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DIAGNOSING BOOTSTRAP SUCCESS

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RUDOLF BERAN

*Department of Statistics, University of California, Berkeley, CA 94720-3860, U.S.A.*
(Received August 8, 1995; revised February 13, 1996)

**Abstract.**
We show that convergence of intuitive bootstrap
distributions to the correct limit distribution is equivalent to a
local asymptotic equivariance property of estimators and to an
asymptotic independence property in the bootstrap world. The first
equivalence implies that bootstrap convergence fails at
superefficiency points in the parameter space. However,
superefficiency is only a sufficient condition for bootstrap
failure. The second equivalence suggests graphical diagnostics for
detecting whether or not the intuitive bootstrap is trustworthy in a
given data analysis. Both criteria for bootstrap convergence are
related to Hájek's (1970, *Zeit. Wahrscheinlichkeitsth*.,
**14**, 323-330) formulation of the convolution theorem and to
Basu's (1955, *Sankhyã*, **15**, 377-380) theorem
on the independence of an ancillary statistic and a complete
sufficient statistic.

*Key words and phrases*:
Bootstrap convergence, local
asymptotic equivariance, local asymptotic sufficiency, asymptotic
independence, superefficiency points, convolution theorem.

**Source**
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