On reparametrization of loss functions used in estimation and the invariance principle

Abstract

The problem addressed in this note concerns the relationship between the minimizers of a given loss function parametrized in two different ways. The so-called “invariance principle” (IP) gives a simple answer to this problem in a special case. However, we consider a situation which is relevant to a number of applications of estimation theory in the signal processing and system identification fields, and where the IP cannot be used. We derive what we term an “extended invariance principle” (EXIP) which is valid under the conditions describing that situation and which reduces to the classical IP in the special conditions under which the latter is known to hold.