We investigate the problem of jointly testing a pair of composite hypotheses and, depending on the test result, estimating a random parameter under distributional uncertainties. Specifically, it is assumed that the distribution of the data given the parameter of interest, is subject to uncertainty. Both, a Bayesian formulation and a Neyman-Pearson-like formulation, are considered. It is shown that the optimal policy induces an $f$-similarity that must be maximized to identify the least favorable distributions. Besides the general results, the implementation is investigated using a band-type uncertainty model. For designing the minimax procedures, existing algorithms are modified to increase convergence speed while maintaining numerical stability. The proposed theory is supplemented by numerical results for both formulations.

翻译：研究联合检验一对复合假设并依据检验结果在分布不确定条件下估计随机参数的问题。具体而言，假定数据在给定感兴趣参数后的分布存在不确定性。本文同时考虑了贝叶斯形式和类奈曼-皮尔逊形式的两种建模方法。研究表明，最优策略会诱导出一种$f$-相似性，该相似性必须通过最大化来识别最不利分布。除了一般性结论外，还采用带状不确定模型探究了实现方法。为设计极小极大流程，对现有算法进行了改进，在保持数值稳定性的同时提升收敛速度。所提出的理论通过两种建模形式的数值结果进行了补充验证。