Estimating the Cumulative incidence Functions under Length-biased Sampling

Consider a population of individuals who experience K dierent causes of death.We observe the ones alive at time t0 and follow them until death or possiblecensoring time. Given this length biased sample, we introduce estimators ofthe cumulative incidence functions of "initial survival times" (i.e. for the entirepopulation) in a competing risks setup. The large sample behavior of our estimatorsis studied and tests for equality of cumulative incidence functions aredeveloped.