  This paper investigates the statistical properties of non-linear
transformations (NLT) of random variables, in order to establish useful tools
for estimation and information theory. Specifically, the paper focuses on
linear regression analysis of the NLT output and derives sufficient general
conditions to establish when the input-output regression coefficient is equal
to the \emph{partial} regression coefficient of the output with respect to a
(additive) part of the input. A special case is represented by zero-mean
Gaussian inputs, obtained as the sum of other zero-mean Gaussian random
variables. The paper shows how this property can be generalized to the
regression coefficient of non-linear transformations of Gaussian-mixtures. Due
to its generality, and the wide use of Gaussians and Gaussian-mixtures to
statistically model several phenomena, this theoretical framework can find
applications in multiple disciplines, such as communication, estimation, and
information theory, when part of the nonlinear transformation input is the
quantity of interest and the other part is the noise. In particular, the paper
shows how the said properties can be exploited to simplify closed-form
computation of the signal-to-noise ratio (SNR), the estimation mean-squared
error (MSE), and bounds on the mutual information in additive non-Gaussian
(possibly non-linear) channels, also establishing relationships among them.
