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贝叶斯同时估计中的序贯分配与可选停止AI 解读

Suppose random variables X 1, X 2, …, with distribution depending on parameter θ1, are observable from population 1, independent of random variables Y 1, Y 2, …, with distribution depending on parameter θ2, and observed from population 2. The simultaneous estimation of parameters θ1 and θ2 is of interest. An experiment is conducted where at each stage either an X or a Y is observed, and the experiment may be stopped at any stage (allocation and decision depending on…

单参数指数族的贝叶斯序贯估计AI 解读

The Bayesian sequential estimation problem for one-parameter exponential families is considered using loss related to the Fisher information and linear sampling cost. Tractible expressions for the Bayes estimator and the posterior expected loss are found, and the myopic, or one-step-ahead, stopping rule is defined. Sufficient conditions are given for optimality of the myopic procedure, and the myopic procedure is shown to be asymptotically optimal in all cases considered.

Dynkin恒等式在泊松过程速率的贝叶斯序贯估计中的应用AI 解读

Conditional on the value of $\theta, \theta > 0$, let $X(t), t \geqslant 0$, be a homogeneous Poisson process. Sequential estimation procedures of the form $(\sigma, \theta(\sigma))$ are considered. To measure loss due to estimation, a family of functions, indexed by $p$, is used: $L(\theta, \hat{\theta}) = \theta^{-p}(\theta - \hat{\theta})^2$, and the cost of sampling involves cost per arrival and cost per unit time. The notion of "monotone case" for total cost functions…

由二元函数加权和导出的置换统计量的渐近正态性AI 解读

Statistics of the form $H_n = \sum d_{ijn}h_n(X_i, X_j)$ are considered, where $X_1, X_2, \cdots$, are independent and identically distributed random variables, the diagonal terms, $d_{iin}$, are equal to zero, and $h_n(x, y)$ is a symmetric real valued function. The asymptotic normality of such statistics is proven and the result then combined with work of Jogdeo on statistics that are weighted sums of bivariate functions of ranks to find sufficient conditions for…

估计泊松过程到达率的贝叶斯序贯程序AI 解读

Let Wn, n = 0, 1, …, be the time until the nth arrival of a Poisson process with rate θ. Using loss L (θ, ) = θ−2(θ − )2 and sampling costs involving cost per arrival and cost per unit time, the Bayes sequential procedure is derived. The large-sample properties of the procedure are then studied in the classical framework, and the optimal stopping time N* is shown to be asymptotically equivalent to n*, that function of θ which minimizes the expected total cost given θ.…

估计泊松过程到达率的贝叶斯序贯程序AI 解读

Let Wn, n = 0, 1, …, be the time until the nth arrival of a Poisson process with rate θ. Using loss L (θ, ) = θ−2(θ − )2 and sampling costs involving cost per arrival and cost per unit time, the Bayes sequential procedure is derived. The large-sample properties of the procedure are then studied in the classical framework, and the optimal stopping time N* is shown to be asymptotically equivalent to n*, that function of θ which minimizes the expected total cost given θ.…

最大后验概率分类AI 解读

The problem of classifying each of $n$ observations to one of two sub-populations is considered. The classification rule examined chooses that classification with maximum posterior probability. Limiting behavior of the rule is given and several examples are presented which show that the rule can lead to classifying all observations to the same subpopulation. Three simulation studies are reported to indicate that this extreme behavior may occur in small samples.

贝叶斯分类的渐近结果AI 解读

We have a population composed of two subpopulations whose probability properties are described by known univariate distribution functions, $G(x)$ and $H(x)$, respectively. The probability of observing an individual from the first population is $\theta$, from the second is $1 - \theta$. We assume $\theta$ is a random variable with a prior distribution on (0, 1) and find the Bayes rule for classifying $n$ observations as from $G$ or from $H$ when the loss function is equal to…