Abstract
In the competing risks problem an important role is played by the cumulative incidence function (CIF), whose value at time t is the probability of failure by time t from a particular type of risk in the presence of other risks. Assume that the lifetime distributions of two populations are uniformly stochastically ordered. Since this ordering may not hold for the empiricals due to sampling variability, it is natural to estimate these distributions under this constraint. This will in turn affect the estimation of the CIFs. This article considers this estimation problem. We do not assume that the risk sets in the two populations are related, give consistent estimators of all the CIFs and study the weak convergence of the resulting processes. We also report the results of a simulation study that show that our restricted estimators outperform the unrestricted ones in terms of mean square error. A real life example is used to illustrate our theoretical results.
Similar content being viewed by others
References
Aly EAA, Kochar SC, McKeague IW (1994) Some tests for comparing cumulative incidence functions and cause-specific hazard rates. J Am Stat Assoc 89: 994–999
Arcones MA, Samaniego FJ (2000) On the asymptotic distribution theory of a class of consistent estimators of a distribution satisfying a uniform stochastic ordering constraint. Ann Stat 28: 116–1150
Block HW, Basu AP (1974) A continuous bivariate exponential extension. J Am Stat Assoc 69: 1031–1037
Breslow N, Crowley J (1974) A large sample study of the life table and product limit estimates under random censorship. Ann Stat 2: 437–453
Chung KL (1974) A course in probability theory. Academic Press, New York
Dyktra R, Kochar S, Robertson T (1991) Statistical inference for uniform stochastic ordering in several populations. Ann Stat 19: 870–888
El Barmi H, Mukerjee H (2004) Consistent estimation of a distribution with type II bias with applications in competing risks problems. Ann Stat 32: 245–267
El Barmi H, Mukerjee H (2006) Restricted estimation of cumulative incidence functions corresponding to competing risks. In: Rojo J (ed) Optimality, second Lehmann symposium, IMS-LNMS, pp 241–252
El Barmi H, Kochar S, Mukerjee H, Samaniego FJ (2004) Inference for subsurvival functions under an order restriction. J Stat Plan Inference 118: 145–165
El Barmi H, Johnson M, Mukerjee H (2010) Estimating cumulative incidence functions when the life distributions are constrained. J Multivar Anal 101: 1903–1909
Fleming TR, Harrington DP (1991) Counting processes and survival analysis. Wiley, New York
Hoel DG (1972) A representation of mortality data by competing risks. Biometrics 28: 475–488
Lin DY (1997) Nonparametric inference for cumulative incidence functions in competing risks studies. Stat Med 16: 901–910
Mukerjee H (1996) Estimation of survival functions under uniform stochastic ordering. J Am Stat Assoc 91: 1684–1689
Peterson AV (1977) Expressing the Kaplan–Meier estimator as a function of empirical survival functions. J Am Stat Assoc 72: 854–858
Rojo J, Samaniego FJ (1991) On nonparametric maximum likelihood estimation of a distribution uniformly stochastically smaller than a standard. Stat Probab Lett 11: 267–271
Rojo J, Samaniego FJ (1993) On estimating a survival curve subject to a uniform stochastic ordering constraint. J Am Stat Assoc 88: 566–572
van der Vart AW, Wellner JA (1996) Weak convergence and empirical processes with applications to statistics. Springer, New York
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Al-Kandari, N.M.A., Aly, EE.A.A. & El Barmi, H. Estimation of cumulative incidence functions when the lifetime distributions are uniformly stochastically ordered. Lifetime Data Anal 18, 19–35 (2012). https://doi.org/10.1007/s10985-011-9204-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10985-011-9204-2