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In this paper we study a bootstrap strategy for estimating the variance of amean taken over large multifactor crossed random effects data sets. We applybootstrap reweighting independently to the levels of each factor, giving eachobservation the product of independently sampled factor weights. No exactbootstrap exists for this problem [McCullagh (2000) Bernoulli 6 285-301]. Weshow that the proposed bootstrap is mildly conservative, meaning biased towardoverestimating the variance, under sufficient conditions that allow veryunbalanced and heteroscedastic inputs. Earlier results for a resamplingbootstrap only apply to two factors and use multinomial weights that are poorlysuited to online computation. The proposed reweighting approach can beimplemented in parallel and online settings. The results for this method applyto any number of factors. The method is illustrated using a 3 factor data setof comment lengths from Facebook.

Bootstrapping data arrays of arbitrary order