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Accepted; International audience; Virtual age models are very useful to analyse recurrent events. Among the strengths of these models is their ability to account for treatment (or intervention) effects after an event occurrence. Despite their flexibility for modeling recurrent events, the number of applications is limited. This seems to be a result of the fact that in the semiparametric setting all the existing results assume the virtual age function that describes the treatment (or intervention) effects to be known. This shortcoming can be overcome by considering semiparametric virtual age models with parametrically specified virtual age functions. Yet, fitting such a model is a difficult task. Indeed, it has recently been shown that for these models the standard profile likelihood method fails to lead to consistent estimators. Here we show that consistent estimators can be constructed by smoothing the profile log-likelihood function appropriately. We show that our general result can be applied to most of the relevant virtual age models of the literature. Our approach shows that empirical process techniques may be a worthwhile alternative to martingale methods for studying asymptotic properties of these inference methods. A simulation study is provided to illustrate our consistency results together with an application to real data.

Cure models have been widely developed to estimate the cure fraction when some subjects never experience the event of interest. However, these models were rarely focused on the estimation of the time-to-cure, that is, the delay elapsed between the diagnosis and "the time from which cure is reached," an important indicator, for instance, to address the question of access to insurance or loans for subjects with personal history of cancer. We propose a new excess hazard regression model that includes the time-to-cure as a covariate-dependent parameter to be estimated. The model is written similarly to a Beta probability distribution function and is shown to be a particular case of the non-mixture cure models. Parameters are estimated through a maximum likelihood approach and simulation studies demonstrate good performance of the model. Illustrative applications to three cancer data sets are provided and some limitations as well as possible extensions of the model are discussed. The proposed model offers a simple and comprehensive way to estimate more accurately the time-to-cure.

In this chapter, a few reliability or survival analysis models involving latent variables are presented. Latent variable models are generally proposed to consider missing information, heterogeneity of observations, measurement errors, dependency, etc. On one hand, the statistical models described are mainly devoted to the study of duration data, and on the other hand, a large number of estimation methods are shown for use to estimate parametrically, semi-parametrically, or nonparametrically the unknown parameters of these models. In the parametric setup, when the maximum likelihood principle is too complex to be handled, an EM or stochastic EM algorithm may be useful. In the nonparametric setup, combining empirical processes are shown with the functional delta-method allowing the derivation of the asymptotic properties of estimators. Some examples are based on Gamma processes to model degradation data and may involve time-dependent latent variables.

We consider a semi-parametric model for recurrent events. The model consists of an unknown hazard rate function, the infinite-dimensional parameter of the model, and a parametrically specified effective age function. We will present a condition on the family of effective age functions under which the profile likelihood function evaluated at the parameter vector $\mathbf{{\theta}}$, say, exceeds the profile likelihood function evaluated at the parameter vector $\tilde{\boldsymbol {\theta}}$, say, with probability $p$. From this we derive a condition under which profile likelihood inference for the finite-dimensional parameter of the model leads to inconsistent estimates. Examples will be presented. In particular, we will provide an example where the profile likelihood function is monotone with probability one regardless of the true data generating process. We also discuss the relation of our results to other semi-parametric models like the accelerated failure time model and Cox’s proportional hazards model.

A system is considered, which is deteriorating over time according to a non homogeneous gamma process with unknown parameters. The system is subject to periodic and instantaneous imperfect maintenance actions (repairs). Each imperfect repair removes a proportion ρ of the accumulated degradation since the previous repair. The parameter ρ hence appears as a measure for the maintenance efficiency. This model is called arithmetic reduction of degradation of order 1. The system is inspected right before each maintenance action, thus providing some multivariate measurement of the successively observed deterioration levels. Based on these data, a semiparametric estimator of ρ is proposed, considering the parameters of the underlying gamma process as nuisance parameters. This estimator is mainly based on the range of admissible ρ ’s, which depends on the data. Under technical assumptions, consistency results are obtained, with surprisingly high convergence rates (up to exponential). The case where several i.i.d. systems are observed is next envisioned. Consistency results are obtained for the efficiency estimator, as the number of systems tends to infinity, with a convergence rate that can be higher or lower than the classical square root rate. Finally, the performances of the estimators are illustrated on a few numerical examples.

