Bootstrapping a sample with unequal selection probabilities

Question

I want to "blow up" a sample, taken with replacement, for which I know the overall sampling probability $\pi_i$ for each item $i$. Is it valid to use bootstrapping and apply inverse probability weighting during the selection (as in the Horvitz-Thompson estimator: weight each item with $1/\pi_i$), or are there any pitfalls? A quick search on Google suggests that this hasn't been fully investigated yet, and the boot package in R allows weights but does not comment on where they are supposed to come from.

The purpose of "blowing up" is, among others, to be able to resample with uniform probability from the blown-up population.

Answer 1

Did you find a satisfactory answer for this question?

I recently found this reference:

http://www.wseas.us/e-library/conferences/2009/hangzhou/ACACOS/ACACOS21.pdf

but I am pretty sure that the issue must have been investigated before. While it is easy to justify the use of observation weights (in practice, by weighting observations, you are hoping to use a better estimate of the unknown distribution function F), I would like to find the relevant background.

Answer 2

You can verify that the "weights" parameter in the boot package is operating as importance weights with a simple simulation.

example <- data.frame(
   meas=c(1,1,5,8,10),
   wts=c(10,10,3,2,1)
)

Unweighted mean:

mean(example$meas)
# output = 5

Weighted mean:

sum(example$meas * example$wts) / sum(example$wts)
# output = 2.346154

Now doing this with bootstrapping:

my.avg <- function(data, indices) {
   d <- data[indices,]
   return(mean(d$meas))
}

Unweighted bootstrapped mean:

mean(boot(example, my.avg, 1000)$t)
# output = 4.8908

Weighted bootstrapped mean:

mean(boot(example, my.avg, 1000, weights=example$wts)$t)
# output = 2.3712