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稳健位置估计AI 解读

Measures of location differentiable at every density in the Hellinger metric are constructed in this paper. Differentiability entitles these location functionals to the label "robust," even though their influence curves need not be bounded and continuous. The latter properties are, in fact, associated with functionals differentiable in the Prokhorov metric. A Hellinger metric concept of minimax robustness of a location measure at a density shape $f$ is developed.…

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多元数据的序列相关检验AI 解读

Consider a sample from a multiple time series that is stationary and Gaussian. A test is presented for independence among the multivariate observations that comprise this sample. The test is a generalization of the Kolmogorov-Smirnov test for serial correlation in a single time series. In the test, pairs of spectral-matrix estimates are compared using the largest-root statistic. The comparisons, which are tested simultaneously, are between estimates obtained from upper and…

分布函数估计AI 解读

It is shown that the limiting distribution of any regular estimator of a continuous cdf on [0, 1] can be represented as a convolution of the Brownian bridge process with another distribution on $C\lbrack 0, 1\rbrack$. The result is related to Hajek's representation for limiting distributions of regular parametric estimators.

改进若干已知的密度导数非参数一致估计量AI 解读

Based on a random sample from a univariate distribution with density $f$, this note exhibits a class of kernel estimators of the $p$th order derivative $f^{(p)}$ of $f, p \geqq 0$ fixed. These estimators improve some known estimators of $f^{(p)}$ by weakening the conditions, sharpening the rates of convergence, or both for the properties of strong consistency, asymptotic unbiasedness and mean square consistency, each uniform on the real line.

两均值之差的序贯贝叶斯估计AI 解读

It is desired to estimate the difference between the means of two independent normal distributions as accurately as possible and in a sequential manner when the total number of observations is fixed. The problem is posed in a Bayesian framework with conjugate prior distributions and squared error loss function. It is shown that the optimal sequential design depends on the ratio of the posterior variances of the two means. There exist constants (dependent on the prior…

经验分布在凸集类上的一致收敛AI 解读

The empirical distribution function $P_n$ converges with probability 1 to a true distribution $P$ in $R^k$, uniformly over measurable convex sets, if and only if $P$ is a countable mixture of distributions, each of which is carried by a flat and gives zero probability to the relative boundaries of convex sets included in the flat.

伯努利试验中的序贯估计AI 解读

The sequential estimation of $p$, the probability of success in a sequence of Bernoulli trials, is considered for the case where loss is taken to be symmetrized relative squared error of estimation, plus a fixed cost $c$ per observation. Using $s_n/n$ as a terminal estimator of $p$, where $s_n$ is the number of successes in $n$ trials, a heuristic rule is derived and shown to perform well for any fixed $0 < p < 1$ as $c \rightarrow 0$. However for any fixed $c > 0$, this…

与似然比有关的过程的弱收敛AI 解读

Material in Chapter VI of Hajek and Sidak's book is extended to a sequential analysis setting: conditions are given under which a sequence of log-likelihood-ratio processes (log-likelihood-ratios for sequential sampling, represented as jump processes in continuous time) converges weakly to a Wiener process with drift, the drift parameter depending on which hypothesis, in a suitable neighborhood of a null hypothesis, prevails. Conditions for convergence of other "test…

无矩条件下停止时间指数有界的一般定理及应用AI 解读

The stopping time $N$ of a sequential test based on an i.i.d. sequence $X, X_1, X_2, \cdots$ with common distribution $P$ is defined as the first integer $n \geqq 1$ such that $l_1 < L_n < l_2$ is violated, where $L_n$ is a statistic depending only on $X_1, \cdots, X_n$. If for all $l_1, l_2$ there exist constants $c > 0, \rho < 1$ such that $P(N > n) < c\rho^n, n = 1, 2, \cdots,$ then $N$ is called exponentially bounded under $P$. In the contrary case $P$ is termed…

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 $A$, $\Pi^r_{j = 1} P\lbrack\cap^{a_j}_{i = 1} \{Z_i \in A\}\rbrack \geqq \Pi^r_{j = 1} P\lbrack\cap^{b_j}_{i = 1} \{Z_i \in A\}\rbrack$…

In establishing weak convergence of von Mises' differentiable statistical functions to a normal distribution usually square integrability conditions with respect to the underlying kernel function are assumed. It is shown that these conditions can be weakened by assuming integrability of the von Mises' functional itself. In addition it is pointed out that in nontrivial cases the conditions of square integrability of the kernel do not hold whereas weak convergence of the von…

