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A question about statistics

  1. Feb 4, 2013 #1
    1. The problem statement, all variables and given/known data

    Let X1, X2, …, Xn be a random sample from a Poisson(λ) distribution. Let [itex]\bar{X}[/itex] be their sample mean and [itex]S^2[/itex] their sample variance.
    a) Show that [itex]\frac{\sqrt{n}[\bar{X}-\lambda]}{\sqrt{\bar{X}}}[/itex] and [itex]\frac{\sqrt{n}[\bar{X}-\lambda]}{S}[/itex] both have a standard normal limiting distribution.
    b) Find the limiting distribution of [itex]\sqrt{n}[\bar{X}-\lambda]^2[/itex]
    c) Find the limiting distribution of [itex]\sqrt{n}[\bar{X}^2-\lambda^2][/itex]


    2. Relevant equations



    3. The attempt at a solution


    a) For [itex]\frac{\sqrt{n}[\bar{X}-\lambda]}{S}[/itex], we know that [itex]\frac{\sqrt{n}[\bar{X}-\lambda]}{S}[/itex] = [itex](\frac{\sqrt{n}[\bar{X}-\lambda]}{\sigma})(\frac{\sigma}{S})[/itex]. Since
    [itex]\frac{\sqrt{n}[\bar{X}-\lambda]}{\sigma}[/itex] appraches N(0,1) in distribution by CLT, and since [itex](\frac{\sigma}{S})[/itex] appraches 1 in probability, the whole thing approaches N(0,1).

    For
    [itex]\frac{\sqrt{n}[\bar{X}-\lambda]}{\sqrt{\bar{X}}}[/itex], we get the same result...since the mean is equal to the variance in a poisson distribution.

    b) I'm a little confused about this one...

    c) From a theorem in my textbook, I know that if [itex]\sqrt{n}(X_n - \theta) \rightarrow N(0,\sigma^2)[/itex] and if there is a differentiable function g(x) at theta where the derivative at theta is not zero...then [itex]\sqrt{n}(g(X_n)-g(\theta)) \rightarrow N(0,\sigma^2(g'(\theta))^2)[/itex].

    So I just need to use this theorem, right? And in this case g(x)=x^2.

    Do you think my answer for a), and c) are correct? Also, can you give me a hint for b)?

    Thanks in advance
     
    Last edited: Feb 4, 2013
  2. jcsd
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