Suppose we have a random sample of size n with multiple censoring. The exac
t Fisher information in the data is derived and expressed in terms of matri
ces when each block of censored data contains at least two order statistics
. The results are applied to determine how much Fisher information about th
e location (scale) parameter is contained in the middle (two tails) of an o
rdered sample. The results show that, for Cauchy, Laplace, logistic, and no
rmal distributions, the middle 30% (extreme half) of the ordered data conta
ins more than 80% of the Fisher information about the location (scale) para
meter. These results provide insight into the behavior of two well-known ro
bust linear estimators of the location parameter.