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\noindent Statistica Sinica {\bf 13}(2003), 97-109
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\centerline{\large\bf EFFECTS OF BAGGING AND BIAS CORRECTION ON}
\vspace{2pt}
\centerline{\large\bf ESTIMATORS DEFINED BY ESTIMATING EQUATIONS}
\vspace{.25cm}
\centerline{Song Xi Chen and Peter Hall}
\vspace{.4cm}
\centerline{\it
National University of Singapore and Australian National University}
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\begin{quotation}
\noindent {\it Abstract:}
Bagging an estimator approximately doubles its bias
through the impact of bagging on quadratic terms in expansions of
the estimator.  This difficulty can be alleviated by bagging a
suitably bias-corrected estimator, however.  In these and other
circumstances, what is the overall impact of bagging and/or bias
correction, and how can it be characterised?  We answer these
questions in the case of general estimators defined by estimating
equations, including for example maximum likelihood and method of
moments estimators.  It is shown that, despite the considerable
variety of estimators that can be constructed by bagging and bias
correction, the number of modes of behaviour is very small.  In
particular, bagging a bias-corrected estimator produces a new
estimator that is second-order equivalent to the original,
unadjusted estimator.  Furthermore, the conventional bagged
estimator, and the standard bias-corrected estimator, represent
virtually equal but opposite adjustments of the conventional
estimator.  In particular, bagging adds back the adjustment
provided by bias correction.  If we bag a doubly bias corrected
estimator, constructed so as to counteract the tendency of bagging
to exacerbate bias, then the result is an estimator that is
second-order equivalent to the standard bias-corrected estimator.
These results do not depend on the manner of bias correction; that
procedure may be implemented using the jackknife, the parametric
bootstrap or the nonparametric bootstrap.  They show that, when
bagging is applied to relatively conventional statistical
problems, it cannot reliably be expected to improve performance.
Its domain is, in effect, restricted to problems such as
regression trees, where variability is so high that it cannot be
plausibly modelled using the approach taken here.\par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Bootstrap, estimating function,
jackknife, maximum likelihood, mean square error, parametric bootstrap.\par
\end{quotation}
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