I am reading about robust statistics.
Actually the rejection point is the thing that I can't understand.
Rejection point is defined as:



But what is the meaning of that? The function IF is defined as:



Which is actually a difference, and if it is small or even zero, this means that we have low error of estimation, if there outlier point is included.
The infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. In our case the S is actually all points (real valued: R) so that IF(x; T, F) = 0.
As I understand IF function actually compares how strong the x value will influence the estimation. For example for mean value the influence will be very big. For median the influence will be zero. So IF function (influence funtion) can measure how well the estimator is, when there is an outlier point.
But let P be the rejection point.

lets --- be the axis, from 0 to infinity.

0------P------>
It is written that that point P, when finite represents the point at whih abservations are cimpletely rejected. I suppose that the set S (see the text above) is actually in this example a set consisting of all points untill P, from 0 to P. So the infinym should be 0. Help me to understand please. I've understood the gross error sensitivity, which is supremum. The bigger the supremum the more unstable the estimator is for some big outlier points.
Thank you