The robustness of a sample min or max for a continuous distribution is basically proportional to the density of the distribution at that extremum. A steep or vertical drop-off at the edge of the distribution is the ideal case.
The article gives a formula for the statistical power of the hypothesis test derived from the sample min. It depends on the function h(x), whose purpose is to establish a lower bound on the density of the distribution at the min, and hence a lower bound on the robustness of the sample min as an estimator.
