Probabilistic mean value theorems for conditioned random variables with applications

In this paper, we propose a probabilistic analogue of the mean value theorem for conditional nonnegative random variables ordered in the hazard rate and reversed hazard rate order, upon conditioning on intervals of the form (t,∞) and [0,t]. This result is then specialized within the proportional hazards model and the proportional reversed hazards model with applications to series systems in reliability theory and to absorption random times of linear birth‐death processes. We also study the comparison of residual entropies and discuss some connections to Wasserstein and stop‐loss distances of random variables. A treatment for the additive hazard rate model is finally provided, with an application to life annuities.     

Consistency and Asymptotic Normality of Estimators in a Proportional Hazards Model with Interval Censoring and Left Truncation

Satten et al. (1998, J. Amer. Statist. Assoc., 93, 318–327) proposed an approach to the proportional hazards model for interval censored data in which parameter estimates are obtained by solving estimating equations which are the score equations for the full data proportional hazards model, averaged over all rankings of imputed failure times consistent with the observed censoring intervals. In this paper, we extend this approach to incorporate data that are left-truncated and right censored (dynamic cohort data). Consistency and asymptotic normality of the estimators obtained in this way are established.     

Cox模型与BP神经网络在处理非线性数据时的性能比较

本文采用BP神经网络、Cox模型和bootstrap方法,比较BP神经网络与Cox模型在处理非线性资料时的性能。两种方法的预测一致性的均数分别为0.7525和0.7706。对于非线性资料,BP神经网络的预测效果优于Cox模型。     

Maximum likelihood estimation in the proportional hazards cure model

The proportional hazards cure model generalizes Cox’s proportional hazards model which allows that a proportion of study subjects may never experience the event of interest. Here nonparametric maximum likelihood approach is proposed to estimating the cumulative hazard and the regression parameters. The asymptotic properties of the resulting estimators are established using the modern empirical process theory. And the estimators for the regression parameters are shown to be semiparametric efficient.     

Some results on the proportional reversed hazards model

The proportional reversed hazards model consists in describing random failure times by a family {[F(x)]^θ, θ>0} of distribution functions, where F(x) is a baseline distribution function. We show various results on this model related to some topics in reliability theory, including ageing notions of random lifetimes, comparisons based on stochastic orders, and relative ageing of distributions.     

Assessment of local influence on Cox’s proportional hazards model

The second order approach of local influence (see [15]) is developed and applied to Cox’s proportional hazards model, and compared with Cook's local influence approach (see [6] and [13]) which was used in this model. To study local influence, we perturb not only all cases simultaneously, but also cases individually to obtain “direction curvature” in directionl and “curvature” for single case. Some examples are used to illustrate these methods. This work is supported by the Youth Science Foundation of Peking University and a research grant from State Educational Committee     

Estimation of a smooth quantile function under the proportional hazards model

The problem of estimating a smooth quantile function, Q(·), at a fixed point p, 0 < p < 1, is treated under a nonparametric smoothness condition on Q. The asymptotic relative deficiency of the sample quantile based on the maximum likelihood estimate of the survival function under the proportional hazards model with respect to kernel type estimators of the quantile is evaluated. The comparison is based on the mean square errors of the estimators. It is shown that the relative deficiency tends to infinity as the sample size, n, tends to infinity.     

Efficient Algorithm for Computing Maximum Likelihood Estimates in Linear Transformation Models

Linear transformation models, which have been extensively studied in survival analysis, include the two special cases: the proportional hazards model and the proportional odds model. Nonparametric maximum likelihood estimation is usually used to derive the efficient estimators. However, due to the large number of nuisance parameters, calculation of the nonparametric maximum likelihood estimator is difficult in practice, except for the proportional hazards model. We propose an efficient algorithm for computing the maximum likelihood estimates, where the dimensionality of the parameter space is dramatically reduced so that only a finite number of equations need to be solved. Moreover, the asymptotic variance is automatically estimated in the computing procedure. Extensive simulation studies indicate that the proposed algorithm works very well for linear transformation models. A real example is presented for an illustration of the new methodology.     

On estimation and inference in a partially linear hazard model with varying coefficients

We study estimation and inference in a marginal proportional hazards model that can handle (1) linear effects, (2) non-linear effects and (3) interactions between covariates. The model under consideration is an amalgamation of three existing marginal proportional hazards models studied in the literature. Developing an estimation and inference procedure with desirable properties for the amalgamated model is rather challenging due to the co-existence of all three effects listed above. Much of the existing literature has avoided the problem by considering narrow versions of the model. The object of this paper is to show that an estimation and inference procedure that accommodates all three effects is within reach. We present a profile partial-likelihood approach for estimating the unknowns in the amalgamated model with the resultant estimators of the unknown parameters being root- \(n\) consistent and the estimated functions achieving optimal convergence rates. Asymptotic normality is also established for the estimators.     

On goodness-of-fit for accelerated life models

A goodness-of-fit test for the generalized Sedyakin's model is proposed when accelerated experiments are carried out under step-stresses. Various alternatives, such as the Cox proportional hazards model with time-varying stresses, are considered. The test power against the neighbouring alternatives is studied.     

