On a modified bootstrap for certain asymptotically nonnormal statistics

The classical bootstrap approximation is known to break down (Babu, 1984), even for "nice" statistics such as a smooth function of a multivariate sample mean for certain "critical" values of the mean vector. A simple modification of the naive bootstrap is suggested to take care of this problem. Simulation results show improvements at or near a critical value while using the modified bootstrap. Asymptotic validity (with rate of convergence) of the modified bootstrap is established for parameter values including the critical values.