Please check our work out, any feedback is always welcome!

To showcase the usefulness of our theory and its stepdown procedure, we apply it to a genome-wide study on oscillatory mouse liver cells, where p is much larger than the sample size.

Importantly, we were able to detect novel rhythmic gene activity not previously reported!

Numerical evidence shows that our test

1. is fairly insensitive to the particular choice of block size q.

2. outperforms other popular tests of independence in high dimensions with large p and dependent Y's in many scenarios.

One of the challenges of this approach is that the block-multiplier bootstrap involves the choice of a tuning parameter (the size of the "big" block).

We provide a rule-of-thumb that enjoys a certain optimality property.

1. Its distribution can be approximated under weak assumptions.

2. It can be combined with a stepdown procedure à la Romano-Wolf to identify individual Ys that violate independence while controlling the FWER.

We considered a max-type statistic, where the maximum is taken over p Chatterjee's rank correlations.

This choice has two key advantages in our setup with large p:

We propose a powerful bootstrap test that controls size uniformly over a large class of data-generating processes.

Importantly, it allows p to be (much) larger than the sample size while at the same time not restricting the dependence among Y1, . . . , Yp in any way.

There are examples in which testing this hypothesis and, in particular, screening out variables that violate independence is of interest. Think of testing whether a treatment indicator has an effect on various outcomes and then select those outcomes on which there is an effect.

More concretely, suppose you are interested in testing whether X is independent of Y1,...,Yp and would like to pinpoint those Y's that violate independence so as to control the family-wise error rate (FWER).