关于渐近最优检验AI 解读

Sequences of tests with error $\exp(-nA)$ of the first type are investigated. It is shown that the error of the second type of such a sequence of tests is bounded by $\exp(- nB)$ where $B$ is determined by the Kullback-Leibler information distance of the hypotheses tested. The information distance between the empirical measure and the null-hypothesis on a finite partition of the sample space is proposed to use as a test statistic. A sufficient condition is given which…

次序统计量线性组合的误差界AI 解读

A Berry-Esseen bound is obtained for trimmed linear combinations of order statistics. These linear combinations are written as the sum of a linear and a quadratic combination of independent exponentially distributed random variables plus a remainder term. The remainder term is shown to be of negligible order and Fourier methods are then employed to handle the linear and quadratic terms. The main theorem is also given in a version that more easily lends itself to applications.

从下方逼近回归函数零点的修正 Robbins–Monro 程序AI 解读

A Robbins-Monro type procedure for estimating the zero of a regression function is discussed. The procedure is a modification of the Robbins-Monro procedure which is designed to approximate the zero from below. An almost sure convergence is proved and it is shown that one can guarantee that the procedure overestimate the zero only finitely many times with probability one.

次序统计量伴随变量之秩的分布与期望AI 解读

Let $(X_i, Y_i)$ be $n$ independent rv's having a common bivariate distribution. When the $X_i$ are arranged in nondecreasing order as the order statistics $X_{r:n} (r = 1,2,\cdots, n)$, the $Y$-variate $Y_{\lbrack r:n\rbrack}$ paired with $X_{r:n}$ is termed the concomitant of the $r$th order statistic. The small-sample theory of the distribution and expected value of the rank $R_{r:n}$ of $Y_{\lbrack r:n\rbrack}$ is studied. In the special case of bivariate normality an…

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 $\mathscr{F}_1(\Sigma) = \{f|f(x) = \mid \Sigma|^{-\frac{1}{2}}q(x'\Sigma^{-1}x), q \in Q\}$…

论 Cramér–Rao 不等式AI 解读

Sufficient conditions for the Cramer-Rao inequality are formulated which do not impose any requirement on the estimator and which are independent of the choice of the densities. A hereditary property is proved for these conditions and the attainability of the lower bound is studied.

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…

最大后验概率分类AI 解读

The problem of classifying each of $n$ observations to one of two sub-populations is considered. The classification rule examined chooses that classification with maximum posterior probability. Limiting behavior of the rule is given and several examples are presented which show that the rule can lead to classifying all observations to the same subpopulation. Three simulation studies are reported to indicate that this extreme behavior may occur in small samples.

多重决策问题的经验贝叶斯方法AI 解读

In the empirical Bayes approach to multiple decision problems, we obtain theorems and lemmas which can be used to obtain asymptotic optimality and rate results in any multiple decision empirical Bayes problem. Applications of these results to a classification problem, a monotone multiple decision, and a selection problem are given. In addition, a special lemma unique to the monotone multiple decision problem gives improved (exact) rate results in that case.

最简单经验贝叶斯决策问题的完全类定理AI 解读

For the problem of empirical Bayes classification into two known probability distributions on a finite outcome space, an essentially complete class of procedures is determined. This class is proven to be minimal essentially complete if there are only two possible outcomes.

损失函数不依赖未知参数时的预测充分性AI 解读

It is shown by Takeuchi and Akahira, 1974, that conditional independence together with a condition of "partial sufficiency" imply "prediction sufficiency" for loss functions not depending on the unknown parameter. We shall here prove that these conditions are necessary as well and thereby obtain a complete description, in terms of conditional expectations, of "prediction sufficiency" for loss functions not depending on the unknown parameter. It turns out that these…

U 统计量与拟合优度检验的大样本理论AI 解读

Let $X_{ni}, i = 1, \cdots, n$ be i.i.d. random variables on an arbitrary measurable space $(\mathscr{X}, B)$. Suppose $\mathscr{L}(X_{ni}) = Q_{n1}, i = 1, \cdots, n$ and let $P_0$ be a fixed probability measure on $(\mathscr{X}, B)$. We consider limiting distribution theory for $U$-statistics $T_n = n^{-1} \sum_{i \neq j} Q(X_{ni}, X_{nj})$ (1) under conditions which imply the product measures $Q_n = Q_{n1} \times \cdots \times Q_{n1}, n$ times, are contiguous to the…