Results 1 to 4 of 4

Math Help - Negative binomial moment estimators

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    36

    Negative binomial moment estimators

    X_1, ... , X_n is random sample from negative binomial distribution NB(r,p) .
    I need to find moment estimators \hat r and \hat p.
    My literature on this topic is very poor, any hints?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    May 2010
    Posts
    1,030
    Thanks
    28
    My notation:
    the sample mean: \bar{X}
    the sample variance:  \sigma^2

    the population mean: E(X)
    the population variance:  Var(X)


    i assume you are trying to do a method of moments estimate of the parameters.
    background info: http://en.wikipedia.org/wiki/Method_...ts_(statistics)

    Quick explanation: equate the sample moments with the theoretical population moments

    \bar{X} = E(X)
     \sigma^2 = Var(X)


    According to Negative binomial distribution - Wikipedia, the free encyclopedia, the moments for this distribution are:

    E(X) = r \frac{p}{1-p}
    Var(X) = r \frac{p^2}{(1-p)^2} = \frac{E^2(X)}{r}

    So

    \frac{E^2(X)}{Var(X)}= r
    To obtain the method of moments estimator, replace all the moments in the above equation with their sample analogues.
    So your method of moment estimate for r is \hat{r} = \bar{X}^2 / \sigma^2

    Can you use a similar approach to find the estimate for p?


    NBI'd check the theoretical moments are correct if you aren't getting sensible answers. wikipedia is sometimes wrong about those
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2010
    Posts
    36
    I have checked NB formulas and used the right ones, is this ok?

    E(X) = \frac{r(1-p)}{p}
    Var(X) = \frac{r(1-p)}{p^2} = \frac{r(1-p)}{p} \cdot \frac{1}{p} =  \frac{E(X)}{p}

    p= E(X) / Var(X)

    \hat{p} = \bar{X} / \sigma^2

    r = \frac{Var(X)}{1-p}

    \hat{r} = (\sigma^2)^2 / \left(\sigma^2 - \bar{X}\right)
    Last edited by losm1; June 19th 2010 at 12:37 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    May 2010
    Posts
    1,030
    Thanks
    28
    i hate wikipedia sometimes...

    Your answer is perfect though
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Moment estimators?
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: November 5th 2010, 05:05 PM
  2. Replies: 3
    Last Post: July 15th 2010, 06:33 AM
  3. Method of Moment Estimators
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: February 24th 2010, 08:13 AM
  4. integration and moment estimators
    Posted in the Advanced Statistics Forum
    Replies: 8
    Last Post: May 6th 2009, 01:27 PM
  5. Replies: 3
    Last Post: February 15th 2009, 11:59 AM

Search Tags


/mathhelpforum @mathhelpforum