### Rina Foygel Barber - Distribution-free inference for estimation and classification

### 3 Answers

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Here the black is the population quantity that you are trying to estimate. How is it defined in population terms. It seems like it is the expecctation of some quantity conditioned on an order statistics from some training sample, but I can't figure out what it is more precisely.

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My intuition for this problem is built around estimation of a survival function, which may be far enough from your problem to be misleading, but, following that analogy, Iwould expect that you won't do much better pointwise, as the monitonicity forces pointwise coverage to hold uniformly as well. Isouldn't expect that you'll get a much more narrow interval pointwise.
The overall convergence property is going to hold uniformly as n goes to infinity, but we are optimistic that the width of the interval might change substantially in our finite sample calculations. It might be just by a constant factor, although our current intuition is that we might swap a log(n) for a log(log(n)). Asymptotically it's probably all the same (i.e. if each point converges, then convergence is uniform) but in finite samples we are hoping to be less conservative - even a constant factor would help us.

written
7 months ago by
Rina Barber

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When we think about the post-selection inference problem, it actually may get a bit trickier because, after observing (X_i, corrupted Y_i) for i=1,...,n, we now know something about the X_i's. So at stage 2, where we want to do inference on the p_i vector (which now becomes the function f(t)), it's no longer safe to assume that the X_i's are iid draws from some population---we've used this part of the data, it's no longer completely random to us. We are not sure how to think about the problem in this setting, i.e. how to formulate the n data points as a discrete version of some underlying continuous process.