  For many complex diseases, prognosis is of essential importance. It has been
shown that, beyond the main effects of genetic (G) and environmental (E) risk
factors, the gene-environment (G$\times$E) interactions also play a critical
role. In practice, the prognosis outcome data can be contaminated, and most of
the existing methods are not robust to data contamination. In the literature,
it has been shown that even a single contaminated observation can lead to
severely biased model estimation. In this study, we describe prognosis using an
accelerated failure time (AFT) model. An exponential squared loss is proposed
to accommodate possible data contamination. A penalization approach is adopted
for regularized estimation and marker selection. The proposed method is
realized using an effective coordinate descent (CD) and minorization
maximization (MM) algorithm. Simulation shows that without contamination, the
proposed method has performance comparable to or better than the unrobust
alternative. With contamination, it outperforms the unrobust alternative and,
under certain scenarios, can be superior to the robust method based on quantile
regression. The proposed method is applied to the analysis of TCGA (The Cancer
Genome Atlas) lung cancer data. It identifies interactions different from those
using the alternatives. The identified marker have important implications and
satisfactory stability.
