Backward Recurrence Time Research Articles

Bounds for the distribution and moments of the forward and backward recurrence times in a renewal process

We obtain a variety of lower and upper bounds for the bivariate tail of the joint distribution between the (forward and backward) recurrence times in a renewal process. Some of these bounds are valid generally, some others in the case where the interarrival distribution F exhibits monotone aging. We also give various bounds for the moments of the recurrence times. Our results improve upon existing bounds for these quantities in the literature. We also discuss the case where the aging of F is non-monotonic. Finally, we present numerical examples to illustrate the performance of the new bounds.

Refinement bounds for the expected number of renewal epochs over a finite interval

ABSTRACT Primary quantities of interest in renewal theory are the expected number of renewals in ( 0 , t ] , known as renewal function (denoted by U ( t ) ), and the expected number of renewals in ( t , t + h ] , given by the difference U ( t + h ) − U ( t ) .One of the most known bounds for the expected number of renewals over a finite interval is the one delivered by Barlow and Proschan [Statistical theory of reliability and life testing. MD: Silver Spring; 1981. Reprint Edition. To begin with]. The essential quantity of this bound is the difference U ( t + h ) − U ( t ) − U ( h ) . In this paper, initially, we offer bounds for the conditional tails of the backward and forward recurrence times, and we give a renewal-type equation for the difference U ( t + h ) − U ( t ) − U ( h ) . Employing those results, we refine the provided bound by Barlow and Proschan [Statistical theory of reliability and life testing. MD: Silver Spring; 1981. Reprint Edition. To begin with]. The results are illustrated with numerical examples.

Recurrence times and the expected number of renewal epochs over a finite interval

This paper initially presents bounds for the joint tail and the conditional tails of the backward and forward recurrence times. Using these bounds, we deliver an improvement to the lower bound for the renewal function given by Brown [Inequalities for distributions with increasing failure rate. In: Gelfand AE, editors. Contributions to the theory and applications of statistics, a volume in honour of Herbert Solomon. Orlando, FL: Academic; 1987. p. 3–17] for IFR inter-arrival times. Finally, this paper proposes a renewal type equation for the expected number of renewals in and, under certain conditions, improves Lorden's [On excess over the boundary. Ann Math Stat. 1970;41(2):520–527] well-known general upper bound for the expected number of renewals in .

Subexponential Upper and Lower Bounds in Wasserstein Distance for Markov Processes

In this article, relying on Foster–Lyapunov drift conditions, we establish subexponential upper and lower bounds on the rate of convergence in the \(\text {L}^p\)-Wasserstein distance for a class of irreducible and aperiodic Markov processes. We further discuss these results in the context of Markov Lévy-type processes. In the lack of irreducibility and/or aperiodicity properties, we obtain exponential ergodicity in the \(\text {L}^p\)-Wasserstein distance for a class of Itô processes under an asymptotic flatness (uniform dissipativity) assumption. Lastly, applications of these results to specific processes are presented, including Langevin tempered diffusion processes, piecewise Ornstein–Uhlenbeck processes with jumps under constant and stationary Markov controls, and backward recurrence time chains, for which we provide a sharp characterization of the rate of convergence via matching upper and lower bounds.

Covariance between the forward recurrence time and the number of renewals

Recurrence times and the number of renewals in $(0,t]$ are fundamental quantities in renewal theory. Firstly, it is proved that the upper orthant order for the pair of the forward and backward recurrence times may result in NWUC (NBUC) interarrivals. It is also demonstrated that, under DFR interarrival times, the backward recurrence time is smaller than the forward recurrence time in the hazard rate order. Lastly, the sign of the covariance between the forward recurrence time and the number of renewals in $(0,t]$ at a fixed time point t and when $t\to \infty $ is studied assuming that the interarrival distribution belongs to certain ageing classes.

Efficiency of Naive Estimators for Accelerated Failure Time Models under Length-Biased Sampling.

In prevalent cohort studies where subjects are recruited at a cross-section, the time to an event may be subject to length-biased sampling, with the observed data being either the forward recurrence time, or the backward recurrence time, or their sum. In the regression setting, assuming a semiparametric accelerated failure time model for the underlying event time, where the intercept parameter is absorbed into the nuisance parameter, it has been shown that the model remains invariant under these observed data set-ups and can be fitted using standard methodology for accelerated failure time model estimation, ignoring the length-bias. However, the efficiency of these estimators is unclear, owing to the fact that the observed covariate distribution, which is also length-biased, may contain information about the regression parameter in the accelerated life model. We demonstrate that if the true covariate distribution is completely unspecified, then the naive estimator based on the conditional likelihood given the covariates is fully efficient for the slope.

