We consider a design problem where experimental conditions (design points $X_i$) are presented in the form of a sequence of i.i.d.\ random variables, generated with an unknown probability measure $\mu$, and only a given proportion $\alpha\in(0,1)$ can be selected. The objective is to select good candidates $X_i$ on the fly and maximize a concave function $\Phi$ of the corresponding information matrix. The optimal solution corresponds to the construction of an optimal bounded design measure $\xi_\alpha^*\leq \mu/\alpha$, with the difficulty that $\mu$ is unknown and $\xi_\alpha^*$ must be constructed online. The construction proposed relies on the definition of a threshold $\tau$ on the directional derivative of $\Phi$ at the current information matrix, the value of $\tau$ being fixed by a certain quantile of the distribution of this directional derivative. Combination with recursive quantile estimation yields a nonlinear two-time-scale stochastic approximation method. It can be applied to very long design sequences since only the current information matrix and estimated quantile need to be stored. Convergence to an optimum design is proved. Various illustrative examples are presented.

中文翻译：

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用于细化实验设计的顺序在线子采样

我们考虑一个设计问题，其中实验条件（设计点 $X_i$）以一系列 iid\ 随机变量的形式呈现，以未知的概率度量 $\mu$ 生成，并且只有给定的比例 $\alpha\in可以选择 (0,1)$。目标是即时选择好的候选者 $X_i$ 并最大化相应信息矩阵的凹函数 $\Phi$。最优解对应于最优有界设计测度 $\xi_\alpha^*\leq\mu/\alpha$ 的构建，困难在于 $\mu$ 未知且 $\xi_\alpha^*$ 必须是在线构建。所提出的构造依赖于在当前信息矩阵中 $\Phi$ 的方向导数的阈值 $\tau$ 的定义，$\tau$ 的值由该方向导数分布的某个分位数固定。结合递归分位数估计产生非线性两时间尺度随机逼近方法。它可以应用于非常长的设计序列，因为只需要存储当前信息矩阵和估计的分位数。证明收敛于最佳设计。提供了各种说明性示例。