On a problem in pharmaceutical statistics and the iteration of a peculiar nonlinear operator in the upper complex halfplane

In this paper we discuss the relationship between a well known problem of pharmaceutical statistics, the bioequivalence problem, and a purely geometrical problem, the construction of a certain set in the upper complex halfplane C+. This set has to obey certain peculiar geometric properties due to the restriction of defining the critical region of an unbiased test for the bioequivalance problem. The case for large nominal level alpha and large sample sizes was solved by Brown, Hwang & Munk [2] whereas the existence of such a set for small alpha and sample sizes is still an open problem. We will review in this paper recent developments and highlight some serious practical consequences if such a set would exist. To this end the concept of coherency of a test is introduced. It is shown that unbiased coherent regions do not exist if sample size and level are small.