# This R code pertains to Sec. 7.4 of E&T.
#
# Let's see how well nonparametric bootstrapping can 
# estimate the standard error of the sample maximum 
# when the data comes from a uniform distribution.
#
# First I'll generate a sample of size 50 from a 
# uniform (0, 1) distribution.
set.seed(321)
x <- runif(50, 0, 1)

#
# Now I'll generate 2000 bootstrap samples and estimate
# the standard error of the sample maximum.
bss <- matrix( sample( x, size=2000*length(x), replace=T ), nrow=2000 )
replicates <- apply( bss, 1, max )
#
# Here's the bootstrap estimate of the standard error:
sd(replicates)

#
# Here's a histogram of the replicates:
hist(replicates)

#
# Here's the true value of the standard error:
sqrt( 50/52 - (50/51)^2 )

#
# Here's a parametric estimate of the standard error:
max(x)*sqrt( 50/52 - (50/51)^2 )

#
# It can be concluded that bootstrapping doesn't work very 
# well in this situation.