Isotonized smooth estimators of a monotone baseline hazard in the Cox model

Abstract

We consider two isotonic smooth estimators for a monotone baseline hazard in the Cox model, a maximum smooth likelihood estimator and a Grenander-type estimator based on the smoothed Breslow estimator for the cumulative baseline hazard. We show that they are both asymptotically normal at rate n m/(2m+1) , where m ≥ 2 denotes the level of smoothness considered, and we relate their limit behavior to kernel smoothed isotonic estimators studied in Lopuhaä and Musta (2016). It turns out that the Grenander-type estimator is asymptotically equivalent to the kernel smoothed isotonic estimators, while the maximum smoothed likelihood estimator exhibits the same asymptotic variance but a different bias. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behavior of the two methods.

Central limit theorems for the $L_{p}$-error of smooth isotonic estimators

“…Proof. The proof is completely similar to that of Lemma A.7 in [36]. Note that condition (8) in that paper follows from our Assumption (A2) and that here λ is a decreasing function.…”

“…To prove (S23), we split the interval [0, 1] in five intervalsI 1 = [0, b), I 2 = [b, b γ ), I 3 = [b γ , 1 − b γ ], I 4 = (1 − b γ , 1 − b] and I 5 = (1 − b, 1]. Then, as in Lemma 3.2 in[36], we show that P Λ * n (t) ≥ Λ s n (t), for all t ∈ I i ≥ 1 − δ/10, i = 1, . .…”

Limit Theory in Monotone Function Estimation

“…The first inequality in (9) is extended to the current status model in [47, see (11.32) and (11.33)] for the direct MLE, whereas the second inequality follows from [47,Theorem 11.3] for the inverse process, for all p ≥ 1. The results in (9) have been extended for p = 2 in [72] to isotonic estimators of a baseline hazard in the Cox model, and for all p ≥ 1, in [8, Theorems 4.1 and 4.5] to the settings of regression on a random design with subgaussian errors, and estimation of a density or monotone failure rate under right censoring.…”

“…Smooth isotonic estimators of a monotone function λ 0 that are constructed by smoothing followed by an isotonization step have been considered in [19,97,36,86], for the regression setting, in [35,91] for estimating a monotone density, and in [74], for estimation of the baseline hazard in the Cox model. Comparisons between isotonized smooth estimators and smoothed isotonic estimators were made in [77] for the regression setting, and in [50] for the current status model.…”

“…One approach for first smoothing and then isotonize, are maximum smoothed likelihood estimators (MSLE), obtained by smoothing the loglikelihood and then maximizing the smoothed likelihood over all monotone functions, see [35,50,74]. For example, in the current status problem (see also Section 2.1), the loglikehood…”