, a non-central F…