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检验两条简单回归直线是否平行的非参数方法AI 解读

This paper provides a means for testing the hypothesis that two simple regression lines are parallel, when the two sets of error terms have two arbitrary unknown continuous distributions. The non-parametric test which is developed here is analogous to the two-sample Wilcoxon test. At the end of the paper, two additional problems in non-parametric regression analysis are briefly referred to.

贝叶斯检验与贝叶斯风险的渐近行为AI 解读

In this paper the asymptotic behaviors of the Bayes test and Bayes risk are studied in both the one-sided and two-sided testing problems, where the independent observations are taken from one member of a one-parameter exponential family. Precise asymptotic expressions are found which show the Bayes procedure to be relatively insensitive to the prior distribution as the sample size increases to infinity.

两样本非线性模型的经验概率图与统计推断AI 解读

Let $X$ and $Y$ be two random variables with continuous distribution functions $F$ and $G$ and means $\mu$ and $\xi$. In a linear model, the crucial property of the contrast $\Delta = \xi - \mu$ is that $X + \Delta =_\mathscr{L} Y$, where $= _\mathscr{L}$ denotes equality in law. When the linear model does not hold, there is no real number $\Delta$ such that $X + \Delta = _\mathscr{L} Y$. However, it is shown that if parameters are allowed to be function valued, there is…

分布函数的 k 阶导数估计AI 解读

Estimation of the $k$th derivative of a df by means of the $k$th-order difference quotients of the empiric df is investigated. In particular, consistency conditions are given, the asymptotic bias, variance, and mean-squared error of the estimator are computed, and means of minimizing the latter are discussed.

位置参数的 Fisher 信息与 Pitman 估计量AI 解读

In this paper we consider estimation of the location parameter $\theta \in R^d$ based on a random sample from $(\theta + X, Y),$ where $X$ is a $d$-dimensional random vector, $Y$ is a random element of some measure space $\mathscr{Y},$ and $(X, Y)$ has a known distribution. We first define the Fisher information $\mathscr{J}(\theta + X, Y)$ and the inverse information $\mathscr{J}^-(\theta + X, Y)$ under no regularity conditions. The properties of these quantities are…

调查抽样中的超可容许性AI 解读

Two different definitions of hyperadmissibility are considered. The relation between hyperadmissibility and admissibility is discussed. Special attention is given to the hyperadmissibility of the Horvitz-Thompson estimator.

The problem of estimating powers of the variance of a normal distribution is considered when loss is essentially squared error. A class of minimax estimators is found by extending the techniques of Stein. It is shown, at least for estimating the variance, that a subclass of the above consists of generalized Bayes estimators.

随机球体的交集与球对称模型的非中心径向误差分布AI 解读

The paper gives an expression suitable for numerical computation and recursive relationships for the expected volume of the intersection of an $n$ dimensional sure sphere and an $n$ dimensional random sphere with a different radius, whose center follows a spherical distribution, the center of which does not coincide with that of the sure sphere. Several expressions are given for the distribution of the squared noncentral radial error when the vector observation follows a…

Wilcoxon 检验对某类样本间依赖的稳健性AI 解读

Robustness properties of the two-sample Wilcoxon test are established when the assumption of independence between samples is weakened to allow pairing. In particular, the test is shown to be (asymptotically) conservative when the bivariate distribution governing the pairing is positively quadrant dependent.

基于加权和的控制图AI 解读

In a continuous production process, samples of fixed size are taken at regular intervals of time and a statistic $X_n$ is computed from the $n$th sample, $n = 1,2, \cdots$. In this paper, we study process inspection schemes which stop the production and take corrective action with $N = \text{first} n \geqq 1$ such that $\sum^n_{i=1} c_{n-i} X_i \geqq h$, where $h$ is a preassigned constant and $c_0 \geqq c_1 \geqq \cdots \geqq c_{k-1} > 0 = c_k = c_{k+1} = \cdots$ is a…

协方差矩阵似然比检验在非零假设下的分布渐近展开AI 解读

In this paper, asymptotic expansions of the non-null distributions of the likelihood ratio criteria are obtained for testing the hypotheses: (a) $H_1: \Sigma = \sigma^2I, \sigma^2 > 0$, (b) $H_2: \Sigma_1 = \Sigma_2$, against alternatives which are close to the hypothesis. These expansions are of chi-square type. The first problem has been considered by Sugiura (1969) but because of the singularity at the hypothesis, his expansion will not be good for alternatives close to…

自回归信号加白噪声模型的估计AI 解读

If $x(\bullet)$ is a time series which may be written as $x(t) = s(t) + n(t)$ where $t$ is an integer, $s(\bullet)$ an autoregressive signal of order $q$ and $n(\bullet)$ white noise, then the model has $q + 2$ parameters. These are (i) the $q$ autoregressive parameters (ii) the residual variance of the autoregressive scheme and (iii) the variance of the white noise. A method is proposed to estimate the $q + 2$ parameters. This method is based on analogies with regression…

