  We investigate the problem of jointly testing two hypotheses and estimating a
random parameter based on data that is observed sequentially by sensors in a
distributed network. In particular, we assume the data to be drawn from a
Gaussian distribution, whose random mean is to be estimated. Forgoing the need
for a fusion center, the processing is performed locally and the sensors
interact with their neighbors following the consensus+innovations approach. We
design the test at the individual sensors such that the performance measures,
namely, error probabilities and mean-squared error, do not exceed pre-defined
levels while the average sample number is minimized. After converting the
constrained problem to an unconstrained problem and the subsequent reduction to
an optimal stopping problem, we solve the latter utilizing dynamic programming.
The solution is shown to be characterized by a set of non-linear Bellman
equations, parametrized by cost coefficients, which are then determined by
linear programming as to fulfill the performance specifications. A numerical
example validates the proposed theory.
