  In this paper, we revisit the LAPUE model with a different focus: we begin by
adopting a new penalty function which gives a smooth transition of the boundary
between lateness and no lateness and demonstrate the LAPUE model based on the
new penalty function has a unique equilibrium and is stable with respect to
(w.r.t.) small perturbation of probability distribution under moderate
conditions. We then move on to discuss statistical robustness of the modified
LAPUE (MLAPUE) model by considering the case that the data to be used for
fitting the density function may be perturbed in practice or there is a
discrepancy between the probability distribution of the underlying uncertainty
constructed with empirical data and the true probability distribution in
future, we investigate how the data perturbation may affect the equilibrium. We
undertake the analysis from two perspectives: (a) a few data are perturbed by
outliers and (b) all data are potentially perturbed. In case (a), we use the
well-known influence function to quantify the sensitivity of the equilibrium by
the outliers and in case (b) we examine the difference between empirical
distributions of the equilibrium based on perturbed data and the equilibrium
based on unperturbed data. To examine the performance of the MLAPUE model and
our theoretical analysis of statistical robustness, we carry out some numerical
experiments, the preliminary results confirm the statistical robustness as
desired.
