Expected value on subset of Pareto

  1. Mar 12, 2008 #1

    xag

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    Hi -
    I have a Pareto distribution X (xm=1, k known)*. High-value samples are filtered out and I want the expected value of the remaining. Namely: E[X|X<a].

    *Wikipedia notations

    Many thanks in advance.
    Xavier
     
  2. jcsd
  3. Mar 13, 2008 #2
    What is the definition of the conditional expectation you are referring to? How did you try to calculate it? Where did you get stuck?
     
  4. Mar 13, 2008 #3

    xag

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    Hi Pere -

    What I'm looking for is a mean value.
    I found out that the samples of interest belong to a larger set that roughly follow a Pareto distribution, and that among this larger set, the samples of interest are the ones whose values are below a certain number, 'a'.

    I can estimate the parameters of the Pareto once for all, and calculate 'a' for each run. But I can't access the samples for each run, so if there was a way to predict it with 'k' (the Pareto shape) and 'a', that would be great.

    My first guess is that the samples of interest also follow a Pareto, but I've not shown that. And in this case, could we express its shape K and scale Xm? Then the mean I'm looking for would be K Xm/(K-1).

    Best regards,
    Xavier
     
  5. Mar 14, 2008 #4

    xag

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    Ok. It seems I got all the way to the finish line and stopped just before :

    I guess I just had to integrate the probability density function between 1 and a (it's defined between 1 and +oo).
    It gives : (a ^(1 - k) - 1) * k / (1 - k)

    Can anyone confirm?
     
  6. Mar 15, 2008 #5
    What you did gives you the probability of a sample with value less than a.

    Try the following instead: First calculate the conditional distribution function[itex]\mathbb{P}(X\leqx| X \leq a)[/itex].
    By definition this is equal to

    [tex]
    \frac{\mathbb{P}(X\leq x \wedge X \leq a)}{\mathbb{P}(X\leq a)}
    [/tex]

    (You already calculated the denominator.)

    You can also easily calculate the numerator. For x>a the numerator is [itex]\mathbb{P}(X \leq a)[/itex] which cancels with denominator giving one. For x>a you calculate it in the same way as you did for [itex]\mathbb{P}(X \leq a)[/itex] with a replaced by x. Then you take the derivative with respect to x to find the conditional density function [itex]\rho_a(x)[/itex] (suported on [1,a]), which you then use to calculate the conditional expectation as
    [tex]\mathbb{E}\left[X|X\leq a\right]=\int_1^a{x\rho_a(x)dx}[/tex]
    A more direct though maybe less obvious formula would be
    [tex]\mathbb{E}\left[X|X\leq a\right]=\frac{\int_1^a{x\rho(x)dx}}{\int_1^a{\rho(x)dx}}[/tex],
    where [itex]\rho(x)[/itex] is the density function of your original Pareto distribution.
    I suggest you try them both and check if they really are the same:smile:
    -Pere
     
    Last edited: Mar 15, 2008
  7. Mar 15, 2008 #6

    xag

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    Forget this post: the thing I integrated was x * dF(x) and moreover the result I gave here is wrong.
     
  8. Mar 15, 2008 #7

    xag

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    Thank you so much.

    In both cases I find [tex] \mathbb{E}\left[X|X\leq a\right]= \frac{k}{k - 1} \frac{a ^ k - a}{a ^ k - 1}[/tex]
     
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