We consider the Cox regression model and prove some properties of the maximum partial likelihood estimator $\hatβ_n$ 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.
We consider the smoothed maximum likelihood estimator and the smoothed Grenander-type estimator for a monotone baseline hazard rate $λ_0$ in the Cox model. We analyze their asymptotic behavior and show that they are asymptotically normal at rate $n^{m/(2m+1)}$, when~$λ_0$ is $m\geq 2$ times continuously differentiable, and that both estimators are asymptotically equivalent. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behavior of the two methods.
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 $m\geq 2$ 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,…
We investigate the problems of drift estimation for a shifted Brownian motion and intensity estimation for a Cox process on a finite interval $[0,T]$, when the risk is given by the energy functional associated to some fractional Sobolev space $H^1_0\subset W^{α,2}\subset L^2$. In both situations, Cramer-Rao lower bounds are obtained, entailing in particular that no unbiased estimators with finite risk in $H^1_0$ exist. By Malliavin calculus techniques, we also study super-efficient Stein type estimators (in the Gaussian case).
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 a…
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. We also 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.
Cox proportional hazards model with measurement error is investigated. In Kukush et al. (2011) [Journal of Statistical Research 45, 77-94] and Chimisov and Kukush (2014) [Modern Stochastics: Theory and Applications 1, 13-32] asymptotic properties of simultaneous estimator $λ_n(\cdot)$, $β_n$ were studied for baseline hazard rate $λ(\cdot)$ and regression parameter $β$, at that the parameter set $Θ=Θ_λ\times Θ_β$ was assumed bounded. In the present paper, the set $Θ_λ$ is unbounded from above and not separated away from $0$. We construct the estimator in two steps: first we derive a strongly co…
The purpose of this article is to provide an adaptive estimator of the baseline function in the Cox model with high-dimensional covariates. We consider a two-step procedure : first, we estimate the regression parameter of the Cox model via a Lasso procedure based on the partial log-likelihood, secondly, we plug this Lasso estimator into a least-squares type criterion and then perform a model selection procedure to obtain an adaptive penalized contrast estimator of the baseline function. Using non-asymptotic estimation results stated for the Lasso estimator of the regression parameter, we estab…
The relationship between a time-dependent covariate and survival times is usually evaluated via the Cox model. Time-dependent covariates are generally available as longitudinal data collected regularly during the course of the study. A frequent problem, however, is the occurence of missing covariate data. A recent approach to estimation in the Cox model in this case jointly models survival and the longitudinal covariate. However, theoretical justification of this approach is still lacking. In this paper we prove existence and consistency of the maximum likelihood estimators in a joint model. T…
The change-plane Cox model is a popular tool for the subgroup analysis of survival data. Despite the rich literature on this model, there has been limited investigation into the asymptotic properties of the estimators of the finite-dimensional parameter. Particularly, the convergence rate, not to mention the asymptotic distribution, has not been fully characterized for the general model where classification is based on multiple covariates. To bridge this theoretical gap, this study proposes a maximum smoothed partial likelihood estimator and establishes the following asymptotic properties. Fir…
We consider high-dimensional inference for potentially misspecified Cox proportional hazard models based on low dimensional results by Lin and Wei [1989]. A de-sparsified Lasso estimator is proposed based on the log partial likelihood function and shown to converge to a pseudo-true parameter vector. Interestingly, the sparsity of the true parameter can be inferred from that of the above limiting parameter. Moreover, each component of the above (non-sparse) estimator is shown to be asymptotically normal with a variance that can be consistently estimated even under model misspecifications. In so…
We propose a change detection method for the famous Cox--Ingersoll--Ross model. This model is widely used in financial mathematics and therefore detecting a change in its parameters is of crucial importance. We develop one- and two-sided testing procedures for both drift parameters of the process. The test process is based on estimators that are motivated by the discrete time least-squares estimators, and its asymptotic distribution under the no-change hypothesis is that of a Brownian bridge. We prove the asymptotic weak consistence of the test, and derive the asymptotic properties of the chan…
Functional covariates are common in many medical, biodemographic, and neuroimaging studies. The aim of this paper is to study functional Cox models with right-censored data in the presence of both functional and scalar covariates. We study the asymptotic properties of the maximum partial likelihood estimator and establish the asymptotic normality and efficiency of the estimator of the finite-dimensional estimator. Under the framework of reproducing kernel Hilbert space, the estimator of the coefficient function for a functional covariate achieves the minimax optimal rate of convergence under a…
We consider a non-stationary Cox-Ingersoll-Ross process. We establish a sharp large deviation principle for the maximum likelihood estimator of its drift parameter.
