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Suppose you only have these information from a sample data: $X_i$ and $w_i$, $i=1,...,N$, where $w_i$'s are the respective sampling weights(not integers).

Is it possible to obtain a valid bootstrap estimate of say, variance of $X_i$?

I understand that I might have to readjust the sampling weights(for weighted data) for each bootstrap replicate to obtain a valid bootstrap estimate. But considering if I only have the above information, are there other ways to do this?

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  • $\begingroup$ There are several different meanings of "sampling weights"; for instance, Stata natively supports three different ones. Could you therefore explain what your weights mean? $\endgroup$ – whuber Feb 27 '14 at 17:05
  • $\begingroup$ hi @whuber, in Stata it would be "pweights" $\endgroup$ – stats_newb Feb 28 '14 at 1:50
  • $\begingroup$ My recollection is that these are probability weights: that is, you selected cases independently and randomly but with different chances of inclusion in the sample. Is that correct? $\endgroup$ – whuber Feb 28 '14 at 14:40
  • $\begingroup$ Hi @whuber, these are weights that are calculated (with post-stratification applied) when sampling population and population of target inference are disparate. $\endgroup$ – stats_newb Mar 3 '14 at 8:24
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Short answer: No.

Marginally longer answer: A single vector of weights tells you nothing about:

  1. Stratification
  2. Clustering
  3. Calibration variables

A proper bootstrap scheme would involve the following (I can refer to my paper on this).

  1. Take a bootstrap sample (with replacement) of clusters within each stratum, independently across strata.
  2. Re-calibrate the sample to the same population totals that were used for the main weights.

Since you don't have strata and clusters, you cannot do 1. Since you don't have the calibration variables, you cannot do 2.

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