  This paper considers a practical scenario where a classical estimation method
might have already been implemented on a certain platform when one tries to
apply more advanced techniques such as moving horizon estimation (MHE). We are
interested to utilize MHE to upgrade, rather than completely discard, the
existing estimation technique. This immediately raises the question how one can
improve the estimation performance gradually based on the pre-estimator. To
this end, we propose a general methodology which incorporates the pre-estimator
with a tuning parameter {\lambda} between 0 and 1 into the quadratic cost
functions that are usually adopted in MHE. We examine the above idea in two
standard MHE frameworks that have been proposed in the existing literature. For
both frameworks, when {\lambda} = 0, the proposed strategy exactly matches the
existing classical estimator; when the value of {\lambda} is increased, the
proposed strategy exhibits a more aggressive normalized forgetting effect
towards the old data, thereby increasing the estimation performance gradually.
