A note on conditional versus joint unconditional weak convergence in bootstrap consistency results

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dc.contributor.authorBücher, Axel
dc.contributor.authorKojadinovic, Ivan
dc.date.accessioned2017-06-10T12:42:56Z
dc.date.available2017-06-10T12:42:56Z
dc.date.issued2017
dc.description.abstractThe consistency of a bootstrap or resampling scheme is classically validated by weak convergence of conditional laws. However, when working with stochastic processes in the space of bounded functions and their weak convergence in the Hoffmann-Jorgensen sense, an obstacle occurs: due to possible non-measurability, neither laws nor conditional laws are well-defined. Starting from an equivalent formulation of weak convergence based on the bounded Lipschitz metric, a classical circumvent is to formulate bootstrap consistency in terms of the latter distance between what might be called a conditional law of the (nonmeasurable) bootstrap process and the law of the limiting process. The main contribution of this note is to provide an equivalent formulation of bootstrap consistency in the space of bounded functions which is more intuitive and easy to work with. Essentially, the equivalent formulation consists of (unconditional) weak convergence of the original process jointly with an arbitrary large number of bootstrap replicates. As a by-product, we provide two equivalent formulations of bootstrap consistency for Rd-valued statistics: the first in terms of (unconditional) weak convergence of the statistic jointly with its bootstrap replicates, the second in terms of convergence in probability of the empirical distribution function of the bootstrap replicates. Finally, the asymptotic validity of bootstrap-based confidence intervals and tests is briefly revisited, with particular emphasis on the, in practice unavoidable, Monte Carlo approximation of conditional quantiles.en
dc.identifier.urihttp://hdl.handle.net/2003/35989
dc.identifier.urihttp://dx.doi.org/10.17877/DE290R-18007
dc.language.isoende
dc.relation.ispartofseriesDiscussion Paper / SFB823;13, 2017en
dc.subjectbootstrapen
dc.subjectconditional weak convergenceen
dc.subjectconfidence Intervalsen
dc.subjectresamplingen
dc.subjectstochastic processesen
dc.subjectweak convergenceen
dc.subject.ddc310
dc.subject.ddc330
dc.subject.ddc620
dc.titleA note on conditional versus joint unconditional weak convergence in bootstrap consistency resultsen
dc.typeTextde
dc.type.publicationtypeworkingPaperde
dcterms.accessRightsopen access

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