Say I have a time series X of length T that is split into two subperiods: X1 = X(0),X(1),...,X(T-n) and X2 = X(T-n+1),X(T-n+2),...,X(T), where the length of X1 > the length of X2.
Now, say I want to use the bootstrap to test whether the mean of X2 is the same as the mean of X1. Can I perform the following procedure:
1. Compute mean of X2, call it m2
2. Generate B bootstrap samples of X1
3a. For each bootstrap sample, I obtain the mean of a randomly selected subsample of the series that has the same length as X2, call it m1*, or...
3b. If the length of X1 is, say, more than 3 times that of X2, I can obtain 3 nonoverlapping means from each bootstrap sample of X1, call them m11*,m12* and m13*
4. Use these bootstrapped means to form a distribution (for 3a, there would be B bootstrapped means, but for 3b, there would be 3B bootstrapped means) and generate the one-tail p-value for X2 from this distribution, e.g. the fraction of m1* larger than m2
I know that when deriving inferences from bootstrap resampling, one must use the same lengths for the bootstrap sample and the original sample. So ideally, X1 and X2 should have the same lengths. However, in practice, this will not always be the case. Would appreciate the help!