Exact Results and Bounds for the Joint Tail and Moments of the Recurrence Times in a Renewal Process

The best known result about the joint distribution of the backward and forward recurrence times in a renewal process concerns the asymptotic behavior for the tail of that bivariate distribution. In the present paper we study the joint behavior of the recurrence times at a fixed time point t, and we obtain both exact results and bounds for their joint tail behavior. We also obtain results about the joint moments of these two random variables and we show in particular that the expectation of the product between the two recurrence times increases with time when the interarrival distribution has a decreasing failure rate. The results are illustrated by some numerical examples.

An Advanced Statistical Method for Point Process Modelling With Missing Event Histories

Researchers are always challenged in developing history-dependent point process models for recurrence events such as system failures when the early event history is missing. The raison d'être in most cases is the estimation of model parameters. Even for simple renewal point processes, the model estimation is difficult since the distribution of backward recurrence time is not explicitly given, except for the limiting distribution case. To fill the gap, this article establishes an effective statistical method for parameter estimation for history-dependent point process models with partial missing history. The proposed method addresses the missing history issue through a data augmentation (DA) technique integrated with a Markov Chain Monte Carlo (MCMC) simulation technique. The key novelty is the creation of an acceptance/rejection criterion to assure the validity of the augmented missing history. Next, the augmented and observed histories are combined to form a `complete' history which is in turn used for model estimation. The estimated parameters are then used to generate new missing history. An iterative procedure is implemented to warrant stationary distributions for the estimates after burn-in. The validity and efficiency of the proposed method are demonstrated using two simulation studies and one real-life case study of modelling municipal water pipe failures from different model estimation perspectives.

Parametric modelling of prevalent cohort data with uncertainty in the measurement of the initial onset date

In prevalent cohort studies with follow-up, if disease duration is the focus, the date of onset must be obtained retrospectively. For some diseases, such as Alzheimer's disease, the very notion of a date of onset is unclear, and it can be assumed that the reported date of onset acts only as a proxy for the unknown true date of onset. When adjusting for onset dates reported with error, the features of left-truncation and potential right-censoring of the failure times must be modeled appropriately. Under the assumptions of a classical measurement error model for the onset times and an underlying parametric failure time model, we propose a maximum likelihood estimator for the failure time distribution parameters which requires only the observed backward recurrence times. Costly and time-consuming follow-up may therefore be avoided. We validate the maximum likelihood estimator on simulated datasets under varying parameter combinations and apply the proposed method to the Canadian Study of Health and Aging dataset.

Prevalent cohort studies and unobserved heterogeneity.

Consider lifetimes originating at a series of calendar times [Formula: see text]. At a certain time [Formula: see text] a cross-sectional sample is taken, generating a sample of current durations (backward recurrence times) of survivors until [Formula: see text] and a prevalent cohort study consisting of survival times left-truncated at the current durations. A Lexis diagram is helpful in visualizing this situation. Survival analysis based on current durations and prevalent cohort studies is now well-established as long as all covariates are observed. The general problems with unobserved covariates have been well understood for ordinary prospective follow-up studies, with the good help of hazard rate models incorporating frailties: as for ordinary regression models, the added noise generates attenuation in the regression parameter estimates. For prevalent cohort studies this attenuation remains, but in addition one needs to take account of the differential selection of the survivors from initiation [Formula: see text] to cross-sectional sampling at [Formula: see text]. This paper intends to survey the recent development of these matters and the consequences for routine use of hazard rate models or accelerated failure time models in the many cases where unobserved heterogeneity may be an issue. The study was inspired by concrete problems in the study of time-to-pregnancy, and we present various simulation results inspired by this particular application.

The covariance of the backward and forward recurrence times in a renewal process: the stationary case and asymptotics for the ordinary case

In a Poisson process, it is well-known that the forward and backward recurrence times at a given time point t are independent random variables. In a renewal process, although the joint distribution of these quantities is known (asymptotically), it seems that very few results regarding their covariance function exist. In the present paper, we study this covariance and, in particular, we state both necessary and sufficient conditions for it to be positive, zero or negative in terms of reliability classifications and the coefficient of variation of the underlying inter-renewal and the associated equilibrium distribution. Our results apply either for an ordinary renewal process in the steady state or for a stationary process.

