  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.
