  The problem of many hypotheses logarithmically asymptotically optimal (LAO)
testing for a model consisting of three or more independent objects is solved.
It is supposed that $M$ probability distributions are known and each object
independently of others follows to one of them. The matrix of asymptotic
interdependencies (reliability--reliability functions) of all possible pairs of
the error probability exponents (reliabilities) in optimal testing for this
model is studied.
  This problem was introduced (and solved for the case of two objects and two
given probability distributions) by Ahlswede and Haroutunian. The model with
two independent objects with $M$ hypotheses was explored by Haroutunian and
Hakobyan.
