Sequential comparison of two binomials. Consider two sequences of binomial trials with probabilities of success p1 and p2 respectively, and let

ρ = (p2/q2) ÷ (p1/q1).

(i) If α<β, no test with fixed numbers of trials m and n for testing H : ρ = ρ0 can have power ≥ β against all alternatives with ρ = ρ1.

(ii) The following is a simple sequential sampling scheme leading to the desired result. Let the trials be performed in pairs of one of each kind, and restrict attention to those pairs in which one of the trials is a success and the other a failure. If experimentation is continued until N such pairs have been observed, the number of pairs in which the successful trial belonged to the first series has the binomial distribution b(π,N) with π = p1q2/(p1q2 +

P2q1)=1/(1 + ρ). A test of arbitrarily high power against ρ1 is therefore obtained by taking N large enough.

(iii) If p1/p2 = λ, use inverse binomial sampling to devise a test of H : λ = λ0 against K : λ>λ0.