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卡方数据的经验贝叶斯方法AI 解读

In a thought-provoking paper, Efron investigated the merit and limitation of an empirical Bayes method to correct selection bias based on Tweedie’s formula first reported in the study by Robbins. The exceptional virtue of Tweedie’s formula for the normal distribution lies in its representation of selection bias as a simple function of the derivative of log marginal likelihood. Since the marginal likelihood and its derivative can be estimated from the data directly without…

具有先后约束的一类序贯控制问题的最优策略AI 解读

Consider the following multi-phase project management problem. Each project is divided into several phases. All projects enter the next phase at the same point chosen by the decision maker based on observations up to that point. Within each phase, one can pursue the projects in any order. When pursuing the project with one unit of resource, the project state changes according to a Markov chain. The probability distribution of the Markov chain is known up to an unknown…

Motivated by an application in computerized adaptive tests, we consider the following sequential design problem.There are $J$ jobs to be processed according to a predetermined order. A single machine is available to process these $J$ jobs. Each job under processing evolves stochastically as a Markov chain and earns rewards as it is processed, not otherwise. The Markov chain has transition probabilities parameterized by an unknown parameter $\theta$. The objective is to…

固定与自适应设计下广义线性模型最大拟似然估计的强一致性AI 解读

Strong consistency for maximum quasi-likelihood estimators of regression parameters in generalized linear regression models is studied. Results parallel to the elegant work of Lai, Robbins and Wei and Lai and Wei on least squares estimation under both fixed and adaptive designs are obtained. Let $y_1,\dots, y_n$ and $x_1,\dots, x_n$ be the observed responses and their corresponding design points ($p \times 1$ vectors), respectively. For fixed designs, it is shown that if the…

不可逆自适应分配规则AI 解读

Motivated by a scheduling problem arising from serial sacrifice experiments, the asymptotic efficiency of irreversible adaptive allocation rules is studied. The asymptotic lower bound for the regret of an adaptive allocation rule is characterized by the minimum of a linear program. Based on a class of one-sided sequential tests, asymptotically efficient rules which achieve the lower bound are constructed. The conditions necessary for this construction are verified in the…