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[statistics] find the efficient estimator: where is my reasoning wrong?

  1. Jan 27, 2012 #1
    1. The problem statement, all variables and given/known data
    Find an efficient estimator for [itex]q(\lambda) := e^{-\lambda}[/itex] in a Poisson model.
    (note: the efficient estimator is the one who reaches the Cramér-Rao lower bound)

    2. Relevant equations
    The Cramér-Rao lower bound:
    Let T be an unbiased estimator for q (as defined above), then [itex]Var(T) \geq \frac{q'(\lambda)^2}{I_n(\lambda)}[/itex], with [itex]I_n(\lambda)[/itex] the Fisher information number for the parameter [itex]\lambda[/itex] in the Poisson model, from the sample [itex]\vec X = (X_1,\cdots,X_n)[/itex].

    Concretely, in this case [itex]Var(T) \geq \frac{ \lambda e^{-2\lambda}}{n} [/itex].

    3. The attempt at a solution
    So first things first: I want an unbiased estimator for q. As a guess, I took the form [itex]T(\vec X) = \alpha^{\sum X_i}[/itex]. Calculating the mean shows that one gets an unbiased estimator if one defines [itex]T(\vec X) := \left( \frac{n-1}{n} \right)^{\sum X_i}[/itex].

    Now my reasoning was as follows: this unbiased esimator only depends on [itex]\sum X_i[/itex], i.e. a complete and sufficient statistic (follows, for example, from the Poisson distribution belonging to the exponential class). Hence by Lehman-Scheffé, T as defined above is an UMVUE. Now, if the Cramér-Rao bound is reached (as the exercise suggests), then it must surely be the UMVUE who reaches it. Hence I thought I had the correct estimator.

    However, when I calculate the variance, I get that [itex]Var(T) = E(T^2) - E(T)^2 = e^{-2 \lambda} \left( e^{\lambda / n} - 1 \right)[/itex]. This is not equal to the lower bound (it is, however, for very large n, but that's not enough, but it at least suggests I didn't make a calculation error).

    Can anybody help? Much appreciated!

    ADDENDUM:
    While writing the red text above, I realised it also depends on n explicitly. Is this important? I don't think so.
     
    Last edited: Jan 27, 2012
  2. jcsd
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