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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.

线性模型中的预检验与收缩M估计AI 解读

In a general univariate linear model, $M$-estimation of a subset of parameters is considered when the complementary subset is plausibly redundant. Along with the classical versions, both the preliminary test and shrinkage versions of the usual $M$-estimators are considered and, in the light of their asymptotic distributional risks, the relative asymptotic risk-efficiency results are studied in detail. Though the shrinkage $M$-estimators may dominate their classical versions,…

线性模型中 Jackknife 方差估计量的异方差稳健性AI 解读

The asymptotic unbiasedness and consistency of three types of jackknife variance estimators in the presence of error variance heteroscedasticity in linear models are studied. The results are given in terms of the number of observations deleted and measures of imbalance of the model. The consistency of a class of Wu's weighted jackknife variance estimators for nonlinear parameters is also studied. A necessary and sufficient condition is given for the asymptotic unbiasedness…

局部渐近线性估计量影响函数的一致估计AI 解读

Consider estimators which behave locally asymptotically like an average of some function taken at the observations. This function is called the influence function and one calls such estimators locally asymptotically linear. It is shown that the influence function of a locally asymptotically linear estimator can be estimated consistently and conversely, that, given a consistent estimator of the influence function, estimators can be constructed which are locally asymptotically…

双重删失数据下生存函数非参数估计量的强一致性AI 解读

A double censoring mechanism is such that each variable $X$ in the sample is observable if and only if $X$ is within the observation interval $\lbrack Z, Y \rbrack$. Otherwise, we can only determine whether $X$ is less than $Z$ or greater than $Y$ and observe $Z$ or $Y$ correspondingly. This kind of censoring occurs often in collecting lifetime data. Our problem is to estimate the survival function of $X, S_X(t) = P \lbrack X > t \rbrack$, from a doubly censored sample,…

随机右删失样本核密度估计的渐近最优带宽选择AI 解读

This paper makes two important contributions to the theory of bandwidth selection for kernel density estimators under right censorship. First, an asymptotic representation of the integrated squared error into easily understood variance and squared bias components is given. Second, it is shown that if the bandwidth is chosen by the data-based method of least-squares cross-validation, then it is asymptotically optimal in a compelling sense. A by-product of the first part is an…

Kullback—Leibler损失与密度估计AI 解读

"Discrimination information," or Kullback-Leibler loss, is an appropriate measure of distance in problems of discrimination. We examine it in the context of nonparametric kernel density estimation and show that its asymptotic properties are profoundly influenced by tail properties of the kernel and of the unknown density. We suggest ways of choosing the kernel so as to reduce loss, and describe the extent to which likelihood cross-validation asymptotically minimises loss.…

负相依随机变量的最优停止价值比较与先知不等式AI 解读

Let $\mathbf{Y} = (Y_1, \cdots, Y_n)$ be random variables satisfying the weak negative dependence condition: $P(Y_i < a_i\mid Y_1 < a_1, \cdots, Y_{i-1}) \leq P(Y_i < a_i)$ for $i = 2, \cdots, n$ and all constants $a_1, \cdots, a_n$. Let $\mathbf{X} = (X_1, \cdots, X_n)$ have independent components, where $X_i$ and $Y_i$ have the same marginal distribution, $i = 1, \cdots, n$. It is shown that $V(\mathbf{X}) \leq V(\mathbf{Y})$, where $V(\mathbf{Y}) = \sup \{EY_t: t \text{is a stopping rule for} Y_1,\cdots, Y_n\}$…

集合分割上的分布AI 解读

A two parameter family of distributions on partitions is obtained by mixing a multinomial with a symmetric Dirichlet distribution. Estimates for the parameters are proposed and studied in various asymptotic limits.

统计证据的信念函数表示AI 解读

In Glenn Shafer's theory of parametric statistical inference, observational evidence and prior evidence are separately represented by belief or commonality functions $Q$ and $R$, which are then combined by Dempster's rule. We characterise, for finite parameter spaces, the functionals $Q$ and $R$ for which statistically independent observations may be combined by Dempster's rule, and those for which Dempster's rule is consistent with Bayes' rule. The functionals are…

局部与非局部效率度量AI 解读

General results on the limiting equivalence of local and nonlocal measures of efficiency are obtained. Why equivalence occurs in so many testing and estimation problems is clarified. Uniformity of the convergence is a key point. The concepts of Frechet- and Hadamard-type differentiability, which imply uniformity, play an important role. The theory is applied to tests based on linear rank statistics, showing equivalence of the local limit of exact Bahadur efficiency and…

平方误差加复杂度成本损失下的正态均值估计AI 解读

Estimating the mean of a $p$-variate normal distribution is considered when the loss is squared error plus a complexity cost. The complexity of estimates is defined using a partition of the parameter space into sets corresponding to models of different complexity. The model implied by the use of an estimate determines the estimate's complexity cost. Complete classes of estimators are developed which consist of preliminary-test estimators. As is the case when loss is just…

正态方差区间估计量的条件性质AI 解读

Both one-sided and two-sided interval estimators are examined using conditional criteria and we find that most common intervals have acceptable conditional properties. In the two-sided case we further examine three well known intervals and find the shortest-unbiased (Neyman-shortest) interval possessing the strongest conditional properties, with the minimum-length interval a close second. In the one-sided case we have the somewhat surprising result that the lower confidence…

