Weighted Chernoff information and optimal loss exponent in context-sensitive hypothesis testing
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2026
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| author | Kelbert, Mark Kalimulina, El'mira Yu. |
| author_facet | Kelbert, Mark Kalimulina, El'mira Yu. |
| contents | We study binary hypothesis testing for i.i.d. observations under a multiplicative context weight. For the optimal weighted total loss, defined as the sum of weighted type-I and type-II losses, we prove the logarithmic asymptotic $$ L_n^* = \exp\{-n D_C^{\mathrm{w}}(\mathbb{P}, \mathbb{Q}) + o(n)\}, \quad n \to \infty, $$ where $D_C^{\mathrm{w}}$ is the weighted Chernoff information. The single-letter form of the exponent relies on a structural assumption that the weight factorises across observations, $φ(x_1^n) = \prod_{i=1}^n φ(x_i)$; this restriction is essential for the single-letter representation and should be distinguished from the weaker qualitative description "multiplicative context weight". The proof embeds the weighted geometric mixtures $φp^αq^{1-α}$ into a likelihood-ratio exponential family and identifies the rate through its log-normaliser. We also derive concentration bounds for the tilted weighted log-likelihood, obtain closed forms for Gaussian, Poisson, and exponential models, and extend the exponent characterisation to finitely many hypotheses. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_08308 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weighted Chernoff information and optimal loss exponent in context-sensitive hypothesis testing Kelbert, Mark Kalimulina, El'mira Yu. Statistics Theory Information Theory Probability 62F03, 60F10 We study binary hypothesis testing for i.i.d. observations under a multiplicative context weight. For the optimal weighted total loss, defined as the sum of weighted type-I and type-II losses, we prove the logarithmic asymptotic $$ L_n^* = \exp\{-n D_C^{\mathrm{w}}(\mathbb{P}, \mathbb{Q}) + o(n)\}, \quad n \to \infty, $$ where $D_C^{\mathrm{w}}$ is the weighted Chernoff information. The single-letter form of the exponent relies on a structural assumption that the weight factorises across observations, $φ(x_1^n) = \prod_{i=1}^n φ(x_i)$; this restriction is essential for the single-letter representation and should be distinguished from the weaker qualitative description "multiplicative context weight". The proof embeds the weighted geometric mixtures $φp^αq^{1-α}$ into a likelihood-ratio exponential family and identifies the rate through its log-normaliser. We also derive concentration bounds for the tilted weighted log-likelihood, obtain closed forms for Gaussian, Poisson, and exponential models, and extend the exponent characterisation to finitely many hypotheses. |
| title | Weighted Chernoff information and optimal loss exponent in context-sensitive hypothesis testing |
| topic | Statistics Theory Information Theory Probability 62F03, 60F10 |
| url | https://arxiv.org/abs/2603.08308 |