A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations

In many cases an optimum or computationally convenient test of a simple hypothesis H₀ against a simple alternative H₁ may be given in the following form. Reject H₀ if Sₙ = ∑ⁿⱼ₌₁ Xⱼ k, where X₁, X₂, ⋯, Xₙ are n independent observations of a chance variable X whose distribution depends on the true hypothesis and where k is some appropriate number. In particular the likelihood ratio test for fixed sample size can be reduced to this form. It is shown that with each test of the above form there is associated an index ρ. If ρ₁ and ρ₂ are the indices corresponding to two alternative tests e = log ρ₁/log ρ₂ measures the relative efficiency of these tests in the following sense. For large samples, a sample of size n with the first test will give about the same probabilities of error as a sample of size $en$ with the second test. To obtain the above result, use is made of the fact that P(Sₙ na) behaves roughly like mⁿ where m is the minimum value assumed by the moment generating function of $X - a$. It is shown that if H₀ and H₁ specify probability distributions of X which are very close to each other, one may approximate ρ by assuming that X is normally distributed.