6 Bias correction

Bias is easily estimated as the difference between the original estimates and the mean of the bootstrap replicates:

   > bias <- apply(boots,2,mean) - fit1$par
   > bias
   [1] 0.001642839 0.006198921

If the bias not excessivly large, and if it appears to be approximately independent of the true parameter values, one way to correct for bias (Efron and Tibshirani, 1993, p. 138) is to subtract the estimated bias from the orginal estimates:

   > fit1$par - bias
   [1] 0.002381547 0.122746777

In many cases, however, especially if there is a large amount of between-population differentiation, there may be considerable bias in the obtained estimate, not only because asymptotic theory doesn't apply, but also because the likelihood function is quite approximate. More sophisticated methods for reducing bias such as that of Cabrera and Watson (1997) may then be needed. I will try to tidy up the code I used in Tufto et al. (1998) and put it into the library. The theoretical justification for that method, however, only apply to one-parametric models.