Smoothed Isotonic Estimators of a Monotone Baseline Hazard in the Cox Model

Lopuhaä, Hendrik P.; Musta, Eni

doi:10.1111/sjos.12321

Cited by 6 publications

(9 citation statements)

References 38 publications

(108 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For some ξ 1 >0, define

E_{n, 1} = \{|\sqrt{p} \sum_{I_{k} \subseteq I} (\frac{1}{n} \sum_{i = 1}^{n} | Z_{i} | e^{γ_{k}^{'} Z_{i}}) - \sqrt{p} \sum_{I_{k} \subseteq I} 𝔼 [| Z | e^{γ_{k}^{'} Z}]| \leq ξ_{1}\},

where the summations are over all subsets I k ={ i 1 ,…, i k } of I ={1,…, p }, and γ k is the vector consisting of coordinates

γ_{k j} = β_{0 j} + ϵ / (2 \sqrt{p})

, for j ∈ I k , and

γ_{k j} = β_{0 j} - ϵ / (2 \sqrt{p})

, for j ∈ I

∖

I k . It is shown in Lopuhaä and Musta (2018) (see the proof of Lemma 3.1) that

ℙ (E_{n, 1}) \to 1

and that in this event we have

\sup_{x \in ℝ} | D_{n}^{(1)} (x; β_{n}^{*}) | \leq \sqrt{p} <...>

…”

Section: Auxiliary Results and Proofsmentioning

confidence: 91%

“…We aim at obtaining bounds on tail probabilities of

Û_{n}

. A polynomial bound has been provided in lemma 6.3 of Lopuhaä & Musta, 2018, but it is not sufficient when dealing with the global error of the estimator. Here we derive exponential bounds which hold on an event E n with

ℙ (E_{n}) \to 1

similar to the one considered in Lopuhaä and Musta (2018).…”

Section: Auxiliary Results and Proofsmentioning

confidence: 99%

“…However, there is currently no theoretical study of the properties of the Kolmogorov–Smirnov type of test for this setting. We expect that the performance of the test will improve if instead of the Grenander estimator we consider a smoothed version of it (see Lopuhaä & Musta, 2018). However, that is outside the scope of this article and will be addressed in the future.…”

Section: Discussionmentioning

confidence: 99%

“…Moreover, in van Geloven et al (2013) it is shown that using nonparametric shape constrained techniques leads to more accurate estimators than the traditional ones. Estimation of the baseline hazard function under monotonicity constraints has been studied in Chung and Chang (1994), Lopuhaä and Nane (2013b), Lopuhaä and Musta (2017, 2018), where pointwise rates of convergence and limit distributions have been investigated. However, for goodness of fit tests, results on the global error of estimates are needed.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Durot

Musta

2020

Statistica Neerlandica

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…For some ξ 1 >0, define

E_{n, 1} = \{|\sqrt{p} \sum_{I_{k} \subseteq I} (\frac{1}{n} \sum_{i = 1}^{n} | Z_{i} | e^{γ_{k}^{'} Z_{i}}) - \sqrt{p} \sum_{I_{k} \subseteq I} 𝔼 [| Z | e^{γ_{k}^{'} Z}]| \leq ξ_{1}\},

where the summations are over all subsets I k ={ i 1 ,…, i k } of I ={1,…, p }, and γ k is the vector consisting of coordinates

γ_{k j} = β_{0 j} + ϵ / (2 \sqrt{p})

, for j ∈ I k , and

γ_{k j} = β_{0 j} - ϵ / (2 \sqrt{p})

, for j ∈ I

∖

I k . It is shown in Lopuhaä and Musta (2018) (see the proof of Lemma 3.1) that

ℙ (E_{n, 1}) \to 1

and that in this event we have

\sup_{x \in ℝ} | D_{n}^{(1)} (x; β_{n}^{*}) | \leq \sqrt{p} <...>

…”

Section: Auxiliary Results and Proofsmentioning

confidence: 91%

“…We aim at obtaining bounds on tail probabilities of

Û_{n}

ℙ (E_{n}) \to 1

similar to the one considered in Lopuhaä and Musta (2018).…”

Section: Auxiliary Results and Proofsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Durot

Musta

2020

Statistica Neerlandica

Self Cite

View full text Add to dashboard Cite

show abstract

“…By working with g j n = c j n −0.55∕(d+4) we respect Li and Datta (2001)'s underlying idea to oversmooth with respect to the optimal bandwidth for estimation, which is proportional to n −1/(d + 4) in d dimensions. For the constant c j we use a simple rule used by Groeneboom and Jongbloed (2015), Lopuhaä and Musta (2017), and Lopuhaä and Musta (2018), among others, that consists in taking each constant equal to the range of the jth covariate for j = 1, … , d.…”

Section: Simulationsmentioning

confidence: 99%

A new lack‐of‐fit test for quantile regression with censored data

Conde-Amboage

Keilegom

González–Manteiga

2021

Scandinavian J Statistics

View full text Add to dashboard Cite

A new lack-of-fit test for quantile regression models will be presented for the case where the response variable is right-censored. The test is based on the cumulative sum of residuals, and it extends the ideas of He and Zhu (2003) to censored quantile regression. It will be shown that the empirical process associated with the test statistic converges to a Gaussian process under the null hypothesis and is consistent. To approximate the critical values of the test, a bootstrap mechanism will be used.A simulation study will be carried out to study the performance of the new test in comparison with other tests available in the literature. Finally, a real data application will be presented to show the good properties of the new lack-of-fit test in practice. K E Y W O R D Sbootstrap calibration, censored data, lack-of-fit tests, quantile regression INTRODUCTIONAlthough mean regression is still a traditional benchmark in regression studies, the quantile approach is receiving increasing attention, because it allows a more complete description of the conditional distribution of the response given the covariate, and it is more robust to deviations from error normality. That is, while classical regression gives only information on the conditional

show abstract

Uniform moment bounds for the standard estimators in the Cox proportional hazard model

Durot

Musta

2021

Communications in Statistics - Theory and Methods

View full text Add to dashboard Cite

We consider the Cox regression model and prove some properties of the maximum partial likelihood estimator βn and of the the Breslow estimator Λ n . The asymptotic properties of these estimators have been widely studied in the literature but we are not aware of a reference where it is shown that they have uniformly bounded moments. These results are needed, for example, when studying global errors of shape restricted estimators of the baseline hazard function.

show abstract

Smoothed Isotonic Estimators of a Monotone Baseline Hazard in the Cox Model

Cited by 6 publications

References 38 publications

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

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

A new lack‐of‐fit test for quantile regression with censored data

Uniform moment bounds for the standard estimators in the Cox proportional hazard model

Contact Info

Product

Resources

About

Smoothed Isotonic Estimators of a Monotone Baseline Hazard in the Cox Model

Cited by 6 publications

References 38 publications

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

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

A new lack‐of‐fit test for quantile regression with censored data

Uniform moment bounds for the standard estimators in the Cox proportional hazard model

Contact Info

Product

Resources

About

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

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