In this paper, we study the Adaptive Elastic Net method for the Cox model. We prove the grouping effect and oracle property of its estimators. Finally, we show these two properties by an empirical analysis and a numerical simulation, respectively.
This paper deals with the proportional hazards model proposed by D. R. Cox in a high-dimensional and sparse setting for a regression parameter. To estimate the regression parameter, the Dantzig selector is applied. The variable selection consistency of the Dantzig selector for the model will be proved. This property enables us to reduce the dimension of the parameter and to construct asymptotically normal estimators for the regression parameter and the cumulative baseline hazard function.
This paper considers maximum likelihood inference for a functional marked point process - the stochastic growth-interaction process - which is an extension of the spatio-temporal growth-interaction process to the stochastic mark setting. As a pilot study we here consider a particular version of this extended process, which has a homogenous Poisson process as unmarked point process and shifted independent Cox-Ingersoll-Ross processes as functional marks. These marks have supports determined by the lifetimes generated by an immigration-death process. By considering a (temporally) discrete sample…
To better understand the interplay of censoring and sparsity we develop finite sample properties of nonparametric Cox proportional hazard's model. Due to high impact of sequencing data, carrying genetic information of each individual, we work with over-parametrized problem and propose general class of group penalties suitable for sparse structured variable selection and estimation. Novel non-asymptotic sandwich bounds for the partial likelihood are developed. We establish how they extend notion of local asymptotic normality (LAN) of Le Cam's. Such non-asymptotic LAN principles are further exte…
Cox proportional hazard regression model is a popular tool to analyze the relationship between a censored lifetime variable with other relevant factors. The semi-parametric Cox model is widely used to study different types of data arising from applied disciplines like medical science, biology, reliability studies and many more. A fully parametric version of the Cox regression model, if properly specified, can yield more efficient parameter estimates leading to better insight generation. However, the existing maximum likelihood approach of generating inference under the fully parametric Cox reg…
The purpose of this paper is to construct confidence intervals for the regression coefficients in high-dimensional Cox proportional hazards regression models where the number of covariates may be larger than the sample size. Our debiased estimator construction is similar to those in Zhang and Zhang (2014) and van de Geer et al. (2014), but the time-dependent covariates and censored risk sets introduce considerable additional challenges. Our theoretical results, which provide conditions under which our confidence intervals are asymptotically valid, are supported by extensive numerical experimen…
We consider a particular Cox process from a Bayesian viewpoint and show that the Bayes estimator of the intensity measure is the so-called Pólya sum kernel, which occurred recently in the context of the construction of the so-called Papangelou processes. More precisely, if the prior, the directing measure of the Cox process, is a Poisson-Gamma random measure, then the posterior is again a Poisson-Gamma random measure and the Bayes estimator of the intensity is the Pólya sum kernel. Moreover, we extend this result to doubly stochastic Poisson-Gamma priors and give conditions under which one can…
In survival analysis, the lifetime under study is not always observed. In certain applications, for some individuals, the value of the lifetime is only known to be smaller or larger than some random duration. This framework represent an extension of standard situations where the lifetime is only left or only right randomly censored. We consider the case where the independent observation units include also some covariates, and we propose two semiparametric regression models. The new models extend the standard Cox proportional hazard model to the situation of a more complex censoring mechanism. …
High throughput genetic sequencing arrays with thousands of measurements per sample and a great amount of related censored clinical data have increased demanding need for better measurement specific model selection. In this paper we establish strong oracle properties of nonconcave penalized methods for nonpolynomial (NP) dimensional data with censoring in the framework of Cox's proportional hazards model. A class of folded-concave penalties are employed and both LASSO and SCAD are discussed specifically. We unveil the question under which dimensionality and correlation restrictions can an orac…
A class of estimating functions is introduced for the regression parameter of the Cox proportional hazards model to allow unknown failure statuses on some study subjects. The consistency and asymptotic normality of the resulting estimators are established under mild conditions. An adaptive estimator which achieves the minimum variance-covariance bound of the class is constructed. Numerical studies demonstrate that the asymptotic approximations are adequate for practical use and that the efficiency gain of the adaptive estimator over the complete-case analysis can be quite substantial. Similar …
