Journal of Iranian Statistical Society
Volume:12 Issue: 1, 2013

Lam (2007) introduces a generalization of renewal processes named Geometric processes، where inter-arrival times are independent and identically distributed up to a multiplicative scale parameter، in a geometric fashion. We here envision a more general scaling، not necessar- ily geometric. The corresponding counting process is named Extended Geometric Process (EGP). Semiparametric estimates are provided and studied for an EGP، which includes consistency results and convergence rates. In a reliability context، arrivals of an EGP may stand for suc- cessive failure times of a system submitted to imperfect repairs. In this context، we study: 1) the mean number of failures on some finite hori- zon time; 2) a replacement policy assessed through a cost function on an infinite horizon time.

Interest in the presence and nature of trend arises frequently in science، public health، technology، and many other areas. In this ar- ticle we discuss the notion of trend in the context of recurrent event processes. We discuss different frameworks within which one can inves- tigate trend and consider various ways in which trends may be manifest. Tests for trend are discussed in detail and the utility of intensity-based models is emphasized for characterizing event processes and understand- ing trends. Simulation studies are conducted to study the effect of het- erogeneity in the investigation of trend. Data from a study of hospi- talization patterns in patients with affective disorder are analysed for illustration.

In the perspective of biomedical applications، consider a re- current event situation with a relatively low degree of recurrence. In this setting، the focus is placed on successive inter-event gap times which are observed in the presence of both a terminal event like death and inde- pendent censoring. The terminal event is potentially related to recurrent events while the censoring process is an independent nuisance that bears on the total observation time i. e. on the sum of the successive gap times. We review different modeling and inferential strategies. We also present a nonparametric estimation method of joint distribution functions and outline the need for future developments.

Nonhomogeneous Poisson processes (NHPPs) are often used to model recurrent events، and there is thus a need to check model fit for such models. We study the problem of obtaining exact goodness-of-fit tests for certain parametric NHPPs، using a method based on Monte Carlo simulation conditional on sufficient statistics. A closely related way of obtaining exact confidence intervals in parametric models is also briefly considered.

We present a Bayesian analysis for recurrent events data using a nonhomogeneous mixed Poisson point process with a dynamic subject-specific frailty function and a dynamic baseline intensity func- tion. The dynamic subject-specific frailty employs a dynamic piecewise constant function with a known pre-specified grid and the baseline in- tensity uses an unknown grid for the piecewise constant function. Imple- mentation of Bayesian inference using a reversible jump Markov Chain Monte Carlo (RJMCMC) algorithm is developed to handle the change of the dimension in the parameter space for models with a random num- ber of change points. A data set provided by Grubbs et al. (1991) with recurrent times to mammary tumors for 59 rats is used to illustrate the application of the new models. We compare several models including constant or piecewise constant subject-specific frailty and a fixed num- ber or a random number for the change points in the baseline using the pseudo-marginal likelihood criterion. We show that models with a ran- dom number of change points in the baseline improve upon that of a fixed number.

A resource-efficient approach to making inferences about the distributional properties of the failure times in a competing risks setting is presented. Efficiency is gained by observing recurrences of the compet- ing risks over a random monitoring period. The resulting model is called the recurrent competing risks model (RCRM) and is coupled with two repair strategies whenever the system fails. Maximum likelihood estima- tors of the parameters of the marginal distribution functions associated with each of the competing risks and also of the system lifetime dis- tribution function are presented. Estimators are derived under perfect and partial repair strategies. Consistency and asymptotic properties of the estimators are obtained. The estimation methods are applied to a data set of failures for cars under warranty. Simulation studies are used to ascertain the small sample properties and the efficiency gains of the resulting estimators.