  The ratio of the hazard functions of two populations or two strata of a
single population plays an important role in time-to-event analysis. Cox
regression is commonly used to estimate the hazard ratio under the assumption
that it is constant in time, which is known as the proportional hazards
assumption. However, this assumption is often violated in practice, and when it
is violated, the parameter estimated by Cox regression is difficult to
interpret. The hazard ratio can be estimated in a nonparametric manner using
smoothing, but smoothing-based estimators are sensitive to the selection of
tuning parameters, and it is often difficult to perform valid inference with
such estimators. In some cases, it is known that the hazard ratio function is
monotone. In this article, we demonstrate that monotonicity of the hazard ratio
function defines an invariant stochastic order, and we study the properties of
this order. Furthermore, we introduce an estimator of the hazard ratio function
under a monotonicity constraint. We demonstrate that our estimator converges in
distribution to a mean-zero limit, and we use this result to construct
asymptotically valid confidence intervals. Finally, we conduct numerical
studies to assess the finite-sample behavior of our estimator, and we use our
methods to estimate the hazard ratio of progression-free survival in pulmonary
adenocarcinoma patients treated with Gefitinib or carboplatin-paclitaxel.
