  The problem of multiple endpoint testing for k endpoints is treated as a 2^k
finite action problem. The loss function chosen is a vector loss function
consisting of two components. The two components lead to a vector risk. One
component of the vector risk is the false rejection rate (FRR), that is, the
expected number of false rejections. The other component is the false
acceptance rate (FAR), that is, the expected number of acceptances for which
the corresponding null hypothesis is false. This loss function is more
stringent than the positive linear combination loss function of Lehmann [Ann.
Math. Statist. 28 (1957) 1-25] and Cohen and Sackrowitz [Ann. Statist. (2005)
33 126-144] in the sense that the class of admissible rules is larger for this
vector risk formulation than for the linear combination risk function. In other
words, fewer procedures are inadmissible for the vector risk formulation. The
statistical model assumed is that the vector of variables Z is multivariate
normal with mean vector \mu and known intraclass covariance matrix \Sigma. The
endpoint hypotheses are H_i:\mu_i=0 vs K_i:\mu_i>0, i=1,...,k. A
characterization of all symmetric Bayes procedures and their limits is
obtained. The characterization leads to a complete class theorem. The complete
class theorem is used to provide a useful necessary condition for admissibility
of a procedure. The main result is that the step-up multiple endpoint procedure
is shown to be inadmissible.