Renewal theory with fat-tailed distributed sojourn times: Typical versus rare

Renewal processes with heavy-tailed power law distributed sojourn times are commonly encountered in physical modelling and so typical fluctuations of observables of interest have been investigated in detail. To describe rare events the rate function approach from large deviation theory does not hold and new tools must be considered. Here we investigate the large deviations of the number of renewals, the forward and backward recurrence time, the occupation time, and the time interval straddling the observation time. We show how non-normalized densities describe these rare fluctuations, and how moments of certain observables are obtained from these limiting laws. Numerical simulations illustrate our results showing the deviations from arcsine, Dynkin, Darling-Kac, L{\'e}vy and Lamperti laws.

Linear Autonomous Compartmental Models as Continuous-Time Markov Chains: Transit-Time and Age Distributions

Linear compartmental models are commonly used in different areas of science, particularly in modeling the cycles of carbon and other biogeochemical elements. The representation of these models as linear autonomous compartmental systems allows different model structures and parameterizations to be compared. In particular, measures such as system age and transit time are useful model diagnostics. However, compact mathematical expressions describing their probability distributions remain to be derived. This paper transfers the theory of open linear autonomous compartmental systems to the theory of absorbing continuous-time Markov chains and concludes that the underlying structure of all open linear autonomous compartmental systems is the phase-type distribution. This probability distribution generalizes the exponential distribution from its application to one-compartment systems to multiple-compartment systems. Furthermore, this paper shows that important system diagnostics have natural probabilistic counterparts. For example, in steady state the system’s transit time coincides with the absorption time of a related Markov chain, whereas the system age and compartment ages correspond with backward recurrence times of an appropriate renewal process. These relations yield simple explicit formulas for the system diagnostics that are applied to one linear and one nonlinear carbon-cycle model in steady state. Earlier results for transit-time and system-age densities of simple systems are found to be special cases of probability density functions of phase-type. The new explicit formulas make costly long-term simulations to obtain and analyze the age structure of open linear autonomous compartmental systems in steady state unnecessary.

Two-time correlation and occupation time for the Brownian bridge and tied-down renewal processes

Tied-down renewal processes are generalisations of the Brownian bridge, where an event (or a zero crossing) occurs both at the origin of time and at the final observation time t. We give an analytical derivation of the two-time correlation function for such processes in the Laplace space of all temporal variables. This yields the exact asymptotic expression of the correlation in the Porod regime of short separations between the two times and in the persistence regime of large separations. We also investigate other quantities, such as the backward and forward recurrence times, as well as the occupation time of the process. The latter has a broad distribution which is determined exactly. Physical implications of these results for the Poland Scheraga and related models are given. These results also give exact answers to questions posed in the past in the context of stochastically evolving surfaces.

Simple and fast overidentified rank estimation for right-censored length-biased data and backward recurrence time.

Length-biased survival data subject to right-censoring are often collected from a prevalent cohort. However, informative right censoring induced by the sampling design creates challenges in methodological development. While certain conditioning arguments could circumvent the problem of informative censoring, related rank estimation methods are typically inefficient because the marginal likelihood of the backward recurrence time is not ancillary. Under a semiparametric accelerated failure time model, an overidentified set of log-rank estimating equations is constructed based on the left-truncated right-censored data and backward recurrence time. Efficient combination of the estimating equations is simplified by exploiting an asymptotic independence property between two sets of estimating equations. A fast algorithm is studied for solving non-smooth, non-monotone estimating equations. Simulation studies confirm that the overidentified rank estimator can have a substantially improved estimation efficiency compared to just-identified rank estimators. The proposed method is applied to a dementia study for illustration.

Calendar-based age replacement policy with dependent renewal cycles

ABSTRACTIn this article, we introduce an age-based replacement policy in which the preventive replacements are restricted to specific calendar times. Under the new policy, the assets are renewed at failure or if their ages are greater than or equal to a replacement age at given calendar times, whichever occurs first. This policy is logistically applicable in industries such as utilities where there are large and geographically diverse populations of deteriorating assets with different installation times. Since preventive replacements are performed at fixed times, the renewal cycles are dependent random variables. Therefore, the classic renewal reward theorem cannot be directly applied. Using the theory of Markov chains with general state space and a suitably defined ergodic measure, we analyze the problem to find the optimal replacement age, minimizing the long-run expected cost per time unit. We further find the limiting distributions of the backward and forward recurrence times for this policy and show how our ergodic measure can be used to analyze more complicated policies. Finally, using a real data set of utility wood poles’ maintenance records, we numerically illustrate some of our results including the importance of defining an appropriate ergodic measure in reducing the computational expense.

Rank-based testing of equal survivorship based on cross-sectional survival data with or without prospective follow-up.

