I've done something alike for a combination of 4 discrete random variables a,b,c,d. The combination was ( (a - b) * (c - d) ). My solution was to take a random sample of 2000, calculate mean y and repeat this 10 million times. From that sampling distribution (mean y) I took 95% CI. Anything close to your problem?

Best, Manfred

Thank you for your answer, but it is not quite close. I am not interested in the mean and 95% confidence interval (CI). To put it another way, my value of interest is y = f(x), where I am sampling x ( x= [0; 20,000]), to produce different values of y. Here, x is a good descriptor of a sample of data, and in the bootstrapping, I am sampling random combinations of my data so I can compare it to specific ("informed") combinations of my data.

Strictly speaking, I should be sampling 1000 times for each x* to estimate the distribution of y* = f(x*). Then I would be able, for my choice of data, x, compare where y = f(x) to the distribution of y*; if y is above the 95% percentile of y*, my choice of data is meaningful (i.e. not a product of chance). But with 20,000 different values of x, bootstrapping each x-value becomes infeasible (each calculation of y = f(x) takes approx. 2½ hours), so I my wish is to produce a result kinda in the sense:

If y' = f(x') > 1.2x^2, I have empirical evidence that the choice of x' is with 5% probability not a product of chance. Where the last term would be a regression of the 95% percentile.

Now, to make a long story short, I got advice from another side, who wisely explained, that we can use a first bootstrapping to estimate a statistic, a second bootstrapping on the first statistic to estimate the variance of the statistic, a third bootstrapping on the second variance, to estimate the variance of the variance, etc.