Asymptotic theory for nonparametric estimation of survival curves under order restrictions

Abstract

We consider two problems in nonparametric survival analysis under the restriction of stochastic ordering. The first problem is that of estimating a survival function $\overline{F}(t)$ under the restriction $\overline{F}(t) \geq $\overline{F}_0 (t)$, all t, where $\overline{F}_0 (t)$ is known. The second problem consists of estimating two unknown survival functions $\overline{F}^{(1)}(t)$ and $\overline{F}^{(2)}(t)$ when it is known that $\overline{F}^{(1)}(t) \geq \overline{F}^{(2)}(t)$, 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 $n^{1/2}$, 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.