Estimation of a hazard rate is important in, for example,
survival analysis. Sometimes additional information on the
qualitative form of the hazard function is available. For
example, if a treatment is found to be effective, the hazard
function for individuals who are offered treatment can be
modelled as a decreasing function of time.

We briefly discuss recent methods for nonparametric estimation
of a hazard function under the constraint of monotonicity.
A constrained estimator can be more accurate than an
unconstrained estimator if the assumption of monotonicity is
a valid one.  We next address the issue of testing whether a
hazard function is monotone or not. Several nonparametric
testing procedures are briefly reviewed: a test based on a
monotonized kernel-type hazard estimator, a test based on
normalized spacings, and a  test based on the convexity of
the cumulative hazard function, among others. We explain more
in detail a nonparametric testing procedure which goes back to
a test statistic proposed by Prochan and Pyke (1967) for
testing for a constant hazard rate. The performance of the
testing procedure is demonstrated via simulations, and the
use of the tests is illustrated on  some datasets.

Part of this talk is based on joint works with James Gifford,
Peter Hall, Li-Shan Huang and Nancy Heckman.