Confidence intervals for bootstrap and FPC

Question

I have a population that I am technically sampling without replacement using a stratified design. The resultant ratio estimator of the sample mean in any stratum $i$ is $\hat{R}_{i} = (\sum y_{i})/(\sum x_{i})$. I also know that $x_{i} \in \theta_{x,i}$ in any stratum, such that I can obtain $\hat{\theta}_{y,i} = \hat{R}_{i}\theta_{x,i}$.

My question, then, is whether I can treat this situation as simple random sampling with replacement, as I'm applying the ratio estimate to the known, total value of $\theta_{x,i}$. In other words, $\theta_{x,i} = N$ and not $\theta_{x,i} = N-n$.

I'm using a non-parametric bootstrap routine to come up with confidence intervals. The $y_{i}$ are rare events, so I cannot assume that the bootstrap distribution can be normally approximated, so I'm just using the 95% highest density region of the bootstrap replicates. I'm curious if this is appropriate, or if I need to include the finite population correction factor somehow, as some strata are sampled pretty well (i.e. $n/N \geq 0.1$).

Thanks @StasK for your suggestions, although I need a little clarification. I did not use the percentile method to obtain confidence intervals. Instead I found the HDR. In other words, if $f(x)$ is the density function of random variable $X$, then I found the region in the sample space of $X$ such that $HDR(f_{\alpha}) = \{x: f(x) \geq f_{\alpha}\}$, where $f_{\alpha}$ is the largest constant such that the $Pr(X \in HDR) \geq 1-\alpha$. I did this to somewhat account for the skewed nature of the bootstrapped distribution. Also, the distribution of the bootstrapped replicates is polymodal - yet another reason why I used the HDR. Do you still think this is inappropriate? If so, then what you're recommending is finding the following interval $(\hat{\theta}-\hat{t}^{(1-\alpha)}\cdot \hat{se},\hat{\theta}-\hat{t}^{(\alpha)}\cdot \hat{se})$. Is that right?

Answer 1

You do need to account for the sampling design and FPCs using rescaled bootstraps. See Rao and Wu 1998 for the basic motivation, and if that's not accessible, Preston 2009 or my Stata paper and code for the survey bootstrap weights. Note though that these papers still only address variance estimation, although Rao & Wu might have touched upon the distribution of the test statistic. For rare events, the ratio estimator will be both skewed and biased. Bootstrap can be used to correct for the bias (it's not very frequently used for this purpose, but should be for survey data like your situation).

I would not use the percentile method here. Instead, I would get the distribution of the $t$-statistic, and apply that distribution to your original data. If you had an SRSWOR within strata, then you can utilize the $s^2(1-n/N)/n$ estimator within each replicate.

Sounds like a fair amount of custom programming to me. Each task per se is not reasonably mainstream and straightforward, and this should be a relatively standard task for survey data analysis. If there's R code from Lumley (or his book), you are lucky. If not, you'll have to code it all up.