Results 1 to 4 of 4

Math Help - Cramér–Rao Lower Bound for a Function of a Parameter

  1. #1
    Newbie
    Joined
    Oct 2008
    Posts
    21

    Cramér–Rao Lower Bound for a Function of a Parameter

    I'm trying to compute the Cramér–Rao Lower Bound for a function of the parameter \lambda of a Poisson distribution.

    The function is  g (\lambda) = e^{-\lambda} .
    Let  \hat{g} be an estimate of  g(\lambda)

    I'm using the formula:

     Var (\hat{g}) \ge \dfrac{(g'(\lambda))^2}{nI(\lambda)}

    Now from estimating  \lambda previously, I know that  nI(\lambda) = \dfrac {n}{\lambda}

    And
     g'(\lambda) = -e^{-\lambda}
     (g'(\lambda))^2 = e^{-2\lambda}

    So
     Var (\hat{g}) \ge \dfrac{e^{-2\lambda}}{\dfrac {n}{\lambda}}

     Var (\hat{g}) \ge \dfrac{\lambda e^{-2\lambda}}{n}

    Is this correct because I tried it another way and got?
     Var (\hat{g}) \ge \dfrac{e^{-\lambda}(1-e^{-\lambda})}{n}
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by alan4cult View Post
    I'm trying to compute the Cramér–Rao Lower Bound for a function of the parameter \lambda of a Poisson distribution.

    The function is  g (\lambda) = e^{-\lambda} .
    Let  \hat{g} be an estimate of  g(\lambda)

    I'm using the formula:

     Var (\hat{g}) \ge \dfrac{(g'(\lambda))^2}{nI(\lambda)}

    Now from estimating  \lambda previously, I know that  nI(\lambda) = \dfrac {n}{\lambda}

    And
     g'(\lambda) = -e^{-\lambda}
     (g'(\lambda))^2 = e^{-2\lambda}

    So
     Var (\hat{g}) \ge \dfrac{e^{-2\lambda}}{\dfrac {n}{\lambda}}

     Var (\hat{g}) \ge \dfrac{\lambda e^{-2\lambda}}{n}

    Is this correct because I tried it another way and got?
     Var (\hat{g}) \ge \dfrac{e^{-\lambda}(1-e^{-\lambda})}{n}
    Well I'm not going to work this out but you first form looks wrong as it has the same dimensions as \lambda (lets say \text{[T]}^{-1}) but it should be a pure number. The second form is a pure number which is what we should expect.

    CB
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Oct 2008
    Posts
    21
    Hi thanks for your reply.

    Can you see what's wrong with my end result by looking at the calculations that led me to it? Did I differentiate incorrectly?

    The derivative of g with respect to  \lambda is  -e^{-2\lambda}

    And then I've just used that in the formula?
    Do you think I'm using the formula incorrectly/out of context?

    About the dimensions thing you're talking about, what does that mean? I haven't encountered dimensions before?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Oct 2008
    Posts
    21
    Anyone got any more ideas?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. cramer-rao lower bound
    Posted in the Calculus Forum
    Replies: 0
    Last Post: July 28th 2010, 12:33 AM
  2. Replies: 0
    Last Post: February 19th 2010, 01:06 AM
  3. Greatest lower bound and lower bounds
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: October 13th 2009, 02:26 PM
  4. Cramer-Rao lower bound for Binomial dist
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: August 9th 2009, 07:24 AM
  5. least upper bound and greatest lower bound
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 22nd 2007, 09:59 AM

Search Tags


/mathhelpforum @mathhelpforum