Hypothesis testing for the common mean of two normal distributions in the presence of an indifference zone

Summary

LetX ₁,...,X _m andY _t,...,Y be independent, random samples from populations which are N(θ,σ ²_x ) and N(θ,σ ²_y ), respectively, with all parameters unknown. In testingH ₀:θ=0 againstH ₁:θ≠0, thet-test based upon either sample is known to be admissible in the two-sample setting. If, however, one testsH ₀ againstH ₁:|θ|≧ε>0, with ε arbitrary, our main results show: (i) the construction of a test which is better than the particulart-test chosen, (ii) eacht-test is admissible under the invariance principle with respect to the group of scale changes, and (iii) there does not exist a test which simultaneously is better than botht-tests.