You mentioned moments -- as I recall some writing in statistics uses that word, but I never have.
Your description of the Radon-Nikodym (R-N) result does not look right. Again, I don't want to write a lecture in graduate probability, but from memory the usual description of that result goes:
Given real valued, integrable function f and sigma algebra s, there exists real value function g measurable with respect to sigma algebra s so that g integrates like f over all sets in s. So, with real valued random variable Y in place of function f, we let
E[Y|s]
be the function g. If s is the sigma algebra generated by real random variable X, then we write
g(X) = E[Y|X] = E[Y|s]
So the R-N result makes immediate the definition of E[Y|X] and shows that it exists.
If you want a perfect, polished presentation, see the two references I gave, Rudin and Neveu.
My main contribution here remains:
f(X) = E[Y|X]
minimizes
E[ (Y - f(X))^2 ]
and is a minimum variance, unbiased estimator of Y. That's a generalization of the start of this thread and a bit amazing.
Of course that makes it possible. Note that it doesn’t say that f is simply measurable. By definition being integrable means E[|f|] < infinity. (And being square-integrable means E[f^2] < infinity, an even stronger condition).
Consider Y = Z/X where Z, X iid ~ N(0,1). Try to calculate E[Y|X], and try to minimize E[ (Y - f(X))^2 ]. Neither will make sense. I am sure both Rudin and Neveu were aware of this trivial result.
You mentioned moments -- as I recall some writing in statistics uses that word, but I never have.
Your description of the Radon-Nikodym (R-N) result does not look right. Again, I don't want to write a lecture in graduate probability, but from memory the usual description of that result goes:
Given real valued, integrable function f and sigma algebra s, there exists real value function g measurable with respect to sigma algebra s so that g integrates like f over all sets in s. So, with real valued random variable Y in place of function f, we let
be the function g. If s is the sigma algebra generated by real random variable X, then we write So the R-N result makes immediate the definition of E[Y|X] and shows that it exists.If you want a perfect, polished presentation, see the two references I gave, Rudin and Neveu.
My main contribution here remains:
minimizes and is a minimum variance, unbiased estimator of Y. That's a generalization of the start of this thread and a bit amazing.