On the L_p error of the Grenander‐type estimator in the Cox model

Abstract

We consider the Cox regression model and study the asymptotic global behavior of the Grenander-type estimator for a monotone baseline hazard function. This model is not included in the general setting of Durot (2007). However, we show that a similar central limit theorem holds for L p-error of the Grenander-type estimator. As an illustration of application of our main result, we propose a test procedure for a Weibull baseline distribution, based on the L p-distance between the Grenander estimator and a parametric estimator of the baseline hazard. Simulation studies are performed to investigate the performance of this test.

On the $L_p$-error of the Grenander-type estimator in the Cox model

“…This idea is somewhat similar to what is done in [9] where conditioning with respect to the design points is needed to obtain a martingale. Details on the proof can be found in Section 5 and in the Supplementary Material [10].…”

“…Afterwards we show that, if we consider the conditional probability P θ given θn , Lemma 2 holds also if t is replaced by a sequence depending on n. In this case, we apply this Lemma with t = U θ(a) and using Lemmas 17 and 18 in [10], we obtain that the remainder term r n satisfies…”

“…The proof follows the ones of Theorems 3 and 4 but it is more technical because the 'true' functions in the Bootstrap version depend on n and are less smooth than the original ones. Some ideas are given in Section 5 and further details in the Supplemental Material [10].…”

“…where A 0 is taken from (10). Note that the smoothed version Ψ s n of Φ n − Φ is needed in order to make the function a → a ξ differentiable.…”

Results on standard estimators in the Cox model

“…formly bounded moments of any order. Such results are needed, for example, when studying global errors of the Grenander-type estimator of a monotone baseline hazard (see Appendix D in Durot and Musta (2018)).…”