We study the hypothesis testing problem where the observed samples need not come from either of the specified hypotheses (distributions). In such a situation, we would like our test to be robust to this misspecification and output the distribution closer in Hellinger distance. If the underlying distribution is close to being equidistant from the hypotheses, then this would not be possible. Our main result is quantifying how close the underlying distribution has to be to either of the hypotheses. We also study the composite testing problem, where each hypothesis is a Hellinger ball around a fixed distribution. A generalized likelihood ratio test is known to work for this problem. We give an alternate test for the same.

翻译：我们研究假设检验问题，其中观测样本可能并非来自任一指定假设（分布）。在此情况下，我们希望检验方法对这种设定错误具有鲁棒性，并能输出Hellinger距离更接近的分布。若潜在分布接近与各假设等距，则此目标将无法实现。我们的主要成果在于量化潜在分布需要与任一假设接近到何种程度。同时我们研究复合检验问题，其中每个假设均为固定分布周围的Hellinger球。已知广义似然比检验适用于该问题，我们为此提出了另一种检验方法。