In this manuscript, we study stochastic comparisons of the second-order statistics from dependent or independent observations with modified proportional hazard rates models. First, we establish the usual stochastic order of the second-order statistics from dependent and heterogeneous observations. Second, sufficient conditions are provided in the hazard rate order of the second-order statistics from independent observations. Then, we investigate the hazard rate order of the second-order statistics arising from two sets of independent multiple-outlier modified proportional hazard rates observations. Finally, some numerical examples are given to illustrate the theoretical findings.
Smooth backfitting has proven to have a number of theoretical and practical advantages in structured regression. Smooth backfitting projects the data down onto the structured space of interest providing a direct link between data and estimator. This paper introduces the ideas of smooth backfitting to survival analysis in a proportional hazard model, where we assume an underlying conditional hazard with multiplicative components. We develop asymptotic theory for the estimator and we use the smooth backfitter in a practical application, where we extend recent advances of in-sample forecasting me…
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…
Many insurance premium principles are defined and various estimation procedures introduced in the literature. In this paper, we focus on the estimation of the excess-of-loss reinsurance premium when the risks are randomly right-censored. The asymptotic normality of the proposed estimator is established under suitable conditions and its performance evaluated through sets of simulated data.
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…
This paper proposes a decorrelation-based approach to test hypotheses and construct confidence intervals for the low dimensional component of high dimensional proportional hazards models. Motivated by the geometric projection principle, we propose new decorrelated score, Wald and partial likelihood ratio statistics. Without assuming model selection consistency, we prove the asymptotic normality of these test statistics, establish their semiparametric optimality. We also develop new procedures for constructing pointwise confidence intervals for the baseline hazard function and baseline survival…
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…
Severe Ionosphere magnetic storms are feared events for integrity and continuity of navigation systems such as EGNOS, the European SBAS (Satellite-Based Augmentation System) complementing GPS and an accurate modelling of this event probability is necessary. Our aim for the work presented in this paper is to give an estimation of the frequency of such extreme magnetic storms per time unit (year) throughout a solar cycle. Thus, we develop an innovative approach based on a proportional hazard model, inspired by the Cox model, with time dependent covariates. The number of storms during a cycle is …
Let $X_1, \ldots , X_n$ be mutually independent exponential random variables with distinct hazard rates $λ_1, \ldots , λ_n > 0$ and let $Y_1, \ldots, Y_n$ be a random sample from the exponential distribution with hazard rate $\bar \lmd = \sum_{i=1}^n \lmd_i/n$. Also let $X_{1:n} < \cdots < X_{n:n}$ and $Y_{1:n} < \cdots < Y_{n:n}$ be their associated order statistics. It is shown that for $1\le i <j \le n$, the generalized spacing $X_{j:\, n} - X_{i:\, n}$ is more dispersed than $Y_{j:\,n} - Y_{i:\, n}$ according to dispersive ordering. This result is used to solve a long standing open problem…
In this paper, we develop robust estimators and tests for one-shot device testing under proportional hazards assumption based on divergence measures. Through a detailed Monte Carlo simulation study and a numerical example, the developed inferential procedures are shown to be more robust than the classical procedures, based on maximum likelihood estimators.
The semi-parametric Cox proportional hazards regression model has been widely used for many years in several applied sciences. However, a fully parametric proportional hazards model, if appropriately assumed, can often lead to more efficient inference. To tackle the extreme non-robustness of the traditional maximum likelihood estimator in the presence of outliers in the data under such fully parametric proportional hazard models, a robust estimation procedure has recently been proposed extending the concept of the minimum density power divergence estimator (MDPDE) under this set-up. In this pa…
The use of massive survival data has become common in survival analysis. In this study, a subsampling algorithm is proposed for the Cox proportional hazards model with time-dependent covariates when the sample is extraordinarily large but computing resources are relatively limited. A subsample estimator is developed by maximizing the weighted partial likelihood; it is shown to have consistency and asymptotic normality. By minimizing the asymptotic mean squared error of the subsample estimator, the optimal subsampling probabilities are formulated with explicit expressions. Simulation studies sh…
Otto-von-Guericke University Magdeburg（马格德堡奥托·冯·格里克大学）；Universidad Adolfo Ibáñez（阿道夫·伊巴涅斯大学）；Universidad de Valparaíso（瓦尔帕莱索大学）
