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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Tuesday, May 15, 2001}
\centerline{Sequoia Hall Rm. 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Arup Bose}
\centerline{\sl Indian Statistical Institute}
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\centerline{\bf Generalised bootstrap and its accuracy}

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Consider estimating the variance of the least squares estimate
in linear regressions. There are several resampling schemes
available in the literature. By establishing representation results,
Liu and Singh(1992) classified these into two groups. Those that are
{\it efficient} but not consistent under heteroscedasticity and those
that are consistent under heteroscedasticity ({\it robust}) but not 
efficient.

Classes of generalised bootstrap are introduced and in some sense all of
the above schemes are special cases of these bootstraps. By establishing
higher order expansions, we distinguish between the estimators within the
robust and the efficient class. First order representation results are 
also
established for high dimensional regression models where the number of
parameters
increases with the sample size.

For the related problem of estimating the entire distribution of the least
squares estimate we establish consistency of the generalised bootstrap. It 
is
known from the existing works that the paired bootstrap (which is robust)
is not
second order accurate. We show that with proper bias correction and
studentisation,
a (smooth) generalisation of the paired bootstrap is second order 
accurate.

We then extend these ideas to estimates obtained by solving martingale
estimating
equations. We establish representation results for the bootstrap estimator
and obtain
some first and second order distribution results. Representation results
are also
obtained for the bootstrap variance estimator.

Finally, we show how these ideas can be implemented in estimating the
distribution
of $M_m$ estmators.

(This is joint work with Dr. Snigdhansu Chatterjee and are from his PhD
thesis.)
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