Weighted Chernoff information and optimal loss exponent in context-sensitive hypothesis testing
Provides theoretical foundations for weighted hypothesis testing, relevant to statisticians and information theorists, but is incremental as it extends known Chernoff information to a weighted setting.
The paper derives the optimal error exponent for context-sensitive hypothesis testing under multiplicative weights, expressed via weighted Chernoff information, and provides explicit formulas for Gaussian and Poisson models.
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.