The delta method is a powerful method to approximate the probability distribution of a function $$\phi$$ of an asymptotically normal statistical estimator $$T_n$$ where n is the sample size. Generally the asymptotic normality of the estimator results from central limit theorems that requires to center $$T_n$$ and the function $$\phi$$ does not depend on n. In this paper, we provide an extension of the classical delta method to the case where both the centering parameter and the function $$\phi$$ depend on the sample size. This method can be used in various statistical applications (repeated measurement, panel data, degradation data, etc.). Starting by pointing out the main classical parametric delta methods, we provide its extension. The extended delta method is illustrated by an application to degradation data where the unknown parameters are estimated using the method of moments.

The aim of the present paper is the stochastic modeling and statistical inference of a component which deteriorates over time, for prediction purpose. The deterioration is due to defects which appear one by one and next independently propagate over time. The motivation comes from an application to passive components within electric power plants, where measurable flaw indications first initiate one at a time and next grow over time. The available data come from inspections at discrete times, where only the largest flaw indication is measured together with the total number of indications on each component. Although detected, too small indications cannot be measured, leading to censored observations. Taking into account this partial information coming from the field, a specific stochastic model is proposed, where the flaw indications initiate according to a Poisson process and next propagate according to competing independent gamma processes. A parametric estimation procedure is developed, tested on simulated data and then applied to the industrial case. The fitted model is next used to make some prediction over the future deterioration of each component and over its residual operating time until a specified critical degradation level is reached. Copyright © 2016 John Wiley & Sons, Ltd.

International audience

Oil and gas exploration and production relies usually on the interpretation of a single seismic image, which is obtained from observed data. However, the statistical nature of seismic data and the various approximations and assumptions are sources of uncertainties which may corrupt the evaluation of parameters. The quantification of these uncertainties is a major issue which supposes to help in decisions that have important social and commercial implications. The residual moveout analysis, which is an important step in seismic data processing is usually performed by a deterministic approach. In this paper we discuss a Bayesian approach to the uncertainty analysis.

The parametric estimation and optimal censoring scheme are considered under the progressive multi-stage Type-I censoring scheme as well as step-stress accelerated lifetime model. Nonparametric estimators, using the information of the observable numbers of failures and numbers of censored units at the censoring times, are used to derive estimates of the reliability function at the censoring times. Then two parametric estimators, the maximum likelihood and the minimum-distance, are used to estimate the unknown Euclidean parameters of a parametric model. We use D-optimality criterion to determine an optimal sequential step-stress plan under progressive Type-I censoring. Simulation studies are also conducted to assess the finite performance of our estimators.

International audience; Mixture models in reliability bring a useful compromise between parametric and nonparametric models, when several failure modes are suspected. The classical methods for estimation in mixture models rarely handle the additional difficulty coming from the fact that lifetime data are often censored, in a deterministic or random way. We present in this paper several iterative methods based on EM and Stochastic EM methodologies, that allow us to estimate parametric or semiparametric mixture models for randomly right censored lifetime data, provided they are identifiable. We consider different levels of completion for the (incomplete) observed data, and provide genuine or EM-like algorithms for several situations. In particular, we show that simulating the missing data coming from the mixture allows to plug a standard R package for survival data analysis in an EM algorithm's M-step. Moreover, in censored semiparametric situations, a stochastic step is the only practical solution allowing computation of nonparametric estimates of the unknown survival function. The effectiveness of the new proposed algorithms are demonstrated in simulation studies and an actual dataset example from aeronautic industry.

International audience; We consider, in the presence of covariates, non independent competing risks that are subject to right censoring. We define a nonparametric estimator of the incident regression function through the generalized product-limit estimator of the conditional censorship distribution function. Under suitable conditions we establish the almost sure uniform convergence of those estimators with appropriate rate.

