# parametric bootstrap for low sample sizes

I believe that this question is sufficiently different from previous related ones to warrant a new post. (I apologize if it has been answered already)

I need to decide between various resampling methods to "best" (highest power and correct type-I error to reject H0 for the right reasons) evaluate auto correlation in binary sequences of small to moderate lengths (20-50) with relatively low incidence rate. The time series are too short to justify any normal approximations, so to e.g. to decide whether or not the observed auto correlation is "real", I would like to generate my empirical Null distribution under the assumption of no auto correlation. With that goal in mind I can either

1. shuffle the sequence repeatedly (fixed number of 0s and 1s), or
2. estimate the probability $\hat{p}$ of an event and repeatedly draw from the respective Bernoulli process.

The latter is -I think- known as the parametric bootstrap. I struggle with the correct choice for the following reasons:

(i) The estimate for the true Bernoulli probability will be rather poor/noisy for low smple sizes. That bias will be part of the generated empirical NULL. (ii) The permutation procedure 1 will generate test statistics with considerably less variation than procedure 2. In fact, the distribution will tend to generate few discrete levels.

What line of reasoning will defend either choice ?

Here is an example:

x=c(0,1,0,0,0,1,1,0,0,1,0,0,0,0,1,0)
#observed acf:
aObs = acf(x,lag.max=1,plot=F)$acf a1=a2=rep(NA,100) for (i in 1:100){ a1[i]=acf(sample(x),lag.max=1,plot=F)$acf