  A common problem faced in clinical studies is that of estimating the effect
of the most effective (e.g., the one having the largest mean) treatment among
$k~(\geq2)$ available treatments. The most effective treatment is adjudged
based on numerical values of some statistic corresponding to the $k$
treatments. A proper design for such problems is the so-called "Drop-the-Losers
Design (DLD)". We consider two treatments whose effects are described by
independent Gaussian distributions having different unknown means and a common
known variance. To select the more effective treatment, the two treatments are
independently administered to $n_1$ subjects each and the treatment
corresponding to the larger sample mean is selected. To study the effect of the
adjudged more effective treatment (i.e., estimating its mean), we consider the
two-stage DLD in which $n_2$ subjects are further administered the adjudged
more effective treatment in the second stage of the design. We obtain some
admissibility and minimaxity results for estimating the mean effect of the
adjudged more effective treatment. The maximum likelihood estimator is shown to
be minimax and admissible. We show that the uniformly minimum variance
conditionally unbiased estimator (UMVCUE) of the selected treatment mean is
inadmissible and obtain an improved estimator. In this process, we also derive
a sufficient condition for inadmissibility of an arbitrary location and
permutation equivariant estimator and provide dominating estimators in cases
where this sufficient condition is satisfied. The mean squared error and the
bias performances of various competing estimators are compared via a simulation
study. A real data example is also provided for illustration purposes.
