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Order Statistics PDF

  1. Oct 3, 2009 #1
    Hello,

    Suppose that we have the following set of independent and identically distributed RVs: [tex]\gamma_1,\,\gamma_2,\,\ldots,\,\gamma_M[/tex]. Arranging them in descending order as: [tex]\gamma_{1:M}\ge\gamma_{2:M}\ge\cdots\ge\gamma_{M:M}[/tex]. Now suppose we select the largest [tex]m\leq M[/tex] order statistics. What is the PDF of the selected set? Mathematically:

    [tex]f_{\gamma_{1:M},\,\ldots,\,\gamma_{m:M}}(\gamma_{1:M},\,\ldots,\,\gamma_{m:M})=??[/tex]

    Thanks in advance
     
  2. jcsd
  3. Oct 4, 2009 #2

    EnumaElish

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    What would f be for m = 1? How do you get there?
     
  4. Oct 4, 2009 #3
    [tex]f_{\gamma_{1:M}}(\gamma)=\frac{d}{d\,\gamma}F_{\gamma_{1:M}}(\gamma)=\frac{d}{d\,\gamma}\text{Pr}\left[\gamma_{1:M}\le\gamma\right]=\frac{d}{d\,\gamma}\text{Pr}\left[\gamma_{1}\le\gamma,\gamma_{2}\le\gamma,\ldots,\gamma_{M}\le\gamma\right]=\frac{d}{d\,\gamma}\left[F_{\gamma}(\gamma)\right]^M[/tex]

    where [tex]F_{\gamma}(\gamma)[/tex] is the CDF of the original set of RVs.

    But when we pick a subset of the [tex]m^{\text{th}}[/tex] largest order statistics, how can we treat the statistics? I mean I have the final answer from books and papers, but I didn't understand how they derive it.
     
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