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.

翻译：本文研究在序贯设置下联合检验多个假设并估计底层分布随机参数的问题。目标是在平均样本量尽可能少、且检测与估计误差均低于预设阈值的条件下，联合推断真实假设与真实参数。基于对底层模型的温和假设，我们提出了一种渐近最优方法——即当容许的检测与估计误差水平趋近于零时达到最优的方法。该渐近最优停止规则的实现计算成本低廉，因此适用于高维数据。我们进一步提出一种投影拟牛顿法，用于优化参数化即时成本函数的系数，从而精确满足约束条件。所提理论通过数值算例得到验证。