I would like to show you a pitfall . From the corrected values we should be able to iterate, further improving the corrections. A pessimist might call this bootstrapping a self-fulfilling prophecy, but we will see.
Truth is stranger than fiction.
I tried bootstrapping the variances. With my first starting value for the ratio ,iterating led to the ratio being infinite. Another starting value led to the ratio being zero. All starting values led to zero or infinity. Eventually I deduced that there must be a metastable starting value.