A CLASS OF ASYMPTOTICALLY AND OPTIMALLY CLOSED SEQUENTIAL TESTS

Let X₁, X₂, … be a data sequence of i. i. d r. v.’ s with probability density function f（x,θ）,θ∈Θ.（T_n, n≥1） is a sequence of statistics for testing H_0: θ∈Θ₀ vs. H₁: θ∈Θ₁\triangleq \Theta-\Theta_0.. Let τ_b=inf（n≥m₀: T_n≥b or n≥m） where b＞0, 0＜m₀＜m＜∞ depend on b. Then we can construct a closed sequential test of H₀, which rejects H₀ if and only if Tτ_b≥b. In this paper, we prove that under certain conditions E_θτb/（-log （α（b））） has an asymptotically lower bound as b→∞, where α（b）= sup (P_θ(T_τb≥b) :θ∈Θ) is the significance level of the test. Especially for multi- dimensional exponential families, the Repeated Significance Test that leads to the lower bound is asymptotically optimal. Some asymptotically properties of the Fixed Sample Size Test are also obtained in this paper.