+ Reply to Thread
Page 1 of 2 1 2 LastLast
Results 1 to 15 of 27

Thread: Consistent Estimators and Consistency in Mean Square

  1. #1
    Points: 1,278, Level: 19
    Level completed: 78%, Points required for next Level: 22

    Posts
    135
    Thanks
    13
    Thanked 0 Times in 0 Posts

    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. #2
    Devorador de queso
    Points: 95,819, Level: 100
    Level completed: 0%, Points required for next Level: 0
    Awards:
    Posting AwardCommunity AwardDiscussion EnderFrequent Poster
    Dason's Avatar
    Location
    Tampa, FL
    Posts
    12,935
    Thanks
    307
    Thanked 2,629 Times in 2,245 Posts

    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. #3
    Points: 1,278, Level: 19
    Level completed: 78%, Points required for next Level: 22

    Posts
    135
    Thanks
    13
    Thanked 0 Times in 0 Posts

    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
    Last edited by MrAnon9; 11-30-2011 at 09:26 PM.

  4. #4
    Devorador de queso
    Points: 95,819, Level: 100
    Level completed: 0%, Points required for next Level: 0
    Awards:
    Posting AwardCommunity AwardDiscussion EnderFrequent Poster
    Dason's Avatar
    Location
    Tampa, FL
    Posts
    12,935
    Thanks
    307
    Thanked 2,629 Times in 2,245 Posts

    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(| \hat{\theta} - \theta| > \epsilon)?

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

  5. #5
    Points: 1,278, Level: 19
    Level completed: 78%, Points required for next Level: 22

    Posts
    135
    Thanks
    13
    Thanked 0 Times in 0 Posts

    Re: Consistent Estimators and Consistency in Mean Square

    Quote Originally Posted by Dason View Post
    So your estimator can only take two values correct? So think about this - for a fixed n what is P(| \hat{\theta} - \theta| > \epsilon)?

    Hints:
    For any n we know that \hat{\theta} can only take on two values. Find what |\hat{\theta} - \theta| 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 | \theta + n - \theta | ? which is equal to n?
    Last edited by MrAnon9; 11-30-2011 at 11:10 PM.

  6. #6
    Devorador de queso
    Points: 95,819, Level: 100
    Level completed: 0%, Points required for next Level: 0
    Awards:
    Posting AwardCommunity AwardDiscussion EnderFrequent Poster
    Dason's Avatar
    Location
    Tampa, FL
    Posts
    12,935
    Thanks
    307
    Thanked 2,629 Times in 2,245 Posts

    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 [tex][/tex] tags can be a little finicky sometimes. I changed one set of them to [math][/math] tags for you instead because the math was screwed up otherwise.

  7. #7
    Points: 1,278, Level: 19
    Level completed: 78%, Points required for next Level: 22

    Posts
    135
    Thanks
    13
    Thanked 0 Times in 0 Posts

    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

    Quote Originally Posted by Dason View Post
    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?
    Last edited by MrAnon9; 11-30-2011 at 11:20 PM.

  8. #8
    Devorador de queso
    Points: 95,819, Level: 100
    Level completed: 0%, Points required for next Level: 0
    Awards:
    Posting AwardCommunity AwardDiscussion EnderFrequent Poster
    Dason's Avatar
    Location
    Tampa, FL
    Posts
    12,935
    Thanks
    307
    Thanked 2,629 Times in 2,245 Posts

    Re: Consistent Estimators and Consistency in Mean Square

    Because if epsilon is less than n then we have... \epsilon < n = |\theta + n - \theta| = difference

  9. #9
    Points: 1,278, Level: 19
    Level completed: 78%, Points required for next Level: 22

    Posts
    135
    Thanks
    13
    Thanked 0 Times in 0 Posts

    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. #10
    TS Contributor
    Points: 22,410, Level: 93
    Level completed: 6%, Points required for next Level: 940

    Posts
    3,020
    Thanks
    12
    Thanked 565 Times in 537 Posts

    Re: Consistent Estimators and Consistency in Mean Square

    For consistency you need to check \lim_{n\to+\infty} \Pr\{|\hat{\theta} - \theta| > \epsilon\} = 0 ~~ \forall \epsilon > 0

    As Dason suggested, you can explicitly calculate the probability:

    \Pr\{|\hat{\theta} - \theta| > \epsilon\} = \left\{\begin{matrix} 0 & \text{if} & \epsilon \geq n \\ \frac {1} {n} & \text{if} & 0 < \epsilon < n \end{matrix}\right.

    \leq \frac {1} {n} ~~ \forall \epsilon > 0

    So you have the bounds 0 \leq \Pr\{|\hat{\theta} - \theta| > \epsilon\} \leq \frac {1} {n} ~~ \forall \epsilon > 0

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

  11. #11
    Points: 1,278, Level: 19
    Level completed: 78%, Points required for next Level: 22

    Posts
    135
    Thanks
    13
    Thanked 0 Times in 0 Posts

    Re: Consistent Estimators and Consistency in Mean Square

    \Pr\{|\hat{\theta} - \theta| > \epsilon\} = \left\{\begin{matrix} 0 & \text{if} & \epsilon \geq n \\ \frac {1} {n} & \text{if} & 0 < \epsilon < n \end{matrix}\right.

    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. #12
    TS Contributor
    Points: 22,410, Level: 93
    Level completed: 6%, Points required for next Level: 940

    Posts
    3,020
    Thanks
    12
    Thanked 565 Times in 537 Posts

    Re: Consistent Estimators and Consistency in Mean Square

    From the definition,

    \Pr\{|\hat{\theta} - \theta| = 0\} = 1 - \frac {1} {n} and \Pr\{|\hat{\theta} - \theta| = n \} = \frac {1} {n}

    Am I right?

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

  13. #13
    Points: 1,278, Level: 19
    Level completed: 78%, Points required for next Level: 22

    Posts
    135
    Thanks
    13
    Thanked 0 Times in 0 Posts

    Re: Consistent Estimators and Consistency in Mean Square

    Can you explain how it's obvious

  14. #14
    Points: 1,278, Level: 19
    Level completed: 78%, Points required for next Level: 22

    Posts
    135
    Thanks
    13
    Thanked 0 Times in 0 Posts

    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 P(0) =  (n-1)/n .. now how can I say that is greater than or equal to n?

  15. #15
    Devorador de queso
    Points: 95,819, Level: 100
    Level completed: 0%, Points required for next Level: 0
    Awards:
    Posting AwardCommunity AwardDiscussion EnderFrequent Poster
    Dason's Avatar
    Location
    Tampa, FL
    Posts
    12,935
    Thanks
    307
    Thanked 2,629 Times in 2,245 Posts

    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 LastLast

           




Posting Permissions

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






Advertise on Talk Stats