Confidence intervals for bootstrap and FPC

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?

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.
• So you're recommending the following interval $(\hat{\theta}-\hat{t}^{(1-\alpha)}\cdot \hat{se},\hat{\theta}-\hat{t}^{(\alpha)}\cdot \hat{se})$. Is that right? – Guest Aug 9 '13 at 19:00