Estimators based on data-driven generalized weighted Cramer-von Mises distances are defined for data that are subject to a possible right censorship. The function used to measure the distance between the data, summarized by the Kaplan–Meier estimator, and the target model is allowed to depend on the sample size and, for example, on the number of censored items. It is shown that the estimators are consistent and asymptotically multivariate normal for every p dimensional parametric family fulfiling some mild regularity conditions. The results are applied to finite mixtures. Simulation results for finite mixtures indicate that the estimators are useful for moderate sample sizes. Furthermore, the simulation results reveal the usefulness of sample size dependent and censoring sensitive distance functions for moderate sample sizes. Moreover, the estimators for the mixing proportion seem to be fairly robust against a ‘symmetric’ contamination model even when censoring is present.

International audience; The objective of this paper is to provide a new estimation method for parametric models under progressive Type-I censoring. First, we propose a Kaplan-Meier nonparametric estimator of the reliability function taken at the censoring times. It is based on the observable number of failures, and the number of censored units occurring from the progressive censoring scheme at the censoring times. This estimator is then shown to asymptotically follow a normal distribution. Next, we propose a minimum-distance method to estimate the unknown Euclidean parameter of a given parametric model. This method leads to consistent, asymptotically normal estimators. The maximum likelihood estimation method based on group-censored samples is discussed next, and the efficiencies of these two methods are compared numerically. Then, based on the established results, we derive a method to obtain the optimal Type-I progressive censoring scheme, Finally we illustrate all these results through a Monte Carlo simulation study, and an illustrative example. © 2006 IEEE.

This paper discusses the condition-based non-periodic maintenance policy for a stochastic deteriorating system influenced by a dynamic environment which is described by a covariates process. The deterioration is modelled by a stochastic univariate process. The process of covariates is assumed to be a time homogeneous finite state space Markov chain. A model similar to the proportional hazards model is used to represent the influence of the covariates. In the framework of a monotone deteriorating system, we derive the optimal maintenance threshold, optimal inspection sequence to minimise the expected maintenance cost per time unit. An adequate maintenance policy in which the inspection schemes depend both on the level of degradation and on the state of covariates is studied. Comparison of the expected costs per time unit under different conditions of covariates and different maintenance policies is given by numerical results of Monte Carlo simulation.

International audience; This paper proposes an inferential method for the semiparametric proportional hazards model for progressively Type-II censored data. We establish martingale properties of counting processes based on progressively Type-II censored data that allow to derive the asymptotic behavior of estimators of the regression parameter, the conditional cumulative hazard rate functions, and the conditional reliability functions. A Monte Carlo study and an example are provided to illustrate the behavior of our estimators and to compare progressive Type-II censoring sampling plans with classical Type-II right censoring sampling plan. © 2008 The Institute of Statistical Mathematics, Tokyo.

Recently, there has been a considerable interest in finite mixture models with semi-/non-parametric component distributions. Identifiability of such model parameters is generally not obvious, and when it occurs, inference methods are rather specific to the mixture model under consideration. Hence, a generalization of the EM algorithm to semiparametric mixture models is proposed. The approach is methodological and can be applied to a wide class of semiparametric mixture models. The behavior of the proposed EM type estimators is studied numerically not only through several Monte-Carlo experiments but also through comparison with alternative methods existing in the literature. In addition to these numerical experiments, applications to real data are provided, showing that the estimation method behaves well, that it is fast and easy to be implemented.

In this paper, we discuss the problem of testing the homogeneity of several populations when the available data are progressively Type-II censored. Defining for each sample a univariate counting process, we can modify all the methods that were developed during the last two decades (see e.g. [P.K. Andersen, Ø. Borgan, R. Gill, N. Keiding, Statistical Models Based on Counting Processes, Springer, New York, 1993]) for use to this problem. An important aspect of these tests is that they are based on either linear or non-linear functionals of a discrepancy process (DP) based on the comparison of the cumulative hazard rate (chr) estimated from each sample with the chr estimated from the whole sample (viz., the aggregation of all the samples), leading to either linear tests or non-linear tests. Both these kinds of tests suffer from some serious drawbacks. For example, it is difficult to extend non-linear tests to the K-sample situation when K⩾3. For this reason, we propose here a new class of non-linear tests, based on a chi-square type functional of the DP, that can be applied to the K-sample problem for any K⩾2.