Generalized time-dependent conditional linear models under left truncation and right censoring

Consider the model f(S(z|X)){\phi(S(z|X))} = \pmbb(z) [(X)\vec]{\pmb{\beta}(z) {\vec{X}}}, where f{\phi} is a known link function, S(·|X) is the survival function of a response Y given a covariate X, [(X)\vec]{\vec{X}} = (1, X, X ² , . . . , X ^p ) and \pmbb(z){\pmb{\beta}(z)} is an unknown vector of time-dependent regression coefficients. The response is subject to left truncation and right censoring. Under this model, which reduces for special choices of f{\phi} to e.g. Cox proportional hazards model or the additive hazards model with time dependent coefficients, we study the estimation of the vector \pmbb(z){\pmb{\beta}(z)} . A least squares approach is proposed and the asymptotic properties of the proposed estimator are established. The estimator is also compared with a competing maximum likelihood based estimator by means of simulations. Finally, the method is applied to a larynx cancer data set.     

Asymptotic properties of the maximum likelihood estimator for the proportional hazards model with doubly censored data

Doubly censored data, which include left as well as right censored observations, are frequently met in practice. Though estimation of the distribution function with doubly censored data has seen much study, relatively little is known about the inference of regression coefficients in the proportional hazards model for doubly censored data. In particular, theoretical properties of the maximum likelihood estimator of the regression coefficients in the proportional hazards model have not been proved yet. In this paper, we show the consistency and asymptotic normality of the maximum likelihood estimator and prove its semiparametric efficiency. The proposed methods are illustrated with simulation studies and analysis of an application from a medical study.     

Using principal components in a proportional hazards model with applications in condition-based maintenance

This paper proposes the application of a principal components proportional hazards regression model in condition-based maintenance (CBM) optimization. The Cox proportional hazards model with time-dependent covariates is considered. Principal component analysis (PCA) can be applied to covariates (measurements) to reduce the number of variables included in the model, as well as to eliminate possible collinearity between the covariates. The main issues and problems in using the proposed methodology are discussed. PCA is applied to a simulated CBM data set and two real data sets obtained from industry: oil analysis data and vibration data. Reasonable results are obtained.     

Statistical analysis of survival and reliability data with multiple crossings of survival functions

We present a statistical model and semiparametric estimation procedure for analysis of survival data with multiple cross-effects (MCE) of survival functions. A goodness-of-fit test for the proportional hazards model against the MCE model is proposed. To cite this article: V. Bagdonavi?ius, M. Nikulin, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

含有治愈部分的可加危险模型的估计

最近可加危险(AH)模型被广泛地应用于生存分析数据,模型的协变量可以假设为时间独立或时间相关的.基于混合治愈模型,有界累计危险治愈模型和"不正确"的比例危险模型.本文将上述的可乘危险模型延伸到可加的危险模型,这里的模型可以允许含治愈部分的生存数据的存在."不正确"的AH模型的识别和参数估计也将在本文给出讨论.     

多元失效时间数据的一般边际半参数危险率模型研究

在生物医学研究中,多元失效时间数据非常常见.该文提出用一般边际半参数危险率回归模型来分析多元失效时间数据.此模型包括了三种常用边际模型:边际比例风险模型、边际加速失效时间模型和边际加速危险模型作为子模型.对于模型中的回归系数,可以通过估计方程的方法来估计它,同时也给出了基准累积危险率函数的估计.得到的估计可以证明是相合的和渐近正态的.     

Additive hazards models for gap time data with multiple causes

Recurrent event data with multiple causes are often observed in biomedical studies. The additive hazards model describes a different aspect of the association between covariates and the failure time than does the proportional hazards model. In this paper, we introduce additive hazards models for the analysis of gap time data of recurrent events with multiple causes. We estimate the regression parameter vector and cumulative baseline cause specific hazard rate function using counting process approach. Asymptotic properties of the estimators are studied. The proposed model is applied to the kidney dialysis data given in Lawless (2003). A simulation study is carried out to assess the performance of the estimates.     

European generic scoring models using survival analysis

Credit scoring discriminates between ‘good’ and ‘bad’ credit risks to assist credit-grantors in making lending decisions. Such discrimination may not be a good indicator of profit, while survival analysis allows profit to be modelled. The paper explores the application of parametric accelerated failure time and proportional hazards models and Cox non-parametric model to the data from the retail card (revolving credit) from three European countries. The predictive performance of three national models is tested for different timescales of default and then compared to that of a single generic model for a timescale of 25 months. It is found that survival analysis national and generic models produce predictive quality, which is very close to the current industry standard—logistic regression. Stratification is investigated as a way of extending Cox non-parametric proportional hazards model to tackle heterogeneous segments in the population.     

Empirical likelihood inference for linear transformation models

Empirical likelihood inference is developed for censored survival data under the linear transformation models, which generalize Cox's [Regression models and life tables (with Discussion), J. Roy. Statist. Soc. Ser. B 34 (1972) 187-220] proportional hazards model. We show that the limiting distribution of the empirical likelihood ratio is a weighted sum of standard chi-squared distribution. Empirical likelihood ratio tests for the regression parameters with and without covariate adjustments are also derived. Simulation studies suggest that the empirical likelihood ratio tests are more accurate (under the null hypothesis) and powerful (under the alternative hypothesis) than the normal approximation based tests of Chen et al. [Semiparametric of transformation models with censored data, Biometrika 89 (2002) 659-668] when the model is different from the proportional hazards model and the proportion of censoring is high.