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August 1996 Asymptotic theory for nonparametric estimation of survival curves under order restrictions
Jens Thomas Præstgaard, Jian Huang
Ann. Statist. 24(4): 1679-1716 (August 1996). DOI: 10.1214/aos/1032298291

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.

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Jens Thomas Præstgaard. Jian Huang. "Asymptotic theory for nonparametric estimation of survival curves under order restrictions." Ann. Statist. 24 (4) 1679 - 1716, August 1996. https://doi.org/10.1214/aos/1032298291

Information

Published: August 1996
First available in Project Euclid: 17 September 2002

zbMATH: 0896.62044
MathSciNet: MR1416656
Digital Object Identifier: 10.1214/aos/1032298291

Subjects:
Primary: 60E20, 62J02
Secondary: 60E12

Rights: Copyright © 1996 Institute of Mathematical Statistics

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Vol.24 • No. 4 • August 1996
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