  The paper studies distributed static parameter (vector) estimation in sensor
networks with nonlinear observation models and noisy inter-sensor
communication. It introduces \emph{separably estimable} observation models that
generalize the observability condition in linear centralized estimation to
nonlinear distributed estimation. It studies two distributed estimation
algorithms in separably estimable models, the $\mathcal{NU}$ (with its linear
counterpart $\mathcal{LU}$) and the $\mathcal{NLU}$. Their update rule combines
a \emph{consensus} step (where each sensor updates the state by weight
averaging it with its neighbors' states) and an \emph{innovation} step (where
each sensor processes its local current observation.) This makes the three
algorithms of the \textit{consensus + innovations} type, very different from
traditional consensus. The paper proves consistency (all sensors reach
consensus almost surely and converge to the true parameter value,) efficiency,
and asymptotic unbiasedness. For $\mathcal{LU}$ and $\mathcal{NU}$, it proves
asymptotic normality and provides convergence rate guarantees. The three
algorithms are characterized by appropriately chosen decaying weight sequences.
Algorithms $\mathcal{LU}$ and $\mathcal{NU}$ are analyzed in the framework of
stochastic approximation theory; algorithm $\mathcal{NLU}$ exhibits mixed
time-scale behavior and biased perturbations, and its analysis requires a
different approach that is developed in the paper.
