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Biometrika 2003 90(3):724-727; doi:10.1093/biomet/90.3.724
© 2003 by Biometrika Trust
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Miscellanea

Identifiability and censored data

Nader Ebrahimi1, Daniel Molefe2 and Zhiliang Ying3

1 Division of Statistics, Northern Illinois University, DeKalb, Illinois 60115, U.S.Anader{at}math.niu.edu 2 Division of Biometry and Risk Assessment, U.S.Food and Drug Administration, Jefferson, Arkansas 72079, U.S.A. dmolefe{at}nctr.fda.gov 3 Department of Statistics, Columbia University, 2990 Broadway, New York, New York 10027, U.S.A. zying{at}stat.columbia.edu

It is well known that, without the assumption of independence between two nonnegative random variables X and Y, the survival function of X is not identifiable on the basis of the joint distribution function of Z = min(X, Y) and {delta} = I(Z = Y).In this paper, we provide a simple condition in the form of conditional distribution of Y given X. We show that our condition is equivalent to the constant-sum condition proposed by Williams & Lagakos (1977). As a result the survival function of X can be identified from the joint distribution of Z and {delta} and the Kaplan–Meier estimator with Greenwood's formula for its variance remains valid. Examples which satisfy the condition are given.

Key Words: Censored observation; Constant-sum condition; Greenwood's formula; Identifiability; Kaplan–Meier estimator; Survival function

Received September 2001. Revised November 2002


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