  We consider the problem of modulation and estimation of a random parameter
$U$ to be conveyed across a discrete memoryless channel. Upper and lower bounds
are derived for the best achievable exponential decay rate of a general moment
of the estimation error, $\bE|\hat{U}-U|^\rho$, $\rho\ge 0$, when both the
modulator and the estimator are subjected to optimization. These exponential
error bounds turn out to be intimately related to error exponents of channel
coding and to channel capacity. While in general, there is some gap between the
upper and the lower bound, they asymptotically coincide both for very small and
for very large values of the moment power $\rho$. This means that our
achievability scheme, which is based on simple quantization of $U$ followed by
channel coding, is nearly optimum in both limits. Some additional properties of
the bounds are discussed and demonstrated, and finally, an extension to the
case of a multidimensional parameter vector is outlined, with the principal
conclusion that our upper and lower bound asymptotically coincide also for a
high dimensionality.
