Innovations in medicine provide us longer and healthier life, leading lower mortality. Sooner rather than later, much greater longevity would be possible for us due to artificial intelligence advances in health care. Similarly, Advanced Driver Assistance Systems (ADAS) in highly automated vehicles may reduce or even eventually eliminate accidents by perceiving dangerous situations, which would minimize the number of accidents and lead to fewer loss claims for insurance companies. To model the survivor function capturing greater longevity as well as the number of claims reflecting less accidents in the long run, in this paper, we study a Cox process whose intensity process is piecewise-constant and decreasing. We derive its ultimate distributional properties, such as the Laplace transform of intensity integral process, the probability generating function of point process, their associated moments and cumulants, and the probability of no more claims for a given time point. In general, this simple model may be applicable in many other areas for modeling the evolution of gradually disappearing events, such as corporate defaults, dividend payments, trade arrivals, employment of a certain job type (e.g., typists) in the labor market, and release of particles. In particular, we discuss some potential applications to insurance.

Unlike the mortality risk on which actuaries have been working for more than a century, the long-term care (LTC) risk is relatively new and as of today hardly mastered. Semi-Markov processes have been identified as an adequate tool to study this risk. Nevertheless, access to data is limited and the associated literature still scarce. Insurers mainly use discrete time methods directly inspired from the study of mortality in order to build experience tables. Those methods however are not perfectly suited for the study of competing risk situations. This article provides a theoretical framework to estimate biometric laws associated with a LTC insurance portfolio. The presented method relies on a continuous-time semi-Markov model with three states: autonomy, disability and death. The process describing the state of disability is defined through its transition intensities. We provide a formula to infer the mortality of autonomous people from the mortality of the whole portfolio, on which we have more reliable knowledge. We then propose a parametric expression for the remaining intensities of the model. In particular, incidence in LTC is described by a logistic formula. Under the assumption that the disabled population is a mixture of two latent populations with respect to the category of pathology that caused LTC, we show that the resulting intensity of mortality in LTC takes a very peculiar form and depends on time spent in the LTC state. Estimation of parameters relies on the maximum likelihood method. Our parametric approach, while inducing model uncertainty, eliminates issues related to segmentation in age categories, smoothing or extrapolation at higher ages and thus proves very convenient for the practitioner. Finally, we provide an application using data from a real LTC insurance portfolio.

Chemotherapy and/or radiotherapy used as conditioning regimens before autologous or allogeneic haematopoietic cell transplantations (HCTs) cause neutropenia, which is the main reason for bloodstream infections. Autologous HCTs are considered to be superior to allogeneic HCTs in terms of infection outcome. A previous analysis suggested that patients with allogeneic HCTs are exposed to a reduced infection hazard and that an unfavourable infection outcome of allogeneic HCTs may be mediated through prolonged neutropenia. Therefore, we investigated whether allogeneic HCTs initially lead to fewer infections. We evaluated data from a prospective non-randomized multi-centre cohort study, with a total of 1616 patients. Of these, 703 patients received autologous and 913 patients received allogeneic HCTs from January 2000 to June 2004. The retrospective analysis used simultaneous confidence bands for the cumulative infection probability in the presence of competing risks. Patients with allogeneic HCTs experienced fewer infections during the early phase of neutropenia. As patients with autologous HCTs are not necessarily subject to antibiotic prophylaxis, a future study should investigate this policy. A limitation of the analysis is that it did not find the effect of crossing cumulative infection probabilities to be significant.

This paper surveys some statistical models of survival data. Competing risks models are described; the unidentifiability of net decrements suggests a sceptical approach to the use of underlying single decrement tables. Approaches based on observations of complete lifetimes (with censoring) are surveyed including the Kaplan-Meier and Nelson-Aalen estimates. Regression models for lifetimes depending on covariates are discussed, in particular the Cox model and partial likelihood estimation.

In this paper we present a general framework for the modelling of the process of corrective and condition-based preventive maintenance actions for complex repairable systems. A new class of models is proposed, the generalized virtual age models. On the one hand, these models generalize Kijima's virtual age models to the case where both preventive and corrective maintenances are present. On the other hand, they generalize the usual competing risks models to imperfect maintenance actions which do not renew the system. A generalized virtual age model is defined by both a sequence of effective ages which characterizes the effects of both types of maintenance according to a classical virtual age model, and a usual competing risks model which characterizes the dependency between the two types of maintenance. Several particular cases of the general model are derived.

Within reliability theory, identifiability problems arise through competing risks. If we have a series system of several components, and if that system is replaced or repaired to as good as new on failure, then the different component failures represent competing risks for the system. It is well known that the underlying component failure distributions cannot be estimated from the observable data (failure time and identity of failed component) without nontestable assumptions such as independence. In practice many systems are not subject to the ‘as good as new’ repair regime. Hence, the objective of this paper is to contrast the identifiability issues arising for different repair regimes. We consider the problem of identifying a model within a given class of probabilistic models for the system. Different models corresponding to different repair strategies are considered: a partial-repair model, where only the failing component is repaired; perfect repair, where all components are as good as new after a failure; and minimal repair, where components are only minimally repaired at failures. We show that on the basis of observing a single socket, the partial-repair model is identifiable, while the perfect- and minimal-repair models are not.

Given a coherent system, let Z be the age of the machine at breakdown, and I the set of parts failed by time Z. Assume that the component lifetimes are independent. Assume further that the lifetime distributions are mutually absolutely continuous and that each possesses a single positive atom at the common essential infimum. We prove that the joint distribution of (Z, I) identifies the lifetime distribution of each part if and only if there is at most one component belonging to all cut sets. If we relax the mutual absolute continuity assumption by allowing isolated intervals of constancy, then a necessary and sufficient condition for identifiability is that no two parts be in parallel.

Given an indexed family of rate functions, we construct a family of independent random variables such that their minimum and the corresponding index form a marked point having the same rates. If the rates have common discontinuities, the random variables are conditionally independent, given a random allocation of these jumps. Between any two adjacent points of a marked point process there is a similar structure. Coherent functions in system lifetime theory provide examples.

Non-identifiability of multivariate lifetime distributions from mortality data is interpreted in this context.

Given a coherent reliability system, let Z be the age of the machine at breakdown, and I the set of parts dead by time Z. We prove that if all lifetime distributions are non-atomic and share the same essential extrema, and if the incidence matrix of the minimal cut sets has rank equal to the number of parts, then the joint distribution of Z and I determines uniquely the lifetime distribution of each part. We present a Newton–Kantorovič iterative method for the computation of those distributions. We deal informally with the relaxation of the assumptions and with the statistical problem where instead of the joint distribution of Z and I we have an empirical estimate of this joint distribution.