Weighted likelihood, in which one solves Horvitz-Thompson or inverse probability weighted (IPW) versions of the likelihood equations, offers a simple and robust method for fitting models to two phase stratified samples. We consider semiparametric models for which solution of infinite dimensional estimating equations leads to $\sqrt{N}$ consistent and asymptotically Gaussian estimators of both Euclidean and nonparametric parameters. If the phase two sample is selected via Bernoulli (i.i.d.) sampling with known sampling probabilities, standard estimating equation theory shows that the influence …
The problem to establish not only the asymptotic distribution results for statistical estimators but also the moment convergence of the estimators has been recognized as an important issue in advanced theories of statistics. One of the main goals of this paper is to present a metod to derive the moment convergence of $Z$-estimators as it has been done for $M$-estimators. Another goal of this paper is to develop a general, unified approach, based on some partial estimation functions which we call "$Z$-process", to the change point problems for ergodic models as well as some models where the Fis…
This paper deals with the problem of global parameter estimation of affine diffusions in $\mathbb{R}_+ \times \mathbb{R}^n$ denoted by $AD(1, n)$ where $n$ is a positive integer which is a subclass of affine diffusions introduced by Duffie et al in [14]. The $AD(1, n)$ model can be applied to the pricing of bond and stock options, which is illustrated for the Vasicek, Cox-Ingersoll-Ross and Heston models. Our first result is about the classification of $AD(1, n)$ processes according to the subcritical, critical and supercritical cases. Then, we give the stationarity and the ergodicity theorems…
We study the absolute penalized maximum partial likelihood estimator in sparse, high-dimensional Cox proportional hazards regression models where the number of time-dependent covariates can be larger than the sample size. We establish oracle inequalities based on natural extensions of the compatibility and cone invertibility factors of the Hessian matrix at the true regression coefficients. Similar results based on an extension of the restricted eigenvalue can be also proved by our method. However, the presented oracle inequalities are sharper since the compatibility and cone invertibility fac…
Thomas' partial likelihood estimator of regression parameters is widely used in the analysis of nested case-control data with Cox's model. This paper proposes a new estimator of the regression parameters, which is consistent and asymptotically normal. Its asymptotic variance is smaller than that of Thomas' estimator away from the null. Unlike some other existing estimators, the proposed estimator does not rely on any more data than strictly necessary for Thomas' estimator and is easily computable from a closed form estimating equation with a unique solution. The variance estimation is obtained…
The stratified proportional intensity model generalizes Cox's proportional intensity model by allowing different groups of the population under study to have distinct baseline intensity functions. In this article, we consider the problem of estimation in this model when the variable indicating the stratum is unobserved for some individuals in the studied sample. In this setting, we construct nonparametric maximum likelihood estimators for the parameters of the stratified model and we establish their consistency and asymptotic normality. Consistent estimators for the limiting variances are also…
Under a single-index regression assumption, we introduce a new semiparametric procedure to estimate a conditional density of a censored response. The regression model can be seen as a generalization of Cox regression model and also as a profitable tool to perform dimension reduction under censoring. This technique extends the results of Delecroix et al. (2003). We derive consistency and asymptotic normality of our estimator of the index parameter by proving its asymptotic equivalence with the (uncomputable) maximum likelihood estimator, using martingales results for counting processes and argu…
By means of two simple convexity arguments we are able to develop a general method for proving consistency and asymptotic normality of estimators that are defined by minimisation of convex criterion functions. This method is then applied to a fair range of different statistical estimation problems, including Cox regression, logistic and Poisson regression, least absolute deviation regression outside model conditions, and pseudo-likelihood estimation for Markov chains. Our paper has two aims. The first is to exposit the method itself, which in many cases, under reasonable regularity conditions,…
From an optimizer's perspective, achieving the global optimum for a general nonconvex problem is often provably NP-hard using the classical worst-case analysis. In the case of Cox's proportional hazards model, by taking its statistical model structures into account, we identify local strong convexity near the global optimum, motivated by which we propose to use two convex programs to optimize the folded-concave penalized Cox's proportional hazards regression. Theoretically, we investigate the statistical and computational tradeoffs of the proposed algorithm and establish the strong oracle prop…
We consider a stable Cox--Ingersoll--Ross process driven by a standard Wiener process and a spectrally positive strictly stable Lévy process, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate based on continuous time observations. We distinguish three cases: subcritical, critical and supercritical. In all cases we prove strong consistency of the MLE in question, in the subcritical case asymptotic normality, and in the supercritical case asymptotic mixed normality are shown as well. In the critical case the description of the asymptotic behavior of…
