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Homework Help: Is there a central limit theorem for Median?

  1. Apr 19, 2010 #1
    1. The problem statement, all variables and given/known data
    Hi, We know the famous central limit theorem for means.
    I wonder if there is a central limit theorem for Median?
    If so under what regularity condition, does the median converge to a normal distribution with mean and variance equal to what?

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 19, 2010 #2

    statdad

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    Homework Helper

    One of the simplest results is this:
    * Assume the model is [itex] F(x - \theta) [/itex]
    * Assume both [itex] F(0) = 1/2 [/itex] and that the density [itex] f(0) > 0 [/itex]

    (these assumptions mean the population median is unique)

    Then the sample median satisifies

    [tex]
    \sqrt n \left(\hat \theta - \theta\right) \rightarrow n(0, \sigma^2)
    [/tex]

    in distribution, where the asymptotic variance is given by

    [tex]
    \sigma^2 = \frac 1 {4f^2(0)}
    [/tex]
     
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