In this article the author considers the statistical hypotheses testing to make decision among hypotheses concerning many families of probability distributions. The statistician would like to control the overall error rate relative to draw statistically valid conclusions from each test, while being as efficient as possible. The familywise error (FWE) rate metric and the hypothesis test procedure while controlling both the type I and II FWEs are generalized. The proposed procedure shows simultaneous more reliability and less conservative error control relative to fixed sample and other recently proposed sequential procedures. Also, the characteristics of logarithmically asymptotically optimal (LAO) hypotheses testing are studied. The purpose of research is to express the optimal functional relation among the reliabilities of LAO hypotheses testing and to judge with FWE metric.