Asymptotic theory for nonparametric estimation of survival curves under order restrictions

We consider two problems in nonparametric survival analysis under the restriction of stochastic ordering. The first problem is that of estimating a survival function F̄(t) under the restriction F̄(t) ≥ F̄0(t), all t, where F̄0(t) is known. The second problem consists of estimating two unknown survival functions F̄(1)(t) and F̄(2)(t) when it is known that F̄(1)(t) ≥ F̄(2)(t), all t. The nonparametric maximum likelihood estimators in these problems were derived by Brunk, Franck, Hansen and Hogg and Dykstra. In the present paper we derive their large-sample distributions. We present two sets of proofs depending on whether or not the data are right-censored. When centered and scaled by n1/2, the estimators converge in distribution to limiting processes related to the concave majorant of Brownian motion. The limiting distributions are not known in closed form, but can be simulated for the purpose of forming asymptotic pointwise confidence limits.