Statistics Seminar: Roelof Helmers -- On the M fewer than N bootstrap approximation to the trimmed mean

We show that the M fewer than N ( N is the real data sample size, M denotes the size of
the bootstrap resample) bootstrap approximation to the distribution of the trimmed mean
is consistent without any conditions on the population distribution F , whereas Efron’s
naive ( i.e.  M=N) bootstrap as well as the normal approximation fails to be consistent
if the population distribution F has gaps at the two quantiles where the trimming
occurs.  We illustrate our asymptotic results with some simulations.  Our results
supplement previous work by P.J.Bickel,F.Gotze, W.R.van Zwet( 1997), ’Resampling fewer
than n observations: gains, losses and remedies for losses, Statistica Sinica, V.7, pp
1-31 and by N.V.Gribkova and R.Helmers (2007)’ On the Edgeworth expansion and the M out
of N bootstrap accuracy for a Studentized trimmed mean, Math.  Meth.  Statist, V.16, pp
142-176.  This is joint work with Nadezhda Gribkova ( St.Petersburg).