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There is a substantial literature on testing for the equality of the cumulative incidence functions associated with one specific cause in a competing risks setting across several populations against specific or all alternatives. In this paper we propose an asymptotically distribution-free test when the alternative is that the incidence functions are linearly ordered, but not equal. The motivation stems from the fact that in many examples such a linear ordering seems reasonable intuitively and is borne out generally from empirical observations. These tests are more powerful when the ordering is justified. We also provide estimators of the incidence functions under this ordering constraint, derive their asymptotic properties for statistical inference purposes, and show improvements over the unrestricted estimators when the order restriction holds., Comment: Published in at http://dx.doi.org/10.1214/193940307000000040 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)

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 failure in the presence of other risks. In some cases there are reasons to believe that the CIFs due to various types of failure are linearly ordered. El Barmi et al. [3] studied the estimation and inference procedures under this ordering when there are only two causes of failure. In this paper we extend the results to the case of $k$ CIFs, where $k\ge3$. Although the analyses are more challenging, we show that most of the results in the 2-sample case carry over to this $k$-sample case., Comment: Published at http://dx.doi.org/10.1214/074921706000000482 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)