Example of using bootstraps to estimate SE of a mean

Just a little game to know better bootstraps. Note that when the number of observations is low, the estimator is biased.

 See here for a solution:
Bondy, Warren; Zlot, William (1976). "The Standard Error of the Mean and the Difference Between Means for Finite Populations". The American Statistician. 30 (2): 96–97.


finalerr <- NULL
reptest <- c(10000, 50000, 100000)
ltest <- c(10, 20, 30, 50, 100, 200, 300, 400, 500)
for (rep in reptest) {
  perror <- NULL
  for (l in ltest) {
    
    taille <- runif(n=l, min=160, max=180)
    
    # The standard error of the series
    # sd(taille)/sqrt(l)
    
    
    m <- rep(NA, rep)
    for (i in 1:rep) m[i] <- mean(taille[sample(x=1:l, size = l, replace = TRUE)])
    
    # The standard deviation of the means obtained using bootstrap
    # sd(m)
    
    perror <- c(perror, 100*(sd(m)-(sd(taille)/sqrt(l)))/(sd(taille)/sqrt(l)))
  }
  finalerr <- cbind(finalerr, perror)
  
}
plot(ltest, finalerr[, 1], las=1, bty="n", type="n", ylim=c(-5, 2), xlab="Number of observations", 
     ylab="Percent of error for SE")
segments(x0=0, x1=500, y0=0, y1=0, lty=2)

 lines(x=ltest, y=finalerr[, 1], col="black")
lines(x=ltest, y=finalerr[, 2], col="blue")
lines(x=ltest, y=finalerr[, 3], col="red")

 legend("bottomright", legend=c("10000", "50000", "100000"), 
       lty=1, col=c("black", "blue", "red"), title = "Number of boostraps")


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