A threshold separates the regimes
The central finding is a threshold in a mathematical model of composite binary hypothesis testing. When the Type I error constraint is required to decay exponentially at a rate below that threshold, the optimal Type II error decays to zero; above it, Type II error converges to one.
The setup deals with two nonempty classes of probability laws, a null class and an alternative class, rather than one fully specified pair. It considers an n-observation sample and imposes the Type I requirement uniformly over the null class while examining the best possible Type II error. In the paper, Type I is the constrained error, while Type II is the error whose optimum is being bounded.
How the bounds are built
To obtain an achievability bound, the preprint uses a joint Rényi projection, a way to select a shared mathematical representative of the two classes, and then builds one projected log-likelihood-ratio threshold test. The construction controls errors uniformly over both classes and does not assume that a least-favourable pair is available.
The converse bound follows a different route. It reduces the composite question to simple binary pairs and optimizes a Rényi converse over orders greater than one. Together, the two constructions put an explicit upper and lower frame around the finite-sample Type II error.
The error picture becomes more precise
In the achievable regime, the result is more than a bound on a finite sample. Under the stated finite-alphabet assumptions, the exact composite Type II exponent is the smallest exponent among the corresponding simple binary pairs. The selected Rényi order is unique, and the exponent is continuously differentiable with respect to the Type I decay rate.
On the other side of the threshold, the paper characterizes the rate at which Type II error approaches one. The governing quantity is the largest strong-converse exponent among the simple pairs. The same class-based testing problem therefore has an asymptotic description in both regimes, although the relevant extremum changes from a smallest exponent to a largest one.
Changing the Type I requirement changes the direction of the relevant KL calculation. With a fixed Type I constraint, rather than one that shrinks exponentially, the Type II exponent uses the KL direction from the null class to the alternative class. The threshold in the exponential-constraint regime instead comes from the alternative-to-null direction.
The framework also tracks terms that disappear if results are reported only on an exponential scale. It states that optimal Type II error has a polynomial prefactor multiplying its dominant exponential decay. That refinement makes the finite-sample estimate more detailed, but the supplied analysis does not identify the leading multiplicative constant.
Useful special cases, with clear limits
Least favourability appears under additional structure. When the stated stochastic-ordering conditions hold, the projected pair attains both worst-case composite error levels for every projected threshold test. For separated one-parameter natural exponential families, the Rényi projection selects the endpoint pair, which is least favourable at every sample size.
A numerical illustration shows how the split is used. In the first affine ternary example, the reported critical rate is 0.094. The example uses 0.033 for an achievable-regime rate and 0.142 for a converse-regime rate, putting the two calculations below and above the reported threshold.
The scope is mathematical, and the assumptions are substantial. Exact exponent results require compact, convex classes with full support on a finite alphabet, while the general achievability theorem requires common-measure, convexity, weak-compactness and support conditions.
The results do not settle every version of the problem. The projected threshold family is not claimed to solve the unrestricted composite problem in general, and finite-sample least favourability is not guaranteed for arbitrary composite classes. The numerical examples are illustrative calculations rather than evidence from real-world data. The document is an arXiv preprint.
Paper data and sources
Original title: Finite Sample Bounds for Composite Hypothesis Testing
Authors: El{í}as Vera-Sig{ü}enza, Amedeo Roberto Esposito
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
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