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经验特征函数及其应用AI 解读

Certain probability properties of $c_n(t)$, the empirical characteristic function $(\operatorname{ecf})$ are investigated. More specifically it is shown under some general restrictions that $c_n(t)$ converges uniformly almost surely to the population characteristic function $c(t).$ The weak convergence of $n^{\frac{1}{2}}(c_n(t) - c(t))$ to a Gaussian complex process is proved. It is suggested that the ecf may be a useful tool in numerous statistical problems. Application of…

复合多项分布似然函数的单峰性：证明I. J. Good的猜想AI 解读

I. J. Good's 1965 conjecture of the unimodality of the likelihood function of a symmetrical compound multinomial distribution is proved by the variation-diminishing property of the Laplace transform. The result is a special case of a several sample version with asymmetrical compounding Dirichlet distributions. The technique of proof is applied to yield similar results for the negative binomial distribution and a two point mixture of Poissons.

固定或缩放半径置信集：固定样本量情形AI 解读

Confidence sets in the form of balls of fixed or scaled radius with respect to an arbitrary pseudometric are considered. Easily computable lower bounds for the radius of these confidence sets are derived. As by-products, bounds for the minimax risk are given and a method of deriving multiple decision procedures from point estimators is obtained.

相关样本情形下的稳健估计AI 解读

To analyze the effect of correlation in random samples on the performance of estimators of location, small correlation approximations for the asymptotic variance are found. Approximately optimal estimators (in the asymptotic minimax sense of Huber) are presented and compared to other estimators in terms of maximum asymptotic variance over the class of $\varepsilon$-contaminated normals. The presence of relatively small correlation can drastically inflate variances, and the…

最近邻密度估计量的相合性性质AI 解读

Let $X_1, X_2,\cdots$ be $R^p$-valued random variables having unknown density function $f$. If $K$ is a density on the unit sphere in $R^p, \{k(n)\}$ a sequence of positive integers such that $k(n) \rightarrow \infty$ and $k(n) = o(n)$, and $R(k, z)$ is the distance from a point $z$ to the $k(n)$th nearest of $X_1,\cdots, X_n$, then $f_n(z) = (nR(k, z)^p)^{-1} \sum K((z - X_i)/R(k, z))$ is a nearest neighbor estimator of $f(z).$ When $K$ is the uniform kernel, $f_n$ is an…

密度函数及其导数估计的界AI 解读

Lower bounds on the radius of various confidence sets for density functions and their derivatives are determined. In each case investigated, the smallest radius obtainable and the radius actually obtained by using known estimates exhibit the same dependency on the fixed sample size $n$. The lower bounds are derived using methods in Meyer (1976). Although only a few combinations of pseudometric and density class are considered here, the techniques illustrated can be used…

有限参数取值下的近似最优序贯检验AI 解读

Combinations of one-sided sequential probability ratio tests (SPRT's) are shown to be "nearly optimal" for problems involving a finite number of possible underlying distributions. Subject to error probability constraints, expected sample sizes (or weighted averages of them) are minimized to within $o(1)$ asymptotically. For sequential decision problems, simple explicit procedures are proposed which "do exactly what a Bayes solution would do" with probability approaching one…

Consider a modified version of the empirical Bayes decision problem where the component problems in the sequence are not identical in that the sample size may vary. In this case there is not a single Bayes envelope $R(\bullet)$, but rather a sequence of envelopes $R^{m(n)}(\bullet)$ where $m(n)$ is the sample size in the $n$th problem. Let $\mathbf{\theta} = (\theta_1, \theta_2, \cdots)$ be a sequence of i.i.d. $G$ random variables and let the conditional distribution of the…

渐进删失似然比统计量的弱收敛及寿命试验应用AI 解读

Progressive censoring schemes are often adopted in clinical trials and life testing problems with a view to monitoring the experiment from the start with the objective of a possible early termination of the experiment depending on the cumulative evidence at its various stages. Along with a basic martingale property, a Wiener process approximation for progressively censored likelihood ratio statistics is established here and the same is incorporated in the formulation of some…

均匀分布次序统计量线性组合渐近正态性的刻画AI 解读

A necessary and sufficient condition for the asymptotic normality of linear combinations of order statistics from the uniform distribution over [0, 1] is derived. The condition implies, that the variances of all weighted extremes are asymptotically zero compared with the total variance of the sum, which conversely, in the case of nonnegative constants, is sufficient for the asymptotic normality.

复合泊松过程的最大似然估计AI 解读

The problem of estimating the compounding distribution of a compound Poisson process from independent observations of the compound process has been analyzed by Tucker (1963). A maximum likelihood method is proposed. The existence, uniqueness and convergence of the resulting estimator are derived. One obtains practical solutions by means of a very simple algorithm which is briefly described. A numerical example is presented in the risk business framework.

非参数模型的描述统计：第三篇，离散程度AI 解读

Measures of dispersion are defined as functionals satisfying certain equivariance and order conditions. In the main part of the paper attention is restricted to symmetric distributions. Different measures are compared in terms of asymptotic relative efficiency, i.e., the inverse ratio of their standardized variances. The efficiency of a trimmed to the untrimmed standard deviation turns out not to have a positive lower bound even over the family of Tukey models. Positive…

充分正则最优性准则下的收敛设计序列AI 解读

For an optimality criterion function $\Phi$ and a design $\xi_n$, approximate $\Phi(\mathbf{M}(\xi))$ near $\xi_n$ by a quadratic Taylor expansion, let $\xi_n + \eta$ minimize this approximation, and let $\xi_{n+1} = \xi_n + \alpha\eta$, with $\alpha$ minimizing $\Phi(\mathbf{M}(\xi_{n+1}))$. If $\Phi$ satisfies regularity conditions, including strict convexity, possession of three continuous derivatives, and finiteness only for nonsingular $\mathbf{M}$, then $\mathbf{M}(\xi_n)$…

高次数多项式回归的最优设计AI 解读

Polynomial regression of degree $n$ on an interval is considered. Optimal designs $\xi_n$ are discussed for various optimality criteria. The behavior of $\xi_n$ for large $n$ is investigated and comparisons of $\xi_n$ with the limiting design $\xi_0$ are made.