In this paper we propose a Multiple kernel testing procedure to infer survival data when several factors (e.g. different treatment groups, gender, medical history) and their interaction are of interest simultaneously. Our method is able to deal with complex data and can be seen as an alternative to the omnipresent Cox model when assumptions such as proportionality cannot be justified. Our methodology combines well-known concepts from Survival Analysis, Machine Learning and Multiple Testing: differently weighted log-rank tests, kernel methods and multiple contrast tests. By that, complex hazard…
The proportional hazards assumption in the commonly used Cox model for censored failure time data is often violated in scientific studies. Yang and Prentice (2005) proposed a novel semiparametric two-sample model that includes the proportional hazards model and the proportional odds model as sub-models, and accommodates crossing survival curves. The model leaves the baseline hazard unspecified and the two model parameters can be interpreted as the short-term and long-term hazard ratios. Inference procedures were developed based on a pseudo score approach. Although extension to accommodate cova…
For statistical inference on regression models with a diverging number of covariates, the existing literature typically makes sparsity assumptions on the inverse of the Fisher information matrix. Such assumptions, however, are often violated under Cox proportion hazards models, leading to biased estimates with under-coverage confidence intervals. We propose a modified debiased lasso approach, which solves a series of quadratic programming problems to approximate the inverse information matrix without posing sparse matrix assumptions. We establish asymptotic results for the estimated regression…
The univariate distorted distribution were introduced in risk theory to represent changes (distortions) in the expected distributions of some risks. Later they were also applied to represent distributions of order statistics, coherent systems, proportional hazard rate (PHR) and proportional reversed hazard rate (PRHR) models, etc. In this paper we extend this concept to the multivariate setup. We show that, in some cases, they are a valid alternative to the copula representations especially when the marginal distributions may not be easily handled. Several relevant examples illustrate the appl…
In engineering systems, it is usually assumed that lifetimes of components are independent and identically distributed (iid). But, the failure of a component results in a higher load on the remaining components and hence causes the distribution of the surviving components change. For modeling this kind of systems, the theory of sequential order statistics (SOS) can be used. Assuming Weibull distribution for lifetimes of components and conditionally proportional hazard rates model as a special case of the SOS theory, the maximum likelihood estimates of the unknown parameters are obtained in dif…
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.
We study large sample properties of Bayesian analysis of the proportional hazard model with neutral to the right process priors on the baseline hazard function. We show that the posterior distribution of the baseline cumulative hazard function and regression coefficients centered at the maximum likelihood estimator is jointly asymptotically equivalent to the sampling distribution of the maximum likelihood estimator.
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…
For time-to-event data with finitely many competing risks, the proportional hazards model has been a popular tool for relating the cause-specific outcomes to covariates [Prentice et al. Biometrics 34 (1978) 541--554]. This article studies an extension of this approach to allow a continuum of competing risks, in which the cause of failure is replaced by a continuous mark only observed at the failure time. We develop inference for the proportional hazards model in which the regression parameters depend nonparametrically on the mark and the baseline hazard depends nonparametrically on both time a…
Cox's proportional hazards model is one of the most popular statistical models to evaluate associations of exposure with a censored failure time outcome. When confounding factors are not fully observed, the exposure hazard ratio estimated using a Cox model is subject to unmeasured confounding bias. To address this, we propose a novel approach for the identification and estimation of the causal hazard ratio in the presence of unmeasured confounding factors. Our approach is based on a binary instrumental variable, and an additional no-interaction assumption in a first stage regression of the tre…
Tests for proportional hazards assumption concerning specified covariates or groups of covariates are proposed. The class of alternatives is wide: log-hazard rates under different values of covariates may cross, approach, go away. The data may be right censored. The limit distribution of the test statistic is derived. Power of the test against approaching alternatives is given. Real data examples are considered.
The Dantzig selector for the proportional hazards model proposed by D.R. Cox is studied in a high-dimensional and sparse setting. We prove the $l_q$ consistency for all $q \geq 1$ of some estimators based on the compatibility factor, the weak cone invertibility factor, and the restricted eigenvalue for certain deterministic matrix which approximates the Hessian matrix of log partial likelihood. Our matrix conditions for these three factors are weaker than those of previous researches.
