  We investigate nonparametric estimation of a monotone baseline hazard and a
decreasing baseline density within the Cox model. Two estimators of a
nondecreasing baseline hazard function are proposed. We derive the
nonparametric maximum likelihood estimator and consider a Grenander type
estimator, defined as the left-hand slope of the greatest convex minorant of
the Breslow estimator. We demonstrate that the two estimators are strong
consistent and asymptotically equivalent and derive their common limit
distribution at a fixed point. Both estimators of a nonincreasing baseline
hazard and their asymptotic properties are acquired in a similar manner.
Furthermore, we introduce a Grenander type estimator for a nonincreasing
baseline density, defined as the left-hand slope of the least concave majorant
of an estimator of the baseline cumulative distribution function, derived from
the Breslow estimator. We show that this estimator is strong consistent and
derive its asymptotic distribution at a fixed point.
