We investigate the problem of jointly testing multiple hypotheses and estimating a random parameter of the underlying distribution in a sequential setup. The aim is to jointly infer the true hypothesis and the true parameter while using on average as few samples as possible and keeping the detection and estimation errors below predefined levels. Based on mild assumptions on the underlying model, we propose an asymptotically optimal procedure, i.e., a procedure that becomes optimal when the tolerated detection and estimation error levels tend to zero. The implementation of the resulting asymptotically optimal stopping rule is computationally cheap and, hence, applicable for high-dimensional data. We further propose a projected quasi-Newton method to optimally choose the coefficients that parameterize the instantaneous cost function such that the constraints are fulfilled with equality. The proposed theory is validated by numerical examples.

中文翻译：

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用于顺序联合检测和估计的渐近最优程序

我们研究了联合测试多个假设并估计顺序设置中基础分布的随机参数的问题。目的是联合推断真实假设和真实参数，同时平均使用尽可能少的样本并将检测和估计误差保持在预定义水平以下。基于对底层模型的温和假设，我们提出了一种渐进最优程序，即当容忍的检测和估计误差水平趋于零时变得最优的程序。由此产生的渐近最优停止规则的实现计算成本低，因此适用于高维数据。我们进一步提出了一种投影拟牛顿方法来最优地选择参数化瞬时成本函数的系数，从而平等地满足约束。所提出的理论通过数值例子得到验证。