The reliability analysis of instrumented safety systems is an important industrial issue. The standard modeling languages (e.g., Fault trees and Markov chains) and methods employed for these studies become difficult to apply mainly because of the increasing complexity of the operating context (e.g., maintenance policies and aging process). Thus, a powerful alternative is Petri nets associated with Monte Carlo simulation (MC). However, obtaining accurate estimators on rare events (system failures) requires very long computing times. To address this issue, common methods are not well-suited to Petri nets whereas the “Methode de Conditionnement Temporel” (MCT), a truncation method, seems to be. Indeed, MCT is independent of the duration distributions involved in a model. However, it is only defined when the rare event consists in reaching an absorbing state. To overcome this limitation, we first propose an extension of MCT (EMCT) to cases of repeated cycles where the failure event is either direct or in competition with other events. Numerical results show that EMCT gives better estimates than MC. Second, we introduce a new computational technique, called Dissociation Method, for systems with independent components. We combine it with both MC and EMCT. Through different numerical examples, we observe a significant improvement of the results.

In the oil and gas industry, obtaining accurate reliability estimators on safety barriers is an important issue that can lead to very long computing times. To address this issue, we propose an extension of a truncation method and we introduce a new computational technique called Dissociation method. Through different numerical examples, we observe a significant improvement of the results obtained on simple Petri net models when applying these Monte Carlo acceleration methods.

We consider a two-component mixture model where one component distribution is known while the mixing proportion and the other component distribution are unknown. These kinds of models were first introduced in biology to study the differences in expression between genes. The various estimation methods proposed till now have all assumed that the unknown distribution belongs to a parametric family. In this paper, we show how this assumption can be relaxed. First, we note that generally the above model is not identifiable, but we show that under moment and symmetry conditions some ‘almost everywhere’ identifiability results can be obtained. Where such identifiability conditions are fulfilled we propose an estimation method for the unknown parameters which is shown to be strongly consistent under mild conditions. We discuss applications of our method to microarray data analysis and to the training data problem. We compare our method to the parametric approach using simulated data and, finally, we apply our method to real data from microarray experiments.

In this paper, we show that if the Euclidean parameter of a semiparametric model can be estimated through an estimating function, we can extend straightforwardly conditions by Dmitrienko and Govindarajulu [2000. Ann. Statist. 28 (5), 1472–1501] in order to prove that the estimator indexed by any regular sequence (sequential estimator), has the same asymptotic behavior as the non-sequential estimator. These conditions also allow us to obtain the asymptotic normality of the stopping rule, for the special case of sequential confidence sets. These results are applied to the proportional hazards model, for which we show that after slight modifications, the classical assumptions given by Andersen and Gill [1982. Ann. Statist. 10(4), 1100–1120] are sufficient to obtain the asymptotic behavior of the sequential version of the well-known [Cox, 1972. J. Roy. Statist. Soc. Ser. B (34), 187–220] partial maximum likelihood estimator. To prove this result we need to establish a strong convergence result for the regression parameter estimator, involving mainly exponential inequalities for both continuous martingales and some basic empirical processes. A typical example of a fixed-width confidence interval is given and illustrated by a Monte Carlo study.

International audience

In this article we introduce a new missing data model, based on a standard parametric Hidden Markov Model (HMM), for which information on the latent Markov chain is given since this one reaches a fixed state (and until it leaves this state). We study, under mild conditions, the consistency and asymptotic normality of the maximum likelihood estimator. We point out also that the underlying Markov chain does not need to be ergodic, and that identifiability of the model is not tractable in a simple way (unlike standard HMMs), but can be studied using various technical arguments.

This work deals with asymptotic properties of the [ αm ]th-order statistic of a type-II progressively censored sample of size m . Such an order statistic, indexed by α ∈[0,1], is called the quantile process. Our main results concern the normalized version of the quantile process for which a weak convergence result is obtained. This result is applied in order to construct non-parametric estimators of quantiles. Monte-Carlo simulations illustrate the behavior of the estimators for limited sample size.

