We address 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. Continuing an earlier work, where the optimal trade-off between the weak-noise estimation performance and the outage probability (threshold-effect breakdown) was studied for a single (scalar) parameter, here we extend the derivation of the weak-noise estimation performance to the case of a multi-dimensional vector parameter. This turns out to be a non-trivial extension, that provides a few insights and it has some interesting implications, which are discussed in depth. Several modifications and extensions of the basic setup are also studied and discussed.