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Math Help - proof consistency

  1. #1
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    proof consistency

    A random variable X has a Poisson - distribution defined for each nonegative integer k, Pr(X= k) = (lamda^(k)/k!)*exp(-lamda). We know that E(X)= lamda and Var(X) = E(X^2) - E^(2) (X) = lamda..
    We draw an i.i.d sample of size n from this distribution: x1, x2,..., xn, and define Zn = Sum (from i=1 to n) xi^(2). Show that lamda(kappa) = (-1+sqrt(1+4Zn)) /2 is a consistent parameter of Lamda.


    I know that for consistency we have to estimate the expected value and the bvariane or find the plim estimator.
    Unfortunatelly i cannot show it using either way. Please if you know show me the proof.


    Many thanks!
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  2. #2
    MHF Contributor matheagle's Avatar
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    Zn = Sum (from i=1 to n) xi^(2) goes to infinity almost surely.
    So, something is wrong with this question.
    You need to divide by n here or in this kappa thingy

    {Zn\over n} = {\sum_{i=1}^n X_i^2\over n}\to \lambda +\lambda^2 almost surely.
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  3. #3
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    If such the case Zn= sum(of xi^2 for i=1 to infinity)/n
    What do i show with that Zn is going to infinity. How is that relevant in showing that Lamda kappa is consistent estimator of parameter lamda.

    Thank you.
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  4. #4
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    Do i need to change inside lamda kappa that Zn is lamda^2 + lamda?

    If i do it then i get that the expected value for lamda kappa is lamda. But i am stuck with the variance.
    How much will be the variance for Zn? (not the one in the original problem but such as you have defiened it)

    Thank you, again.
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