We consider non-parametric estimation of cumulative hazard functions and reliability functions of progressively type-II right censored data. As shown in the book of Balakrishnan and Aggarwala (Progressive Censoring, Birkhauser, Basel, 2000), many results of classical order statistics can be generalized to this kind of statistics. These authors proposed also many inferential methods for parametric models. In this paper we show that non-parametric maximum likelihood estimators (NPMLE) may also be derived under such censoring schemes. These estimators are obtained in a reliability context but they can also be extended to arbitrary continuous distribution functions. Since the large sample properties of the NPMLE depend on counting processes based upon generalized order statistics that are generated by progressive censoring, we need to establish some basic properties of these processes (e.g. martingales properties and weak consistency). Finally, the non-parametric estimator of the reliability is compared with two parametric estimators for a real data set and additionally, some Monte-Carlo simulations are provided.

ACT; International audience; no abstract

International audience; In this paper, we consider a semiparametric model for lifetime data with competing risks and missing causes of death. We assume that an additive hazards model holds for each cause-specific hazard rate function and that a random right censoring occurs. Our goal is to estimate the regression parameters as well as the functional parameters such as the baseline and cause-specific cumulative hazard rate functions / cumulative incidence functions. We first introduce preliminary estimators of the unknown (Euclidean and functional) parameters when cause of death indicators are missing completely at random (MCAR). These estimators are obtained using the observations with known cause of failure. The advantage of considering the MCAR model is that the information given by the observed lifetimes with unknown failure cause can be used to improve the preliminary estimates in order to attain an asymptotic optimality criterion. This is the main purpose of our work. However, since it is often more realistic to consider a missing at random (MAR) mechanism, we also derive estimators of the regression and functional parameters under the MAR model. We study the large sample properties of our estimators through martingales and empirical process techniques. We also provide a simulation study to compare the behavior of our three types of estimators under the different mechanisms of missingness. It is shown that our improved estimators under MCAR assumption are quite robust if only the MAR assumption holds. Finally, three illustrations on real datasets are also given.

In oil and gas industry, the reliability analysis of High Integrity Protection Systems is an important issue. The standard modeling languages and the traditional methods employed for these studies are difficult to apply mainly because of the complexity of the operating context of these equipment. Thus, a powerful alternative is Petri nets associated with the Monte Carlo simulation (MC). However, obtaining accurate estimators on rare events (system failures) calls for very long computing times. To address this issue, the common methods are not well-suited to Petri Nets whereas the "Me thode de Conditionnement Temporel" (MCT) seems to be. Indeed, this method does not require to know the model distributions, however, it is only defined when the rare event is an absorbing state. To overcome this limitation, we first propose an extension of MCT (EMCT) to simple cases which represent repeated cycles where the failure event is either direct or in competition with other events. The first results show that EMCT gives better estimates than MC for a similar computing time. Second, we introduce a new computational technique, called Dissociation method, which is valid only if the components of the system are independent. We combine it with both MC and EMCT. Through different numerical examples, we observe a significant improvement of the obtained results.

In this paper we investigate the asymptotic properties of estimators obtained for the semiparametric additive accelerated life model proposed by Bagdonavicius & Nikulin (1995). This model generalizes the well known additive hazards model of survival analysis and is close to the general transformation model (see Dabrowska & Doksum, 1988). Asymptotic properties of the estimator of the regression parameter and the estimator of the reliability function are given in the case of right censoring for discretized data and a numerical example illustrates these results.

It is shown that for independent and identically distributed random vectors, for which the components are independent and exponentially distributed with a common shift, we can construct unbiased estimators of their density, derived from the Uniform Minimum Variance Unbiased Estimator (UMVUE) of their distribution function. As direct applications of the UMVUEs of the density functions we present a Chi-square goodness of fit test of the model, and give two tables of the UMVUEs of some commonly used functions of the unknown parameters of the multivariate exponential model considered in this paper.

