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Imagine the following setup:

  • I have N specimens that come from the same population (e.g., they are from the dame material)
  • For each specimen i, I have a single estimate of a given property in terms of its mean value $x_i$ and a confidence interval $[x_i-CI_i, x+CI_i]$
  • So I have N pairs of slightly different ($x_i,CI_i$), for $i=1,...,N$, due to the random nature of producing specimens
  • Now, I want to estimate my confidence on the mean of the mean properties of the N specimens, i.e., $E(x_i)$, the mean of the group of N specimens.

I thought to simply bootstrap on the N mean values of each specimen and build a confidence interval from that. But then I feel it will not be taking into consideration the information contained on the confidence interval of each of their individual mean estimates, which is an information I have.

I started considering making a weighted bootstrap, weighting each original sample by its confidence interval range (e.g. basically divide each $x_i$ by a normalized weight computed from their $CI_i$).

But I could not find a solid reference (paper, book) on that and since I am not a statistician I am afraid of getting into some pitfall.

What are your thoughts (and references) on that?

Thanks in advance!

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  • $\begingroup$ Why don't you use parametric bootstrap to bootstrap new samples out of the distributions parametrized by x_i and the corresponding scale (related to what you call CI)? Then you can compute whatever statistic you want ... $\endgroup$
    – Ggjj11
    Commented Apr 3 at 12:52
  • $\begingroup$ And did you come across inverse variance weighting already? en.m.wikipedia.org/wiki/Inverse-variance_weighting (not related to bootstrapping) $\endgroup$
    – Ggjj11
    Commented Apr 3 at 12:58
  • $\begingroup$ @Ggjj11 thanks for the insights! Regarding the inverse variance weighting, I did not know it, thanks for that. So from your inputs I envisioned two possible approaches: 1) instead of a bootstrap sample being made from random sampling with replacement from the set of N values of $x_i$, I would take one random sample out of each distribution parametrized by each pair of $(x_i,CI_i)$; 2) make a traditional bootstrap, using the original set of $x_i$, but then study the inverse-variance weighted average instead of the simple average. Would you know if one is theorectically sounder? $\endgroup$
    – ren1
    Commented Apr 3 at 14:44
  • $\begingroup$ I think it is important to note that you are trying to find an estimator $\hat{E}[y]=f(x_1[y],...x_N[y])$ where the $x_i=\hat{E}[y]$ are estimated mean values already. Now there is different choices for $f$ and they may result in a higher or lower variance $Var[\hat{E}[y]]$. The least variance of the mean estimator will be given by inverse variance weighting (which also gives you a formula for the variance of the estimated mean). You will directly be able to see it in simulations as well e.g. (x_1=20, sigma(x_1) =60), (x_2=70, sigma(x_2)=1) should intuitively have a better estimate around 70 $\endgroup$
    – Ggjj11
    Commented Apr 3 at 18:11
  • $\begingroup$ Also look for sensor data fusion en.m.wikipedia.org/wiki/Sensor_fusion $\endgroup$
    – Ggjj11
    Commented Apr 3 at 18:14

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