  We consider a remote source coding problem subject to a {distortion
function}. Contrary to the use of the classical separable distortion criterion,
herein we consider the more general, $f$-separable distortion measure and study
its implications on the characterization of the minimum achievable rates (also
called $f$-separable indirect rate distortion function (iRDF)) under both
excess and average distortion constraints. First, we provide a single-letter
characterization of the optimal rates subject to an excess distortion using
properties of the $f$-separable distortion. Our main result is a single-letter
characterization of the $f$-separable iRDF subject to an average distortion
constraint. As a consequence of the previous results, we also show a series of
equalities that hold using either indirect or classical RDF under $f$-separable
excess or average distortions. We corroborate our results with two application
examples in which new closed-form solutions are derived, and based on these, we
also recover known special cases.
