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Calculating Variances of Functions of Sample Mean

  1. Oct 11, 2012 #1
    1. Essentially what I'm trying to do is find the asymptotic distributions for
    a)
    Y2
    b) 1/Y and
    c) eY where
    Y = sample mean of a random iid sample of size n.
    E(X) = u; V(X) = σ2




    2. Relevant equations
    a) [tex]Y^2=Y*Y[/tex] which converges in probability to [tex]u^2[/tex],

    [tex]V(Y*Y)=\sigma^4 + 2\sigma^2u^2[/tex]

    So, [tex]\sqrt{n}*(Y^2 - u^2)[/tex] converges in probability to [tex]N(0,\sigma^4 + 2\sigma^2u^2) [/tex]

    So, [tex]Y^2\sim N(u^2,\frac{\sigma^4 + 2\sigma^2u^2}{n})[/tex]

    Is that right?



    b) [tex]\frac{1}{Y}[/tex] converges in probability to [tex]\frac{1}{E(X)} = \frac{1}{u}[/tex]
    [tex]V(\frac{1}{x}) = \frac{1}{σ^2}[/tex] ??

    Thus,
    [tex]\sqrt{n}(\frac{1}{Y} - \frac{1}{u})[/tex] converges in distribution to [tex]N(0,V(\frac{1}{x})*\frac{1}{n})[/tex]
    What is V(1/X) ?


    Am I on the right track?
     
    Last edited: Oct 11, 2012
  2. jcsd
  3. Oct 11, 2012 #2
    Solved. Just had to remember the delta method.
     
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