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Estimation theory, Incidence functions, Competing risks, Asymptotic distribution (Probability theory)


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 CIFs 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 CIFs 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 CIFs 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 MSEs than the unrestricted ones in all cases.


NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Statistical Planning and Inference. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Statistical Planning and Inference, January 2004, Vol. 118 Issue 1/2, p145-165 and can be found online at:

*At the time of publicationSubhash C. Kochar was affiliated with the Indian Statistical Institute.



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