# Thread: Consistent Estimators and Consistency in Mean Square

1. ## Consistent Estimators and Consistency in Mean Square

Hello,

I have an estimator = f(X) and I know that f(X) has two possible values under this probability function:

P(f(X) = / ) = n-1 / n

P(f(X) = + n / = 1/n

I need to show that f(X) is a consistent estimator of but is not consistent in mean square. How can I do that? I know that definitions to use but I am struggling to use them for this question.

Regards.

2. ## Re: Consistent Estimators and Consistency in Mean Square

To start: What are the definitions of consistent and consistent in mean square? Do you have any intuition about what they each mean? What ideas have you tried to show either of these?

3. ## Re: Consistent Estimators and Consistency in Mean Square

As far as I know, for consistency:

Show that P ( | - | > ) = 0 as n tends to infinity

And for consistency in mean square:

Show that the estimator is asympotically unbiased (bias tends to 0 as n tends to infinity) and then that the variance tends to 0 as n tends to infinity.

But I don't know how to apply this for the question

4. ## Re: Consistent Estimators and Consistency in Mean Square

So your estimator can only take two values correct? So think about this - for a fixed n what is P()?

Hints:
For any n we know that can only take on two values. Find what is for both of these values. Is there a way to get either one of them below epsilon by increasing n?

5. ## Re: Consistent Estimators and Consistency in Mean Square

Originally Posted by Dason
So your estimator can only take two values correct? So think about this - for a fixed n what is P()?

Hints:
For any n we know that can only take on two values. Find what is for both of these values. Is there a way to get either one of them below epsilon by increasing n?
can better either or + n, right? so then I have for the first case | | and second case ? which is equal to n?

6. ## Re: Consistent Estimators and Consistency in Mean Square

So in the first case the difference is clearly less than epsilon. And in the second case for any n greater than epsilon the difference is greater than epsilon.

So the probability that the absolute value of the difference is greater than epsilon is just the probability that you're in the second case. And what does that probability go to as n goes to infinity?

Edit: Also the  tags can be a little finicky sometimes. I changed one set of them to  tags for you instead because the math was screwed up otherwise.

7. ## Re: Consistent Estimators and Consistency in Mean Square

Since the probability is 1/n, this tends to 0 as n tends to infinity?

Ok, this is kind of making sense but I don't feel I have a complete full understanding of it because I'm kind of confused of what epsilon is

Originally Posted by Dason
And in the second case for any n greater than epsilon the difference is greater than epsilon.
This is the part I don't understand. Why am I allowed to say for any n greater than epsilon?

8. ## Re: Consistent Estimators and Consistency in Mean Square

Because if epsilon is less than n then we have...

9. ## Re: Consistent Estimators and Consistency in Mean Square

So that is a general rule I can apply, that when n is greater than epsilon .. I evaluate whether this also holds for the absolute value of the difference?

10. ## Re: Consistent Estimators and Consistency in Mean Square

For consistency you need to check

As Dason suggested, you can explicitly calculate the probability:

So you have the bounds

And by the squeeze theorem, the limit converge to 0.

11. ## Re: Consistent Estimators and Consistency in Mean Square

How do I know that epsilon is greater than n in the first case and in between 0 and n in the second case?

12. ## Re: Consistent Estimators and Consistency in Mean Square

From the definition,

and

Am I right?

If the definition above is correct, then the result is obvious??

13. ## Re: Consistent Estimators and Consistency in Mean Square

Can you explain how it's obvious

14. ## Re: Consistent Estimators and Consistency in Mean Square

This is the only part I do not understand.. The relationship between and n, in the first case .. now how can I say that is greater than or equal to n?

15. ## Re: Consistent Estimators and Consistency in Mean Square

You don't. Epsilon is just a small constant. But you have control over n. So for any epsilon that you pick I can choose an n such that n is greater than epsilon. I don't know why you're talking about making epsilon greater than or equal to n because that is just silly talk.

+ Reply to Thread
Page 1 of 2 1 2 Last

 Tweet

#### Posting Permissions

• You may not post new threads
• You may not post replies
• You may not post attachments
• You may not edit your posts