Theoretical Statistics and Mathematics Unit, ISI Delhi

Estimation of cumulative incidence functions in competing risks studies under an order restriction

by Hammou El Barmi, S. C. Kochar, Hari Mukerjee and Francisco J. Samaniego

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 $t$ for a particular type of failure
in the presence of other risks. Its estimation and asymptotic distribution theory have been
studied by many. In some cases there are reasons to believe that the CIF’s due to two types of
failure are order restricted. Several procedures have appeared in the literature for testing for
such orders. In this paper we initiate the study of estimation of two CIF’s subject to a type of
stochastic ordering, both when there are just two causes of failure and when there are more than
two causes of failure, treating those other than the two of interest as a censoring mechanism.
We do not assume independence of the two types of failure of interest, however, these are
assumed to be independent of the other causes in the censored case. Weak convergence results
for the estimators have been derived. It is shown that when the order restriction is strict, the
asymptotic distributions are the same as those for the empirical estimators without the order
restriction. Thus we get the restricted estimators “free of charge”, at least in the asymptotic
sense. When the two CIF’s are equal, the asymptotic MSE is reduced by using the order
restriction. For finite sample sizes simulations seem to indicate that the restricted estimators
have uniformly smaller MSE’s than the unrestricted ones in all cases.

isid/ms/2002/19 [fulltext]

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