Say I have some second-order statistic $m(x)$ where $x$ is a data vector of length $n$. Let's also assume that the limiting distribution of $x$ is gaussian-ish, but generally unknown, so that the assumptions that enable one to derive the usual error analysis expressions do not hold. In this case, if I want to get an estimate in the uncertainty in the measure of $m$, I will have to simulate it, using a bootstrap or something. So, I generate 1,000 unique realizations of $x$, $x_i$ (1 $\leq$ $i$ $\leq$ 1000), and use the distribution of all $m(x_i)$ to get an idea of the error in $m$
Now, since $m$ is second-order, it is preferable to draw without replacement when generating the 1,000 resamples of $x$. This is all fine, and the bootstrapping routines I've implemented work well. Here is my problem:
- I have to choose a size of the resample
- If I choose that size to be $n$, then all $x_i$ will be identical since I'm sampling without replacement
- So, the resample size must obviously be smaller than the size of $x$
Problem is, if I choose the size of each $x_i$ to be high, say $0.9n$, then my error is going to be very small. If I choose the size to be small, $0.1n$, then the error can blow up. So, I can effectively make the error in $m$ whatever I'd like, which obviously isn't right...
What do I do at this point, while maintaining integrity?!