条件可接受的重新定中心集合估计量AI 解读

The usual confidence sphere for a multivariate normal mean can be uniformly improved upon, in terms of coverage probability, by recentering it at a Stein-type estimator. However, these improved sets can have poor conditional performance. Using the theory of relevant betting procedures, which provides an objective means for assessing the conditional performance of a statistical procedure, a criterion for conditional acceptability can be established. A method of constructing…

测量误差及相关模型中有限期望直径置信集的不存在性AI 解读

Confidence intervals are widely used in statistical practice as indicators of precision for related point estimators or as estimators in their own right. In the present paper it is shown that for some models, including most linear and nonlinear errors-in-variables regression models, and for certain estimation problems arising in the context of classical linear models, such as the inverse regression problem, it is impossible to construct confidence intervals for key…

Kiefer–Wolfowitz 等价定理在哈达玛变换光学问题中的应用AI 解读

Let $\Omega = \{\mathbf{x} = (x_1, \cdots, x_n)^T: 0 \leq x_i \leq 1\}$ be the unit cube in $R^n$. For any probability measure $\xi$ on $\Omega$, let $\mathbf{M}(\xi) = \int_\Omega\mathbf{xx}^T \xi(d\mathbf{x})$. Harwit and Sloane (1976) conjectured that if $\mathbf{X}^\ast$ is the incidence matrix of a balanced incomplete block design (BIBD) with $n$ treatments and $n$ blocks of size $(n + 1)/2$, then $\mathbf{X}^\ast$ minimizes $\operatorname{tr}(\mathbf{X}^T\mathbf{X})^{-1}$…

若干类全局 Cramér–Rao 界AI 解读

This paper considers Cramer-Rao type bounds for the estimation error of a parameter in a Bayesian setup. This class of bounds, introduced by Van Trees, proved useful in various stochastic communications and control problems. Two issues are considered in this paper. The first deals with a comparison of the tightness of several different versions of the bound in the multivariate case. The second introduces several useful generalizations of the original version of the bound.

关于买方问题的一则注记AI 解读

Bather considered a buyer's problem and reduced it to an optimal stopping problem. In this note it is shown that this optimal stopping problem can be related to the sequential testing of the sign of the drift parameter of a Wiener process problem considered by Chernoff. Hence the asymptotic expansions for the sequential testing problem can be used in the buyer's problem.

独立随机变量序列中变点极大似然估计量的分布近似AI 解读

The problem of estimating the change-point in a sequence of independent random variables is considered. As the sample sizes before and after the change-point tend to infinity, Hinkley (1970) showed that the maximum likelihood estimate of the change-point converges in distribution to that of the change-point based on an infinite sample. Letting the amount of change in distribution approach 0, it is shown that the distribution, suitably normalized, of the maximum likelihood…

Kaplan–Meier 估计量的一致收敛札记AI 解读

Let $\{X_n, n \geq 1\}$ be i.i.d with $P(X_i \leq u) = F(u)$ and $\{U_n, n \geq 1\}$ be i.i.d. with $P(U_i \leq u) = G(u). \hat{F}_n(t)$ is the Kaplan-Meier estimator based on the censored data $(\tilde{X}_i = X_i \wedge U_i, \delta_i = 1_{(X_i \leq U_i)}, 1 \leq i \leq n)$. In this note, it is shown that for $T_n = \max_{1 \leq i \leq n} \tilde{X}_i$, $\mathrm{pr}-\lim_{n \rightarrow \infty} \sup_{t \leq T_n} |\hat{F}_n(t) - F(t)| = 0.$ Hence, the largest interval on which…

随机过程的定性稳健性AI 解读

In this paper we generalize Hampel's concept of qualitative robustness of a sequence of estimators to the case of stochastic processes with non-i.i.d. observations, defining appropriate metrics between samples. We also present a different approach to qualitative robustness which formalizes the notion of resistance. We give two definitions based on this approach: strong and weak resistance. We show that for estimating a finite dimensional real parameter, $\pi$-robustness is…

两个正态总体共同均值的可容许极小极大估计AI 解读

Consider the problem of estimating the common mean $\mu$ of two normal populations with unknown variances $\sigma^2_1$ and $\sigma^2_2$ under the quadratic loss $(\hat{\mu} - \mu)^2/\sigma^2_1$. A family of minimax estimators with smaller risk than the sample mean in the first population is given, out of which admissible minimax estimators are developed. A class of better estimators of $\mu$ under squared-error loss, which is wider than found by Bhattacharya, is obtained.

随机变量集合内部依赖程度的度量AI 解读

To measure dependence in a set of random variables, a multivariate analog of maximal correlation is considered. This consists of transforming each of the variables so that the largest partial sums of the eigenvalues of the resulting correlation matrix is maximized. A "maximalized" measure of association obtained in this manner permits statements to be made about the strength of internal dependence exhibited by the random variables. It is shown, under a weak regularity…

正相依的一种偏序AI 解读

In this paper we introduce a partial ordering for positive dependent bivariate distributions. Our main result shows that tests of independence based on rank statistics such as Spearman's rho, Kendall's tau, Fisher-Yates' normal score statistic, van der Waerden's statistic and the quadrant statistic become more powerful under increasing positive dependence. In other words, these measures of positive dependence preserve the ordering stochastically in samples whenever it is…