两状态分量复合决策问题的渐近解：贝叶斯方法与比例参数的弥散先验AI 解读

Gilliland and Hannan (1974, Section 3) consider a general finite state compact risk component and reduce the problem of treating the asymptotic excess compound risk of Bayes compound rules to the question of $L_1$ consistency of certain induced estimators. This present paper considers the two state case and for several classes of diffuse symmetric priors on proportions establishes the $L_1$ consistency with rate. The rate $O(n^{-\frac{1}{2}})$ uniform in state sequences is…

对称稳定分布位置参数的渐近有效估计AI 解读

A well-known characteristic function representation of the family of symmetric stable distributions $\mathscr{F}$ indexes them with a location, scale, and type parameter. A sample of size $n$ is taken from an unknown member of $\mathscr{F}$. In this paper, an estimator of the location parameter is constructed which is maximum probability. This means that the estimator conventionally normalized converges in distribution to a normal distribution with zero mean and variance the…

已知方差时最大正态均值的序贯估计AI 解读

Given a procedure $\hat{\theta}^\ast$ for estimating the largest mean $\theta^\ast$ of $k$ normal populations with common known variance, it is desired to choose the common sample size $n$ so that the mean squared error (M.S.E.) of $\hat{\theta}^\ast$ does not exceed a given bound, $r$, regardless of the configuration of values of the $k$ means. Let $\Delta_1 \geqq \cdots \geqq \Delta_k = 0$ be the ordered values of $(\theta^\ast - \theta_i)$, where $\theta_1, \cdots, \theta_k$…

位置向量两个坐标的最佳不变估计量不可容许性AI 解读

Let $X = (X_1, X_2, X_3)$ be a random vector with density $f(x - \theta)$, where $\theta = (\theta_1, \theta_2, \theta_3)$ is unknown. It is desired to estimate $(\theta_1, \theta_2)$ using an estimator $(\delta_1(X), \delta_2(X))$, and under a loss function $L(\delta_1 - \theta_1, \delta_2 - \theta_2)$. (Note that $\theta_3$ is a nuisance parameter.) Under certain conditions on $f$ and $L$, it is shown that the best invariant estimator of $(\theta_1, \theta_2)$ is…

线性模型中的参数估计AI 解读

The first lecture in this series is devoted to a survey of contributions during the last five years to estimation of parameters by linear functions of observations in the Gauss-Markoff model. Some new results are also given. The classes of BLE (Bayes linear estimators) and ALE (admissible linear estimators) are characterized when the loss function is quadratic. It is shown that ALE's are either BLE's or limits of BLE's. Biased estimators like ridge and shrunken estimators…

多元容斥公式与相依随机变量的次序统计量AI 解读

A variant of the general multivariate inclusion-exclusion formula of Meyer (1969) is derived for the case where $K$ classes of events are considered and specific subsets of the events, one from each class, are related to one another by set inclusion. This result, in turn, yields a formula for the cumulative distribution function of any subset of order statistics from dependent random variables in terms of cumulative distribution functions of subsets of the unordered…

凹回归中的一致性AI 解读

For each $t$ in some subinterval $T$ of the real line let $F_t$ be a distribution function with mean $m(t)$. Suppose $m(t)$ is concave. Let $t_1, t_2, \cdots$ be a sequence of points in $T$ and let $Y_1, Y_2, \cdots$ be an independent sequence of random variables such that the distribution function of $Y_k$ is $F_{t_k}$. We consider estimators $m_n(t) = m_n(t; Y_1, \cdots, Y_n)$ which are concave in $t$ and which minimize $\sum^n_{i=1} \lbrack m_n(t_i; Y_1, \cdots, Y_n) - Y_i\rbrack^2$…

固定成本序贯抽样方案中的无偏估计AI 解读

Fixed cost sequential sampling schemes are introduced in this article. In these schemes units are observed sequentially according to a given sampling method until the total cost reaches a preassigned value; it is assumed that the cost of examining each unit is unknown in advance. It is shown how the notion of sufficiency in sampling can be used to construct unbiased estimators of population parameters under these schemes.

论克拉默–拉奥下界的达到AI 解读

It is often stated that the variance of an unbiased estimator of a function of a real parameter can attain the Cramer-Rao lower bound only if the family of distributions is a one-parameter exponential family. A rigorous proof of this statement, subject to certain regularity conditions, has been given by Wijsman. However, in general, the statement is not true. Assuming a revised set of regularity conditions it is shown here that there exists a more general class of…

有限总体中的一致最小方差估计AI 解读

In the literature one finds (at least) two approaches towards proving that the sample mean is uniformly minimum variance (UMV), among unbiased estimates that "ignore the labels," for the finite population mean: The "traditional approach" and the "scale-load approach." The identity of results under the two approaches extends to a more general setting, as shown in this paper: The Horvitz-Thompson estimate is UMV unbiased for any given fixed effective size design.