  We focus on the problem of modulating a parameter onto a power-limited signal
transmitted over a discrete-time Gaussian channel and estimating this parameter
at the receiver. Considering the well-known threshold effect in non-linear
modulation systems, our approach is the following: instead of deriving upper
and lower bounds on the total estimation error, which weigh both weak-noise
errors and anomalous errors beyond the threshold, we separate the two kinds of
errors. In particular, we derive upper and lower bounds on the best achievable
trade-off between the exponential decay rate of the weak-noise expected error
cost and the exponential decay rate of the probability of the anomalous error
event, also referred to as the outage event. This outage event is left to be
defined as part of the communication system design problem. Our achievability
scheme, which is based on lattice codes, meets the lower bound at the high
signal-to-noise (SNR) limit and for a certain range of trade-offs between the
weak--noise error cost and the outage exponent.
