🚀 go-pugleaf

Georg W�tschner <georg.wuetschner@unibw-muenchen.de> wrote in message news:<3EDF7781.4030100@unibw-muenchen.de>...
> Hi folk,
> I hope you allow me as an engineer a basic question:
> a random variable (RV) takes on a certain value, which I have to
> estimate. To this end I have a set of values, each of which was produced
> by a given function of the RV plus noise.
> Now an estimator E1 estimates the value of my RV with a certain variance V1.
> An _independant_ second estimator E2 does the same, but has a much worse
> set of values and therefore estimates with a bigger variance V2.
> 
> What would interest me is a statement on whether I can improve the
> overall performance of my estimation by somehow combining the 2
> estimators or if I allways make it worse. That is, is it possible to
> achieve a variance lower than V2 when combining the result of E1 and
> that of E2 (say weighted average or something more sophisticated)??
> Can this wanted relation between V1, V2 and Vcomb. be mathematically
> expressed? What are the conditions/constrictions?
> 
> [ snip ]
> 
> Thanks a lot,
> 
> Georg

To avoid confusion with the expected value operator E, 
call the estimates X1 and X2. Then we have 
E(X1|RV) = RV, Var(X1|RV) = V1, 
E(X2|RV) = RV, Var(X2|RV) = V2, 
with X1|RV independent of X2|RV. 
Let Xcomb = P1*X1 + P2*X2, where 
P1 = (1/V1)/(1/V1 + 1/V2), 
P2 = (1/V2)/(1/V1 + 1/V2). 
Then E(Xcomb|RV) = RV, 
and Var(Xcomb|RV) = V1*V2/(V1+V2), 
which is less than both V1 and V2. 
But note that V1 and V2 must be known 
(as opposed to estimated).