  This paper studies recursive composite hypothesis testing in a network of
sparsely connected agents. The network objective is to test a simple null
hypothesis against a composite alternative concerning the state of the field,
modeled as a vector of (continuous) unknown parameters determining the
parametric family of probability measures induced on the agents' observation
spaces under the hypotheses. Specifically, under the alternative hypothesis,
each agent sequentially observes an independent and identically distributed
time-series consisting of a (nonlinear) function of the true but unknown
parameter corrupted by Gaussian noise, whereas, under the null, they obtain
noise only. Two distributed recursive generalized likelihood ratio test type
algorithms of the \emph{consensus+innovations} form are proposed, namely
$\mathcal{CIGLRT-L}$ and $\mathcal{CIGLRT-NL}$, in which the agents estimate
the underlying parameter and in parallel also update their test decision
statistics by simultaneously processing the latest local sensed information and
information obtained from neighboring agents. For $\mathcal{CIGLRT-NL}$, for a
broad class of nonlinear observation models and under a global observability
condition, algorithm parameters which ensure asymptotically decaying
probabilities of errors~(probability of miss and probability of false
detection) are characterized. For $\mathcal{CIGLRT-L}$, a linear observation
model is considered and upper bounds on large deviations decay exponent for the
error probabilities are obtained.
