Results 1 to 7 of 7

Math Help - sampling distributions of estimators

  1. #1
    Member
    Joined
    Nov 2008
    From
    MD
    Posts
    165

    sampling distributions of estimators

    suppose that a random variable X has a geometric distribution for which the parameter p is unknown (0<p<1). Show that the only unbiased estimator of p is the estimator delta(X) such that delta(0)=1 and delta(X)=0 for X>0

    any clues on this?? thank you so much in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Sambit's Avatar
    Joined
    Oct 2010
    Posts
    355
    It was answered earlier.
    Quote Originally Posted by SpringFan25 View Post
    suppose you have 1 realisation of x and your estimator is c(x)

    For your estimator to be unbiased you require:
    E(c(x)) = p

    \sum c(x) \times p(1-p)^{x-1} =p
    where the summation is taken over all possible values of x (1,2,3,4,5,6...)

    Now, consider the following function:

    c(x)=1 if x=1
    c(x)=0 otherwise

    The expected value of our function is then

    \sum c(x) \times p(1-p)^x =p
    =(f(1)\times p)  + (f(2)\times p(1-p)) + (f(3)\times p(1-p)^2)  +...
    =(1 \times p)  + (0 \times p(1-p)) + (0 \times p(1-p)^2)  +...
    =p
    as required

    So your estimator is c(x) where c(x) was defined above.

    In reality, this estimator is no practical use. But it is an unbiased which is all the question asked for. I did assume that your sample is a single realisation from the distribution. if you have a sample with multiple data points, just discard all but one of them and the proceedure still works.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by Sambit View Post
    It was answered earlier.
    But that post did not answer the question which was to show that the given estimator was the unique unbiased estimator, that post just showed that the given estimator was unbiased (and there is a typo in the latter part of the post).

    It looks like the proof relies on the uniquness of a power series expansion of a (constant) function on the closed unit interval.

    CB
    Last edited by CaptainBlack; April 11th 2011 at 04:55 AM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Oct 2009
    Posts
    340
    Quote Originally Posted by CaptainBlack View Post
    But that post did not answer the question which was to show that the given estimator was the unique unbiased estimator, that post just showed that the given estimator was unbiased (and there is a typo in the latter part of the post).

    It looks like the proof relies on the uniquness of a power series expansion of a (constant) function on the closed unit interval.

    CB
    X is complete, so all functions of X are almost-surely unique unbiased for their expectations. If OP can appeal to completeness, then the problem is done.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by theodds View Post
    X is complete, so all functions of X are almost-surely unique unbiased for their expectations. If OP can appeal to completeness, then the problem is done.
    That makes no sense at all.

    CB
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member
    Joined
    Oct 2009
    Posts
    340
    What part exactly doesn't make sense? X is distributed according to an exponential family of full rank, so X is complete, and because X is complete it only admits one unbiased estimator of p....
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member Sambit's Avatar
    Joined
    Oct 2010
    Posts
    355
    Quote Originally Posted by CaptainBlack View Post
    That makes no sense at all.

    CB
    I don't understand why. If you go through that page you will see the proof of MVUE is done there.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. sampling distributions and estimators
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: April 11th 2011, 12:09 AM
  2. Sampling Distributions Help
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: February 2nd 2010, 06:12 PM
  3. Sampling Distributions
    Posted in the Statistics Forum
    Replies: 0
    Last Post: January 31st 2010, 01:20 AM
  4. Sampling distributions help
    Posted in the Statistics Forum
    Replies: 2
    Last Post: May 25th 2009, 08:19 AM
  5. Sampling distributions
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: October 22nd 2007, 10:28 AM

Search Tags


/mathhelpforum @mathhelpforum