International audience

A new estimation method for the two-component mixture model introduced in \cite{Van13} is proposed. This model consists of a two-component mixture of linear regressions in which one component is entirely known while the proportion, the slope, the intercept and the error distribution of the other component are unknown. In spite of good performance for datasets of reasonable size, the method proposed in \cite{Van13} suffers from a serious drawback when the sample size becomes large as it is based on the optimization of a contrast function whose pointwise computation requires O(n^2) operations. The range of applicability of the method derived in this work is substantially larger as it relies on a method-of-moments estimator free of tuning parameters whose computation requires O(n) operations. From a theoretical perspective, the asymptotic normality of both the estimator of the Euclidean parameter vector and of the semiparametric estimator of the c.d.f.\ of the error is proved under weak conditions not involving zero-symmetry assumptions. In addition, an approximate confidence band for the c.d.f.\ of the error can be computed using a weighted bootstrap whose asymptotic validity is proved. The finite-sample performance of the resulting estimation procedure is studied under various scenarios through Monte Carlo experiments. The proposed method is illustrated on three real datasets of size $n=150$, 51 and 176,343, respectively. Two extensions of the considered model are discussed in the final section: a model with an additional scale parameter for the first component, and a model with more than one explanatory variable., 43 pages, 4 figures, 5 tables

ACT; International audience; no abstract

In this paper we consider a competing risks model including covariates in which the observations are subject to random right censoring. Without any assumption of independence of the competing risks, and based on a nonparametric kernel-type estimator of the incident regression function, an estimator of the conditional regression function is proposed. We show that at a given covariate value and under suitable conditions the nonparametric estimator of the regression function is asymptotically normal. A simulation study is provided showing that our estimators have good behaviour for moderate sample sizes.Nous consid´erons dans ce papier un mod`ele de risques comp´etitifs dans lequel les observations sont soumises `a une censure al´eatoire `a droite en pr´esence de covariables. Sans aucune hypoth`ese d’ind´ependance sur les risques comp´etitifs, un estimateur non param´etrique de la fonction de r´epartition conditionnelle incidente est propos´e. Cet estimateur est obtenu via celui d’un estimateur non param´etrique de type noyau de la fonction de r´egression incidente. Nous d´emontrons que pour une valeure fix´ee de la covariable, et sous certaines conditions, l’estimateur non param´etrique de la fonction de r´egression incidente est asymptotiquement normal. Des simulations illustrent le bon comportement de nos estimateurs pour des tailles mod´er´ees d’´echantillons.Key words: Competing risks; Nonparametric estimation; Kernel method; Regression function; right censoring.

International audience; The aim of this paper is to discuss the problem of modelling and optimising condition-based maintenance policies for a deteriorating system in presence of covariates. The deterioration is modelled by a non-monotone stochastic process. The covariates process is assumed to be a time-homogenous Markov chain with finite state space. A model similar to the proportional hazards model is used to show the influence of covariates on the deterioration. In the framework of the system under consideration, an appropriate inspection/replacement policy which minimises the expected average maintenance cost is derived. The average cost under different conditions of covariates and different maintenance policies is analysed through simulation experiments to compare the policies performances.

ACT; International audience; no abstract

ACT; International audience; no abstract

We consider a degradation model which is the sum of two independent processes: an homogeneous gamma process and a Brownian motion. This model is called perturbed gamma process. Based on independent copies of the perturbed gamma process observed at irregular instants we propose to estimate the unknown parameters of the model using the moment method. Some general conditions allow to derive the asymptotic behavior of the estimators. We also show that these general conditions are fulfilled for some specific observation schemes. Finally, we illustrate our method by a numerical study and an application to a real data set.

International audience; In this paper we consider a two-component mixture model one component of which has a known distribution while the other is only known to be symmetric. The mixture proportion is also an unknown parameter of the model. This mixture model class has proved to be useful to analyze gene expression data coming from microarray analysis. In this paper is proposed a general estimation method leading to a joint central limit result for all the estimators. Applications to basic testing problems related to this class of models are proposed, and the corresponding inference procedures are illustrated through some simulation studies.

COM; International audience; no abstract

Estimating the unknown parameters of a reliability mixture model may be a more or less intricate problem, especially if durations are censored. We present several iterative methods based on Monte Carlo simulation that allow to fit parametric or semiparametric mixture models provided they are identifiable. We show for example that the well-known data augmentation algorithm may be used successfully to fit semiparametric mixture models under right censoring. Our methods are illustrated by a reliability example.