I have two independent samples of observations. From each sample I produce a statistic. Let's denote these as $\theta_1$ and $\theta_2$. I'd like to test the hypothesis that $H_0: \Theta_1=\Theta_2$, but I have these two constraints:
- There is no analytical estimate of the distributions of $\theta_1$ and $\theta_2$ (the statistic is a product of some (computationally expensive) algorithm that operates on each sample.
- Even under $H_0$, exchanging observations between the two samples in not sensible. Therefore, a permutation approach might reject $H_0$ erroneously.
My current idea was to bootstrap $\theta_1$ and $\theta_2$ independently and then estimate the distribution of $\Theta_2-\Theta_1$ from these two bootstrapped distributions by means of convolution.
Q1: Is this a valid approach?
Q2: Any reason why not the extend this to jackknifing (instead of bootstrapping) as well?
Q3: Any references to such 'two-samples' bootstrap?
Q4: Any recommended alternatives?