  We investigate the problem of jointly testing multiple hypotheses and
estimating a random parameter of the underlying distribution in a sequential
setup. The aim is to jointly infer the true hypothesis and the true parameter
while using on average as few samples as possible and keeping the detection and
estimation errors below predefined levels. Based on mild assumptions on the
underlying model, we propose an asymptotically optimal procedure, i.e., a
procedure that becomes optimal when the tolerated detection and estimation
error levels tend to zero. The implementation of the resulting asymptotically
optimal stopping rule is computationally cheap and, hence, applicable for
high-dimensional data. We further propose a projected quasi-Newton method to
optimally choose the coefficients that parameterize the instantaneous cost
function such that the constraints are fulfilled with equality. The proposed
theory is validated by numerical examples.
