Hypothesis Testing with Finite Statistics

Let X₁, X₂, ⋯ be a sequence of independent identically distributed random variables drawn according to a probability measure P. The two-hypothesis testing problem H₀: P = P₀ vs. H₁: P = P₁ is investigated under the constraint that the data must be summarized after each observation by an m-valued statistic Tₙε \1, 2, ⋯, m\, where Tₙ is updated according to the rule Tₙ₊₁ = fₙ(Tₙ, Xₙ₊₁). An algorithm with a four-valued statistic is described which achieves a limiting probability of error zero under either hypothesis. It is also demonstrated that a four-valued statistic is sufficient to resolve composite hypothesis testing problems which may be reduced to the form H₀:p > p₀ vs. H₁:p < p₀ where X₁, X₂, ⋯ is a Bernoulli sequence with bias p.