Asymptotic theory for nonparametric estimation of survival curves under order restrictions

Jens Thomas Præstgaard, Jian Huang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

15 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)1679-1716
Number of pages38
JournalAnnals of Statistics
Issue number4
Publication statusPublished - 1 Aug 1996
Externally publishedYes


  • Concave majorant
  • Nonparametric survival analysis
  • Order restrictions

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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