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Distribution of Log of Variance

  1. Nov 12, 2013 #1
    1. The problem statement, all variables and given/known data
    If [itex] Y_1, Y_2, ...[/itex] are iid with cdf [itex] F_Y[/itex] find a large sample approximation for the distribution of [itex]\log(S^2_N)[/itex], where [itex]S^2_N[/itex] is the sample variance.


    2. Relevant equations



    3. The attempt at a solution
    The law of large numbers states that for large N [itex]S^2_N[/itex] converges in probability to [itex]\sigma^2[/itex]. However, because I don't know the distribution of the Y's I don't know what [itex]\sigma^2[/itex] is.

    Also [itex]\log S^2_N = \log(\frac{1}{N} \sum_{i=1}^{N} (Y_i - \mu) ^2) = \log(\frac{1}{N}) + \log(\sum_{i=1}^{N} (Y_i - \mu) ^2)[/itex], but I am not sure if this helps me find an approximation for the distribution.
     
  2. jcsd
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