We consider context-sensitive (binary) hypothesis testing for i.i.d. observations under a multiplicative weight function. We establish the logarithmic asymptotic, as the sample size grows, of the optimal total loss (sum of type-I and type-II losses) and express the corresponding error exponent through a weighted Chernoff information between the competing distributions. Our approach embeds weighted geometric mixtures into an exponential family and identifies the exponent as the maximizer of its log-normaliser. We also provide concentration bounds for a tilted weighted log-likelihood and derive explicit expressions for Gaussian and Poisson models, as well as further parametric examples.

翻译：我们考虑在乘法权重函数下对独立同分布观测进行上下文敏感（二元）假设检验。随着样本量的增长，我们建立了最优总损失（第一类与第二类损失之和）的对数渐近性，并通过竞争分布之间的加权Chernoff信息表达了相应的误差指数。我们的方法将加权几何混合嵌入指数族，并将该指数确定为其对数正规化子的最大化值。我们还为倾斜加权对数似然提供了集中界，并针对高斯与泊松模型以及其他参数化示例推导了显式表达式。