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Unformatted text preview: On the Uniform Asymptotic Validity of Subsampling and the Bootstrap Joseph P. Romano Departments of Economics and Statistics Stanford University [email protected] Azeem M. Shaikh Department of Economics University of Chicago [email protected] April 13, 2010 Abstract This paper provides conditions under which subsampling and the bootstrap can be used to construct estimates of the quantiles of the distribution of a root that behave well uniformly over a large class of distributions P . These results are then applied (i) to construct confidence regions that behave well uniformly over P in the sense that the coverage probability tends to at least the nominal level uniformly over P and (ii) to construct tests that behave well uniformly over P in the sense that the size tends to no greater than the nominal level uniformly over P . Without these stronger notions of convergence, the asymptotic approximations to the coverage probability or size may be poor even in very large samples. Specific applications include the multivariate mean, testing moment inequalities, multiple testing, the empirical process, and Ustatistics. KEYWORDS: Bootstrap, Empirical Process, Moment Inequalities, Multiple Testing, Subsam pling, Uniformity, UStatistic 1 1 Introduction Let X ( n ) = ( X 1 ,...,X n ) be an i.i.d. sequence of random variables with distribution P ∈ P and denote by J n ( x,P ) the distribution of a realvalued root R n = R n ( X ( n ) ,P ) under P . In statistics and econometrics, it is often of interest to estimate certain quantiles of J n ( x,P ). Two commonly used methods for this purpose are subsampling and the bootstrap. This paper provides conditions under which these estimates behave well uniformly over P . More precisely, for α ∈ (0 , 1), we provide conditions under which subsampling and the bootstrap may be used to construct estimates ˆ c n (1 α ) of the 1 α quantiles of J n ( x,P ) satisfying liminf n →∞ inf P ∈ P P { R n ≤ ˆ c n (1 α ) } ≥ 1 α . (1) Similarly, for α ∈ (0 , 1), we provide conditions under which subsampling and the bootstrap may be used to construct estimates ˆ c n ( α ) of the α quantiles of J n ( x,P ) satisfying liminf n →∞ inf P ∈ P P { R n ≥ ˆ c n ( α ) } ≥ 1 α . (2) We also provide stronger conditions under which liminf n →∞ inf P ∈ P P { ˆ c n ( α ) ≤ R n ≤ ˆ c n (1 α ) } ≥ 1 2 α . (3) In many cases, it is possible to replace the liminf n →∞ and ≥ in (1), (2), and (3) with lim n →∞ and =, respectively. These results differ from those usually stated in the literature in that they require the convergence to hold uniformly over P instead of just pointwise over P . The importance of this stronger notion of convergence when applying these results is discussed further below....
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 Fall '08
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 Economics, Cumulative distribution function, CN, Jn, lim inf

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