A (tight) upper bound for the length of confidence intervals with conditional coverage
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Kivaranovic and Leeb (2020) showed that confidence intervals based on a truncated normal distribution have infinite expected length. In this paper, we generalize the results of Kivaranovic and Leeb (2020) by considering a larger class of conditional distributions of normal random variables of which the truncated normal distribution is a limit case of. We derive an upper bound for the length of confidence intervals based on a distribution from this class. This means the length of these intervals is always bounded. Furthermore, we show that this upper bound is tight if the conditioning event is bounded. We apply our result to two popular selective inference procedures which are known as data carving (Fithian et al. (2015)) and Lasso selection with a randomized response (Tian et al. (2018b)) and show that these two procedures dominate the sample splitting strategy in terms of interval length.
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