Small Sample Confidence Intervals for Survival Functions under the Proportional Hazards Model
We develop a saddlepoint-based method for generating small sample confidence bands for the population surviival function from the Kaplan-Meier (KM), the product limit (PL), and Abdushukurov-Cheng-Lin (ACL) survival function estimators, under the proportional hazards model. In the process we derive the exact distribution of these estimators and developed mid-ppopulation tolerance bands for said estimators. Our saddlepoint method depends upon the Mellin transform of the zero-truncated survival estimator which we derive for the KM, PL, and ACL estimators. These transforms are inverted via saddlepoint approximations to yield highly accurate approximations to the cumulative distribution functions of the respective cumulative hazard function estimators and these distribution functions are then inverted to produce our saddlepoint confidence bands. For the KM, PL and ACL estimators we compare our saddlepoint confidence bands with those obtained from competing large sample methods as well as those obtained from the exact distribution. In our simulation studies we found that the saddlepoint confidence bands are very close to the confidence bands derived from the exact distribution, while being much easier to compute, and outperform the competing large sample methods in terms of coverage probability.
R. L. Paige and E. Abdurasul, "Small Sample Confidence Intervals for Survival Functions under the Proportional Hazards Model," Communications in Statistics - Theory and Methods, vol. 47, no. 24, pp. 6108-6124, Taylor & Francis, Dec 2018.
The definitive version is available at https://doi.org/10.1080/03610926.2017.1406514
Mathematics and Statistics
Keywords and Phrases
Probability distributions; ACL estimator; Kaplan-Meier estimators; Mellin transform; Proportional hazards; Saddle-point approximation; Distribution functions; Saddlepoint approximations
International Standard Serial Number (ISSN)
Article - Journal
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01 Dec 2018