  Stochastic processes often exhibit sudden systematic changes in pattern a
short time before certain failure events. Examples include increase in medical
costs before death and decrease in CD4 counts before AIDS diagnosis. To study
such terminal behavior of stochastic processes, a natural and direct way is to
align the processes using failure events as time origins. This paper studies
backward stochastic processes counting time backward from failure events, and
proposes one-sample nonparametric estimation of the mean of backward processes
when follow-up is subject to left truncation and right censoring. We will
discuss benefits of including prevalent cohort data to enlarge the identifiable
region and large sample properties of the proposed estimator with related
extensions. A SEER--Medicare linked data set is used to illustrate the proposed
methodologies.
