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多元泊松均值估计及相关问题的完全类定理AI 解读

Basic decision theory for discrete random variables of the multivariate geometric (power series) type is developed. Some properties of Bayes estimators that carry over in the limit to admissible estimators are obtained. A stepwise generalized Bayes representation of admissible estimators is developed with estimation of the mean of a multivariate Poisson random variable in mind. The development carries over to estimation of the mean of a multivariate negative Binomial random…

多元 Poisson 均值的全部可容许线性估计量AI 解读

Admissible linear estimators $Mx + \gamma$ must be pointwise limits of Bayes estimators. Using properties of Bayes estimators preserved by taking limits, the structure of $M$ and $\gamma$ can be determined. Among $M, \gamma$ with this structure, a necessary and sufficient condition for admissibility is obtained. This condition is applied to the case of linear (mixture) models. It is shown that only the most trivial such models admit linear estimators of full rank which are…

序贯假设检验的完全类AI 解读

We consider problems of sequential testing when the loss function is the sum of a component due to an error in the terminal decision and a cost of observation component. In all cases we establish a characterization of a complete class or an essentially complete class. In order to obtain such results for testing a null hypothesis against an alternative hypothesis we establish complete class results for testing the closure of the null hypothesis against the closure of the…

序贯 Bahadur 效率AI 解读

The notion of Bahadur efficiency for test statistics is extended to the sequential case and illustrated in the specific context of testing one-sided hypotheses about a normal mean. An analog of Bahadur's theorem on the asymptotic optimality of the likelihood ratio statistic is seen to hold in the normal case. Some possible definitions of attained level for a sequential experiment are considered.

未知协方差与任意二次损失下正态均值向量的极小极大估计AI 解读

Let $X$ be an observation from a $p$-variate normal distribution $(p \geqq 3)$ with mean vector $\theta$ and unknown positive definite covariance matrix $\not\Sigma$. It is desired to estimate $\theta$ under the quadratic loss $L(\delta, \theta, \not\Sigma) = (\delta - \theta)^tQ(\delta - \theta)/\operatorname{tr} (Q\not\Sigma)$, where $Q$ is a known positive definite matrix. Estimators of the following form are considered: $\delta^c(X, W) = (I - c\alpha Q^{-1}W^{-1}/(X^tW^{-1}X))X,$…

严格单调似然比下的完全类定理及其应用AI 解读

Suppose a random variable has a density belonging to a one parameter family which has strict monotone likelihood ratio. For inference regarding the parameter (or a monotone function of the parameter) consider the loss function to be bowl shaped for each fixed parameter and also to have each action be a "point of increase" or a "point of decrease" for some value of the parameter. Under these conditions, given any nonmonotone decision procedure, a unique monotone procedure is…

共同均值的点估计、置信估计与区组间信息的回收AI 解读

Consider the problem of estimating a common mean of two independent normal distributions, each with unknown variances. Note that the problem of recovery of interblock information in balanced incomplete blocks designs is such a problem. Suppose a random sample of size $m$ is drawn from the first population and a random sample of size $n$ is drawn from the second population. We first show that the sample mean of the first population can be improved on (with an unbiased…

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