具有指数型子集最小值的多元分布AI 解读

The multivariate distribution of a set of random variables has exponential minimums if the minimum over each subset of the variables has an exponential distribution. Such distributions are shown equivalent to the more strongly structured multivariate exponential distributions described by Marshall and Olkin in 1967 in the sense that a multivariate exponential distribution can be found that gives the same marginal distribution for each minimum. The basic application of the…

位置模型中渐近有效的自适应秩估计AI 解读

This paper describes a new construction of uniformly asymptotically efficient rank estimates in the one and two-sample location models. The method adopted differs from van Eeden's (1970) earlier construction in three respects. First, the whole sample, rather than a vanishingly small fraction of the sample, is used in estimating the efficient score function. Secondly, a Fourier series estimator is used for the score function rather than a window estimator. Thirdly, the…

单样本秩统计量的不变性原理AI 解读

Analogous to the Donsker theorem on partial cumulative sums of independent random variables, for a broad class of one-sample rank order statistics, weak convergence to Brownian motion processes is studied here. A simple proof of the asymptotic normality of these statistics for random sample sizes is also presented. Some asymptotic results on renewal theory for one-sample rank order statistics are derived.

不放回抽样中样本和的概率不等式AI 解读

Upper bounds are established for the probability that, in sampling without replacement from a finite population, the sample sum exceeds its expected value by a specified amount. These are obtained as corollaries of two main results. Firstly, a useful upper bound is derived for the moment generating function of the sum, leading to an exponential probability inequality and related moment inequalities. Secondly, maximal inequalities are obtained, extending Kolmogorov's…

改进等变估计量的方法AI 解读

Techniques for improving on equivariant estimators are described. They may be applied, although without assurance of success, whatever be the family of underlying distributions. The loss function is required to satisfy an intuitively reasonable condition but is otherwise arbitrary. One of these techniques amounts to a sample space, orbit-by-orbit analysis of the conditional expected loss given the orbit. It yields, when successful, a "testimator". A second technique obtains…

非参数统计中的 Edgeworth 展开AI 解读

This is a survey of recent work on Edgeworth expansions for $(M)$ estimates, rank tests and some other statistics arising in nonparametric models. A Berry-Esseen theorem for $U$-statistics which seems to be new is also proved.

秩向量在 Bahadur 意义下的渐近充分性AI 解读

We shall consider the hypothesis of randomness under which two samples $X_1, \cdots, X_n$ and $Y_1, \cdots, Y_m$ have an identical but arbitrary continuous distribution. The vector of ranks $(R_1, \cdots, R_{n+m})$ will be shown to be asymptotically sufficient in the Bahadur sense for testing randomness against a general class of two-sample alternatives, simple ones as well as composite ones. In other words, the best exact slope will be attainable by rank statistics,…

正态分布协方差矩阵检验的性质AI 解读

The unbiasedness and the monotonicity property of the power functions of a class of tests for the equality of covariance matrices of two $p$-variate normal distributions have been studied. For testing $\Sigma = I_p$ in a $p$-variate normal distribution with mean vector $\mu$ and covariance matrix $\Sigma$, a class of tests is proposed and their power functions and admissibility are studied.

可交换先验分布下的抽样理论估计AI 解读

Godambe (1955) and Godambe and Joshi (1965) established the joint optimality of the Horvitz-Thompson estimates and the corresponding sampling designs with respect to a class of prior distributions which were "product measures." In the present paper the optimality is established replacing the "product measures" by prior distributions which are exchangeable in appropriately transformed variates. Furthermore, the optimality criterion is justified in terms of Chebychev's…

若干序贯程序停止时间的渐近正态性AI 解读

Two problems of sequential estimation, viz. the estimation of the mean of a normal distribution with unknown variance and the estimation of a binomial proportion are studied as the cost per observation tends to 0. For the first problem the asymptotic distribution of the stopping time of a procedure due to Robbins (1959) is shown to be normal. For the second problem the stopping time of a modification of Wald's (1951) procedure is asymptotically normal when the parameter is…

基于 Wilcoxon 型估计的序贯置信区间AI 解读

For the location parameter family of distributions $F(x - \theta)$ under some regularity conditions, a confidence interval for $\theta$ of fixed width $2d$ and given confidence coefficient $1 - \alpha$ in the limit as $d$ tends to zero is obtained using Hodges-Lehmann estimates based on Wilcoxon statistics. An upper bound on the average sample size is also given.

终止概率不为一时的最优对称性序贯检验AI 解读

A sequential test of a statistical hypothesis $H_0$ versus $H_1$ is said to be a test of Robbins type if there is a positive probability that the test will not stop if $H_0$ is true. Tests of this nature were introduced for testing the Bernoulli case by Darling and Robbins [1]; an earlier paper of Farrell [2] deals implicitly with the asymptotic expected sample size of such tests for testing the hypothesis $\theta = 0$ in the parametrized family of generalized density…