Example of bootstrap to estimate se of a set of data... just for fun

N <- 1000
A <- rnorm(N)
sd(A)/sqrt(N)
# Vraie valeur: 
1/sqrt(N)

t <- NULL
for (rep in rep(c(100, 200, 300, 400, 500, 1000), 10)) {
  print(rep)
  s <- NULL
  for (i in 1:10000) {
    tirage <- sample(A, size=rep, replace = TRUE)
    s <- c(s, mean(tirage))
  }
  
  t <- c(t, sd(s))
}
dta <- data.frame(group=rep(c(100, 200, 300, 400, 500, 1000), 10), 
                  mean=t)
boxplot(mean ~ group, data=dta, las=1, ylab="SE", xlab="Number of bootstraps", ylim=c(0, 0.15))
segments(x0=1, x1=7, y0=1/sqrt(N), y1=1/sqrt(N), col="red", lty=3, lwd=2)


In red, the true SE. The SE estimation by bootstrap is upper biased.

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