We study the behavior of mixture stopping rules in the one-sided sequential
hypothesis testing problem with a simple null hypothesis and a composite
alternative hypothesis. When the alternative hypothesis consists of a finite
set of probability measures, we show how to select a particular mixing
distribution in order to obtain a nearly minimax mixture test in the sense of
minimizing the maximal Kullback-Leibler information.

Several variations of the Shiryaev-Roberts detection procedure in the context
of the simple changepoint problem are considered: starting the procedure at
$R_0=0$ (the original Shiryaev-Roberts procedure), at $R_0=r$ for fixed $r>0$,
and at $R_0$ that has a quasi-stationary distribution. Comparisons of operating
characteristics are made. The differences fade as the average run length to
false alarm tends to infinity.

The CUSUM procedure is known to be optimal for detecting a change in
distribution under a minimax scenario, whereas the Shiryaev-Roberts procedure
is optimal for detecting a change that occurs at a distant time horizon. As a
simpler alternative to the conventional Monte Carlo approach, we propose a
numerical method for the systematic comparison of the two detection schemes in
both settings, i.e., minimax and for detecting changes that occur in the
distant future.