Statistics: Consistent Estimators

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1. The problem statement, all variables and given/known data
Q1) Theorem:
An asymptotically unbiased estimator 'theta hat' for 'theta' is a consistent estimator of 'theta' IF
lim Var(theta hat) = 0
n->inf

Now my question is, if the limit is NOT zero, can we conclude that the estimator is NOT consistent? (i.e. is the theorem actually "if and only if", or is the theorem just one way?)




Q2) http://www.geocities.com/asdfasdf23135/stat9.JPG

I'm OK with part a, but I am stuck badly in part b. The only theorem I learned about consistency is the one above. Using the theorem, how can we prove consistency or inconsistency of each of the two estimators? I am having trouble computing and simplifying the variances...


2. Relevant equations
N/A

3. The attempt at a solution
Q1) I've seen the proof for the case of the theorem as stated.
Let A=P(|theta hat - theta|>epsilon) and B=Var(theta hat)/epsilon^2
At the end of the proof we have 0<A<B and if V(theta hat)->0 as n->inf, then B->0, so by squeeze theorem A->0 which proves convergence in probability (i.e. proves consistency).

I tried to modfiy the proof for the converse, but failed. For the case that lim V(theta hat) is not equal to zero, it SEEMS to me that (by looking at the above proof and modifying the last step) the estimator can be consistent or inconsistent (i.e. the theorem is inconclusive) since A may tend to zero or it may not, so we can't say for sure.

How can we prove rigorously that "for an unbiased estimator, if its variance does not tend to zero, then it's not a consistent estimator." Is this is a true statement?



Q2) Var(aX+b) = a^2 Var(X)
So the variance of the first estimator is [1/(n-1)^2]Var[...] where ... is the summation stuff. I am stuck right here. How can I calculate Var[...]? The terms are not even independent...and (Xi-Xbar) is squared, which creates more trouble in computing the variance


Thanks for helping!
 

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