Bootstrap for nonstandard cases

Scale invariant statistics like Student's t, involving mean absolute deviations instead of standard deviations, and statistics which are distributed asymptotically like linear combinations of chi-squares or F-statistics are considered. Bootstrap procedures in estimating their distributions are described. The error in the bootstrap approximation of the sampling distribution, in the latter type, is of the order O_p(n^{- 3 1}). It is demonstrated that the error term cannot be improved. In such cases Bartlett correction is preferable, which is known to give an approximation with an error term of the order O(n^-2). Bootstrap procedures for errors-in-variables regression coefficients are also described. In all the cases mentioned above, the Edgeworth expansions of multivariate means, which do not satisfy standard regularity assumptions, play an important role.