Statistics and Its Interface

Volume 13 (2020)

Number 2

A composite nonparametric product limit approach for estimating the distribution of survival times under length-biased and right-censored data

Pages: 221 – 235

DOI: https://dx.doi.org/10.4310/SII.2020.v13.n2.a7

Authors

Shuqin Fan (School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China; and Department of Management Sciences, City University of Hong Kong)

Wei Zhao (Department of Biostatistics and Bioinformatics, Emory University, Atlanta, Georgia, U.S.A.; School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China; and Department of Management Sciences, City University of Hong Kong)

Alan T. K. Wan (Department of Management Sciences, City University of Hong Kong)

Yong Zhou (Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China)

Abstract

This paper considers a composite nonparametric product limit estimator for estimating the distribution of survival times when the data are length-biased and right censored. Our method takes into account auxiliary information that frequently arises in survival analysis, and is easier to implement than existing methods for estimating survival functions. We derive a strong representation of the proposed estimator, establish its consistency and asymptotic normality, and derive its convergence rate of approximation. As well, we prove that auxiliary information improves the asymptotic efficiency of the proposed estimator, and provide the values of the composite weights that result in the largest efficiency gain. Our proposed estimator fares well in comparison with other more complex methods in finite samples and offers a clear advantage with respect to computational time.

Keywords

biased data, composite nonparametric estimator, almost sure representation, right-censored, strong consistency, product-limit estimator

Received 8 February 2019

Received revised 20 September 2019

Accepted 5 November 2019