Exact Group Sequential Methods for Estimating a Binomial Proportion

Algorithm 1

∇ Let 𝑘←0,  𝑙₀←Ψ_lb(ℐ_init) and 𝑢₀←Ψ_ub(ℐ_init).

∇ Let 𝒮₀←{ℐ_init} if 𝑢₀>𝛿. Otherwise, let 𝒮₀ be empty.

∇ While 𝒮_𝑘 is nonempty, 𝑙_𝑘<𝛿 and 𝑢_𝑘 is greater than max{𝑙_𝑘+𝜂,𝛿}, do the following:

   ⋄ Split each interval in 𝒮_𝑘 as two new intervals of equal length.

       Let 𝑆_𝑘 denote the set of all new intervals obtained from this splitting procedure.

   ⋄ Eliminate any interval ℐ from 𝑆_𝑘 such that Ψ_ub(ℐ)≤𝛿.

   ⋄ Let 𝒮_𝑘+1 be the set 𝑆_𝑘 processed by the above elimination procedure.

   ⋄ Let 𝑙_𝑘+1←max_{ℐ∈𝒮𝑘+1}Ψ_lb(ℐ) and 𝑢_𝑘+1←max_{ℐ∈𝒮𝑘+1}Ψ_ub(ℐ). Let 𝑘←𝑘+1.

∇ If 𝒮_𝑘 is empty and 𝑙_𝑘<𝛿, then declare max Ψ (ℐ_init)≤𝛿.

 Otherwise, declare max Ψ(ℐ_init)>𝛿.