Statistics: Consistent Estimators

[kingwinner]

Homework Statement

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...

Homework Equations

N/A

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!

 

[mighty2000]

it is important to approach this question with a critical and analytical mindset. Let's break down the forum post and address each question separately.

Q1) The theorem in question states that an asymptotically unbiased estimator is a consistent estimator if the limit of its variance tends to zero as n (the sample size) tends to infinity. The question posed is whether this theorem is a "if and only if" statement, meaning that if the limit is not zero, can we conclude that the estimator is not consistent?

The answer to this question is yes, the theorem is a "if and only if" statement. In other words, if the limit of the variance is not zero, then we can conclude that the estimator is not consistent. This is because the proof of the theorem relies on the squeeze theorem, which states that if a function is squeezed between two other functions that approach the same limit, then the squeezed function must also approach that limit. In this case, the function A (defined in the post) represents the probability that the estimator is far from the true value, and the function B represents the variance of the estimator divided by a constant. If B tends to zero, then A must also tend to zero, and vice versa. Therefore, if the variance does not tend to zero, then the estimator cannot be consistent.

Q2) In part b of the question, the task is to use the given theorem to prove the consistency or inconsistency of two estimators. The theorem itself is not applicable in this case, as it only applies to asymptotically unbiased estimators. However, we can use the properties of variance to prove the consistency or inconsistency of the estimators.

To calculate the variance of the first estimator, we can use the property Var(aX+b) = a^2 Var(X). However, as stated in the post, the terms in the summation are not independent, so this property cannot be directly applied. Instead, we can use the definition of variance Var(X) = E[(X-E[X])^2] and expand the squared term, which will result in a summation of terms that can be simplified and calculated.

Once the variance is calculated, we can then use the given theorem to determine the consistency or inconsistency of the estimator. If the limit of the variance tends to zero, then the estimator is consistent. If it does not tend to zero, then the estimator is inconsistent.

In conclusion, as a scientist,