Existing linear rank statistics cannot be applied to cross-sectional survival data without follow-up since all subjects are essentially censored. However, partial survival information are available from backward recurrence times and are frequently collected from health surveys without prospective follow-up. Under length-biased sampling, a class of linear rank statistics is proposed based only on backward recurrence times without any prospective follow-up. When follow-up data are available, the proposed rank statistic and a conventional rank statistic that utilizes follow-up information from the same sample are shown to be asymptotically independent. We discuss four ways to combine these two statistics when follow-up is present. Simulations show that all combined statistics have substantially improved power compared with conventional rank statistics, and a Mantel-Haenszel test performed the best among the proposal statistics. The method is applied to a cross-sectional health survey without follow-up and a study of Alzheimer's disease with prospective follow-up.

Prevalence of colonoscopy before age 50

ObjectiveDescribe the prevalence of colonoscopy before age 50, when guidelines recommend initiation of colorectal cancer screening for average risk individuals. MethodWe assembled administrative health records that captured receipt of colonoscopy between 40 and 49-years of age for a cohort of 204,758 50-year-old members of four US health plans and used backward recurrence time models to estimate trends in receipt of colonoscopy before age 50 and variation in early colonoscopy by age and sex. We also used self-reported receipt of colonoscopy from 27,157 40- to 49-year-old respondents to the 2010 National Health Interview Survey (NHIS) to estimate the association between early colonoscopy and sex, race/ethnicity, and geographic location based on logistic regression models that accounted for the complex NHIS sampling design. ResultsAbout 5% of the health plan cohort had a record of colonoscopy before age 50. Receipt of early colonoscopy increased significantly from 1999 to 2010 (test for linear trend, p<0.0001), was more likely among women than men (RR=1.9, 95% CI 1.14–1.24) and in the east coast health plan compared to west coast and Hawaii plans. The NHIS analysis found that early colonoscopy was more likely in Northeastern residents compared to residents in the West (odds ratio=1.75, 95% CI 1.28–2.39). ConclusionColonoscopy before age 50 is increasingly common.

Accounting for length-bias and selection effects in estimating the distribution of menstrual cycle length.

Prospective pregnancy studies are a valuable source of longitudinal data on menstrual cycle length. However, care is needed when making inferences of such renewal processes. For example, accounting for the sampling plan is necessary for unbiased estimation of the menstrual cycle length distribution for the study population. If couples can enroll when they learn of the study as opposed to waiting for the start of a new menstrual cycle, then due to length-bias, the enrollment cycle will be stochastically larger than the general run of cycles, a typical property of prevalent cohort studies. Furthermore, the probability of enrollment can depend on the length of time since a woman's last menstrual period (a backward recurrence time), resulting in selection effects. We focus on accounting for length-bias and selection effects in the likelihood for enrollment menstrual cycle length, using a recursive two-stage approach wherein we first estimate the probability of enrollment as a function of the backward recurrence time and then use it in a likelihood with sampling weights that account for length-bias and selection effects. To broaden the applicability of our methods, we augment our model to incorporate a couple-specific random effect and time-independent covariate. A simulation study quantifies performance for two scenarios of enrollment probability when proper account is taken of sampling plan features. In addition, we estimate the probability of enrollment and the distribution of menstrual cycle length for the study population of the Longitudinal Investigation of Fertility and the Environment Study.

Probability Model of Forward Birth Interval and Its Application

In renewal theory approach, it is well known that the limiting forms of the probability density function of backward recurrence time and forward recurrence time which are similar to open birth interval and forward birth interval are identical on the assumption that the renewal densities do not change over time. The forward birth interval defined as the time between the survey date and the date of next birth posterior to the survey date. Forward birth interval is a good index for current change in fertility behavior. The present model has been derived on the assumption that females are not exposed to the risk of conception immediately after the termination of Post-Partum Amenorrhea (PPA). However they may be exposed to the risk of conception at different point of time after the termination of PPA because of some socio-cultural factors or contraceptive practices. In this probability model for forward birth interval regardless of parity assuming that renewal density does not change over time and females are exposed to the risk of conception at different point of time. In this model, fecundability (λ) and the duration of time from the point of termination of PPA to the state of exposure as random variable (µ) which follows exponential distribution. The maximum likelihood estimation technique has been used for the estimation of parameters λ and µ through derived model. The estimated values of λ and µ are 1.1051 and 2.841 respectively. The variance of estimated λ and µ are 0.067 and 0.79 respectively. The co-variance in between estimated λ and µ is -0.026.With these estimates the expected frequencies for the distribution and χ2 = 0.6057 is highly significant. Thus, the derived probability model explains the fertility behavior of observed data satisfactorily well.