Generalised bootstrap and its accuracy
by
Arup Bose
Indian Statistical Institute, Calcutta, India
Coauthors: Snigdhansu Chatterjee (ISI, Calcutta, India)

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 efficient but not consistent under heteroscedasticity and those that are consistent under heteroscedasticity 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 roder 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.

Date received: October 31, 2000