Rao-Blackwell Theorem

#1




I was reading this theorem again and again, but I really don't understand the meaning of "for all theta" in this theorem (circled in red). I thought that theta is supposed to be an unkown CONSTANT. Then how would it make sense to say "for all theta"? How would it change the theorem if the "for all theta" wasn't there? Why is this "for all theta" absolutely necessary?
[If it were to say "for all theta hat", then it would at least make some sense to me since theta hat is a random variable...at least it's a variable and it makes sense to talk about "for all", not a constant like theta]

Could someone please explain? Thank you for any help!
 

Masteras

TS Contributor
#2
there are more than one estimators of a population quantity or a parameter. that is why it means for all theta. the equality is achived when the best theta is the theta star.
 
#3
there are more than one estimators of a population quantity or a parameter. that is why it means for all theta. the equality is achived when the best theta is the theta star.
Yes, there are more than one estimators of a population parameter.
So shouldn't they say "for all theta hat"? theta hat is the estimator, and theta is the population parameter which is a constant.
 

vinux

Dark Knight
#4
I was reading this theorem again and again, but I really don't understand the meaning of "for all theta" in this theorem (circled in red). I thought that theta is supposed to be an unkown CONSTANT. Then how would it make sense to say "for all theta"? How would it change the theorem if the "for all theta" wasn't there? Why is this "for all theta" absolutely necessary?

Could someone please explain? Thank you for any help!
The reason for the sentence "for all theta" could be..
the definition of MSE is

This involves the parameter theta
 
#5
The reason for the sentence "for all theta" could be..
the definition of MSE is

This involves the parameter theta
But "for all theta" what does it mean in the context of the theorem?
The parameter theta is a constant number, theta hat is a variable, so shouldn't the theorem say "for all theta hat"?
 

Martingale

TS Contributor
#8
A minor point: I think that Theta-hat needs (originally) to be defined as an unbiased (or consistent) estimator of Theta...
The theorem is true even when Theta-hat is not unbiased, though whenever I state the theorem I include that Theta-hat is an unbiased estimator since I only use the theorem when I'm looking for UMVUE's
 
#9
But "for all theta" what does it mean in the context of the theorem? I don't understand the meaning of it. The "for all theta" part just looks like to be extraneous information to me, without it it wouldn't change the meaning of the theorem...
 

Dragan

Super Moderator
#10
.....though whenever I state the theorem I include that Theta-hat is an unbiased estimator since I only use the theorem when I'm looking for UMVUE's
In my memory, this is the case of when I've ever observed the theorem to be useful...correct me if I'm wrong.
 
#12
But how can you we it to find MVUE's if the theorem doesn't say "for all theta hat"?
If it says "for all theta hat", then the new estimator theta hat* will have a smaller variance than ANY estimator (i.e. the "smallest" variance), then theta hat* must be the MVUE. But the thing is that the theorem says "for all theta", what does it mean and how is it going to help us find the MVUE?
 
#13
Let's consider the following example:



1) To apply the thoerem, we first need a sufficient statistic. This is easy by the factorization theorem. Next, we need unbiased estimator for theta which is harder to find. Is there a general way to pinpoint the correct one?

In the solutions they found an unbiased estimator by first computing E(Yi^2). It looks like there is some "GUESSING" factor in here. How can we think of computing E(Yi^2) in the first place? There are many possible functions of the sufficient statistic. Why choose the particular one that they've chosen? Where to get this inspiration? Is there a systmatic way to find such unbiased esimtators?



2) What I understand about the theorem now is that it does NOT give the esimator with the smallEST variance (only smallER), but in the above example, they immediately concluded from the theorem that it is the MINIMUM variance unbiased estimator. How come?


Thanks for explaining!
 

vinux

Dark Knight
#14
For the first one..

Check the moment expression.. Here the second order moment will be in terms of theta..
( I think...first order will be in terms of sqrt(theta) )
Regarding sufficient statistic.. you got the sufficient statistic using factorization theorem. So I don't see much choice of sufficient statistic . ( or you can find some transformation of existing statistic).