In many public health problems, an important goal is to identify the effect of some treatment/intervention on the risk of failure for the whole population. A marginal proportional hazards regression model is often used to analyze such an effect. When dependent censoring is explained by many auxiliary covariates, we utilize two working models to condense high-dimensional covariates to achieve dimension reduction. Then the estimator of the treatment effect is obtained by maximizing a pseudo-likelihood function over a sieve space. Such an estimator is shown to be consistent and asymptotically nor…
Several works have been undertaken in the context of proportional reversed hazard rate (PRHR) since last few decades. But any specific statistical methodology for the PRHR hypothesis is absent in the literature. In this paper, a two-sample nonparametric test based on two independent samples is proposed for verifying the PRHR assumption. Based on a consistent U-statistic three statistical methodologies have been developed exploiting U-statistics theory, jackknife empirical likelihood and adjusted jackknife empirical likelihood method. A simulation study has been performed to assess the merit of…
We consider a class of semiparametric regression models which are one-parameter extensions of the Cox [J. Roy. Statist. Soc. Ser. B 34 (1972) 187-220] model for right-censored univariate failure times. These models assume that the hazard given the covariates and a random frailty unique to each individual has the proportional hazards form multiplied by the frailty. The frailty is assumed to have mean 1 within a known one-parameter family of distributions. Inference is based on a nonparametric likelihood. The behavior of the likelihood maximizer is studied under general conditions where the fitt…
Fan, Gijbels and King [Ann. Statist. 25 (1997) 1661--1690] considered the estimation of the risk function $ψ(x)$ in the proportional hazards model. Their proposed estimator is based on integrating the estimated derivative function obtained through a local version of the partial likelihood. They proved the large sample properties of the derivative function, but the large sample properties of the estimator for the risk function itself were not established. In this paper, we consider direct estimation of the relative risk function $ψ(x_2)-ψ(x_1)$ for any location normalization point $x_1$. The ma…
\noindent Randomized nomination sampling (RNS) is a rank-based sampling technique which has been shown to be effective in several nonparametric studies involving environmental and ecological applications. In this paper, we investigate parametric inference using RNS design for estimating the unknown vector of parameters $\boldsymbolθ$ in the proportional hazard rate and proportional reverse hazard rate models. We examine both maximum likelihood (ML) and method of moments (MM) methods and investigate the relative precision of our proposed RNS-based estimators compared with those based on simple …
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…
Most statistical tests for treatment effects used in randomized clinical trials with survival outcomes are based on the proportional hazards assumption, which often fails in practice. Data from early exploratory studies may provide evidence of non-proportional hazards which can guide the choice of alternative tests in the design of practice-changing confirmatory trials. We study a test to detect treatment effects in a late-stage trial which accounts for the deviations from proportional hazards suggested by early-stage data. Conditional on early-stage data, among all tests which control the fre…
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…
We present a new estimator of the restricted mean survival time in randomized trials where there is right censoring that may depend on treatment and baseline variables. The proposed estimator leverages prognostic baseline variables to obtain equal or better asymptotic precision compared to traditional estimators. Under regularity conditions and random censoring within strata of treatment and baseline variables, the proposed estimator has the following features: (i) it is interpretable under violations of the proportional hazards assumption; (ii) it is consistent and at least as precise as the …
In this paper, we have obtained conditions on parameters that result in dispersive ordering and star ordering among two unequal sets of random variables from Proportional hazard rate and Proportional reversed hazard rate family of distributions. The n-independent random variables under observation belonging to a multiple-outlier model are given as corollary. The conditions obtained involve simple inequalities among the parameters and class of life distribution corresponding to ageing. Some stochastic ordering results for sample minimum and maximum with dependency based on Archimedean copula ha…
In many biomedical applications, outcome is measured as a ``time-to-event'' (eg. disease progression or death). To assess the connection between features of a patient and this outcome, it is common to assume a proportional hazards model, and fit a proportional hazards regression (or Cox regression). To fit this model, a log-concave objective function known as the ``partial likelihood'' is maximized. For moderate-sized datasets, an efficient Newton-Raphson algorithm that leverages the structure of the objective can be employed. However, in large datasets this approach has two issues: 1) The com…
A class of random hazard rates, which is defined as a mixture of an indicator kernel convolved with a completely random measure, is of interest. We provide an explicit characterization of the posterior distribution of this mixture hazard rate model via a finite mixture of S-paths. A closed and tractable Bayes estimator for the hazard rate is derived to be a finite sum over S-paths. The path characterization or the estimator is proved to be a Rao--Blackwellization of an existing partition characterization or partition-sum estimator. This accentuates the importance of S-paths in Bayesian modelin…
Although proportional hazard rate model is a very popular model to analyze failure time data, sometimes it becomes important to study the additive hazard rate model. Again, sometimes the concept of the hazard rate function is abstract, in comparison to the concept of mean residual life function. A new model called `dynamic additive mean residual life model' where the covariates are time-dependent has been defined in the literature. Here we study the closure properties of the model for different positive and negative ageing classes under certain condition(s). Quite a few examples are presented …
The lifetimes of subjects which are left-censored lie below a threshold value or a limit of detection. A popular tool used to handle left-censored data is the reversed hazard rate. In this work, we study the properties and develop characterizations of a class of distributions based on proportional reversed hazard rates used for analyzing left censored data. These characterizations are applied to simulate samples as well as analyze real data using distributions belonging to this class.
