Example of using re-centerring in the bootstrap estimate of bias

Given a sample  X1, X2, ... X10 from U(0,1);  the true mean is mu = 0.5

The statistic log(\bar X) estimates log(0.5). But it is biased. 


log( \bar X) as an estimate of log(mu) has bias:

E[ log (\barX) ] - log(mu) = true bias [you can integrate to get this or use
simulation -- Monte Carlo]

We estimate this true bias using (Monte Carlo) bootstrap estimate

   1/N \sum log( \bar Y_i^* )  - log( \bar X ) 

where Y_i^* is a bootstrap sample from the 
empirical distribution based on the Xi.

Control function accelaration: estimate the bias with better Monte Carlo:
[re-centering the estimator]

1/N \sum log( \bar Y_i^* ) -  log ( \bar \bar Y^* )

R-code

> set.seed(1234)
> xvec <- runif(10)
> log( mean(xvec) )
[1] -0.7149299
> log( 0.5)
[1] -0.6931472
> for(i in 1:1000) {
+ bootvec <- sample( xvec, size=10, replace=TRUE)
+ result[i] <- log( mean(bootvec) )
+ center[i] <- mean(bootvec)
+ }
> log( mean(center[1:1000]))
[1] -0.7104151
> mean(result[1:1000])
[1] -0.7253017
> -0.7552906 - -0.7471847  ##bias
[1] -0.0081059
> -0.7509244
[1] -0.7509244
> -0.7509244 - -0.6931472
[1] -0.0577772
> mean(result[1:1000])
[1] -0.7552906
> mean(result[1:1000]) - log(mean(xvec) )
[1] -0.004366129
>