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针对似然原则的一个反例AI 解读

SUMMARY A counter-example against the likelihood principle is strengthened and shown to hold also for a finite sample space.

回归估计量的可容许性AI 解读

SUMMARY Admissibility of the regression estimator is established for two types of sampling designs: single stage sampling using probability proportional to a size variable and double sampling.

样本均值作为有限总体均值估计量的联合可容许性AI 解读

When samples are taken independently from different populations, the sample means are jointly admissible for the population means with the squared error as loss function. The result supplements a previous result that when there are many variate values associated with the population units, the sample means of the variate values are jointly admissible for the population means for general loss functions.

有限总体估计量的一则札记AI 解读

Two previous results of the author, regarding the admissibility of the sample mean as estimator of the population mean in the class of all estimators for arbitrary sampling designs with the squared error as loss function, and for a wide class of loss functions, are shown to remain valid in the more general case when the population means of many variates are simultaneously under estimation.

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…

关于 Birnbaum 似然原则理论的一则注记AI 解读

The principles of sufficiency and conditionality are shown by Birnbaum [1] to imply the principle of likelihood. But in his formulation it is necessary to adopt in the continuous case, a definition of sufficiency which differs from the conventional definition. The objections against the new definition are set out and it is shown that the conventional definition can be adhered to by taking into consideration that in practice there is always a finite limit to the accuracy with…

关于 Birnbaum 似然原则理论的札记AI 解读

The principles of sufficiency and conditionality are shown by Birnbaum [1] to imply the principle of likelihood. But in his formulation it is necessary to adopt in the continuous case, a definition of sufficiency which differs from the conventional definition. The objections against the new definition are set out and it is shown that the conventional definition can be adhered to by taking into consideration that in practice there is always a finite limit to the accuracy with…

Uniformly most accurate level $1 - \alpha$ confidence procedures for a linear function $\mu + \lambda\sigma^2$ with known $\lambda$ for the parameters of a normal distribution defined by Land were previously shown for both the one-sided and two-sided procedures to be always intervals for $\nu \geqq 2, \nu$ being the number of degrees of freedom for estimating $\sigma^2$. These results are shown in this paper to hold also in the case $\nu = 1$. During the course of the…

已知前 N 阶矩时随机概率的极小极大估计AI 解读

$X$ and $\Theta$ are random variables; for given $\Theta = \theta$, the conditional distribution of $X$ is binomial with parameters $N$ and $\theta$; the first $N$ moments of $\Theta$ are known. An estimate of $\Theta$ is made based on the observed value of $X$, the risk being defined in terms of squared error loss. It is shown that as conjectured by H. Robbins, the ratio of the Bayes risk to the minimax risk for all possible distributions of $\Theta$ uniformly tends to…

有限总体均值置信区间集的强可容许性AI 解读

In a previous paper the admissibility was proved of a generalized version of the confidence sets, commonly used in practice, which are based on the sample mean and the sample standard deviation. A stronger definition of admissibility is obtained, if instead of the length of the interval for each individual observable sample $s$, only the expected length for all samples together for each $x \in R_N$, is taken into consideration for defining the permissible alternatives to the…

单侧贝叶斯区间与频率学派置信区间不相容的一则说明AI 解读

Summary With a proper prior distribution the one-sided Bayes confidence interval for the scalar parameter θ in the density function f(x, θ) never coincides with the corresponding confidence interval based on frequency, whether the experiment is of fixed sample size or sequential.

正态均值与方差线性函数的双侧置信区间札记AI 解读

The confidence sets for linear functions $\mu + \lambda\sigma^2$ of the mean $\mu$ and variance $\sigma^2$ of a normal distribution, defined in terms of the uniformly most powerful unbiased level $\alpha$ tests of hypotheses of form $H_0(\lambda, m): \mu + \lambda\sigma^2 = m$ against the two-sided alternative $H_1(\lambda, m): \mu + \lambda\sigma^2 \neq m$ for $-\infty < m < \infty$, for fixed $\alpha$ and $\lambda$, are shown to be intervals if the number of degrees of…

有限总体抽样中的分布无关充分性AI 解读

Summary The concept of distribution-free sufficiency, (DF-sufficiency for short) for estimation in survey sampling, which has been introduced by Godambe (1966), is investigated. Uncorrelatedness in Godambe's definition is shown to be equivalent to statistical independence, as is claimed on the author's behalf in the appendix to Godambe's paper. But the second result claimed there, that every DF-sufficient estimate must necessarily be of the form a(s).∑i∈sxi+b(s) is false, as…