  We consider the problem of jointly testing multiple hypotheses and estimating
a random parameter of the underlying distribution. This problem is investigated
in a sequential setup under mild assumptions on the underlying random process.
The optimal method minimizes the expected number of samples while ensuring that
the average detection/estimation errors do not exceed a certain level. After
converting the constrained problem to an unconstrained one, we characterize the
general solution by a non-linear Bellman equation, which is parametrized by a
set of cost coefficients. A strong connection between the derivatives of the
cost function with respect to the coefficients and the detection/estimation
errors of the sequential procedure is derived. Based on this fundamental
property, we further show that for suitably chosen cost coefficients the
solutions of the constrained and the unconstrained problem coincide. We present
two approaches to finding the optimal coefficients. For the first approach, the
final optimization problem is converted into a linear program, whereas the
second approach solves it with a projected gradient ascent. To illustrate the
theoretical results, we consider two problems for which the optimal schemes are
designed numerically. Using Monte Carlo simulations, it is validated that the
numerical results agree with the theory.
