# File "Rboot5".

B = 1000

n = 100

# Generate some data.
x = runif(n)

thetahatlist = NULL

# Run the bootstrap, with B resamplings of n resamples each.

for (b in 1:B) {

    # Do n bootstrap resamples.
    resampled = rep(0,n)
    for (i in 1:n) {
	j = floor( runif(1,1,n+1) )  # uniform on {1,2,...,n}
	resampled[i] = x[j]
    }

    # Compute thetahat for this resample.
    thetahat = max(resampled)

    thetahatlist = c(thetahatlist, thetahat)

}

# Compute the bootstrap estimate of mean, thetastar.
thetastar = sum(thetahatlist) / B

# Compute the bootstrap estimate of bias.
biasest = thetastar - max(x)

print(biasest)


# OR, TO COMPUTE THE BIAS NUMERICALLY:
# n=100
# densofmax = function(y) { n * y^(n-1) }
# integr = function(y) { y * densofmax(y) }
# meanofmax = integrate(integr, 0, 1)$value
# bias = meanofmax - 1
# bias