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Math Help - upperbound on expected value

  1. #1
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    upperbound on expected value

    Hi

    I found a proof, but can not follow. The critical part is the following:

    \mathbb{E}\left[\frac 1 n \sum_{i=1}^n x_i \left(1-\frac 1 n \sum_{i=1}^n w_i +\left(\frac 1 n \sum_{i=1}^n w_i\right)^2-\left(\frac 1 n \sum_{i=1}^n w_i\right)^3+...\right)\right]-X
    where
    \mathbb{E}\left[x_i\right]=X, \mathbb{E}\left[w_i\right]=0

    I need an approximation / upperbound to n. The authors assert that it is in order of n^{-1}. Can anyone explain it?

    Thanking you in anticipation!
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  2. #2
    MHF Contributor
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    how about some more information?

    Like, the preceeding line and what it is you are trying to find the expected value of.
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  3. #3
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    What I'm trying to do is, estimate the bias of a self normalized importance sampler, that is
    \hat{\mu}=\frac{\sum_{i=1}^n w(z_i) h(z_i)}{\sum_{i=1}^n w(z_i) }

    for sake of simplicity let's see it as
    \hat{\mu}=\frac{\sum_{i=1}^n x_i}{\sum_{i=1}^n y_i}

    the bias is given bei
    Bias=\mathbb{E}[\hat{\mu}]-X

    a taylor expansion for \frac x y around the expected values X and 1 leads to the given equation above.

    I'm interested in an estimation depending on n. there are some books asserting that it is
    |Bias|\leq \frac C n
    for some constant C \geq 0.

    Can you give me a more detailed explanation?
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