  Adaptive designs are commonly used in clinical and drug development studies
for optimum utilization of available resources. In this article, we consider
the problem of estimating the effect of the selected (better) treatment using a
two-stage adaptive design. Consider two treatments with their effectiveness
characterized by two normal distributions having different unknown means and a
common unknown variance. The treatment associated with the larger mean effect
is labeled as the better treatment. In the first stage of the design, each of
the two treatments is independently administered to different sets of $n_1$
subjects, and the treatment with the larger sample mean is chosen as the better
treatment. In the second stage, the selected treatment is further administered
to $n_2$ additional subjects. In this article, we deal with the problem of
estimating the mean of the selected treatment using the above adaptive design.
We extend the result of \cite{cohen1989two} by obtaining the uniformly minimum
variance conditionally unbiased estimator (UMVCUE) of the mean effect of the
selected treatment when multiple observations are available in the second
stage. We show that the maximum likelihood estimator (a weighted sample average
based on the first and the second stage data) is minimax and admissible for
estimating the mean effect of the selected treatment. We also propose some
plug-in estimators obtained by plugging in the pooled sample variance in place
of the common variance $\sigma^2$, in some of the estimators proposed by
\cite{misra2022estimation} for the situations where $\sigma^2$ is known. The
performances of various estimators of the mean effect of the selected treatment
are compared via a simulation study. For the illustration purpose, we also
provide a real-data application.
