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.