Backward Recurrence Time Research Articles

4. Accelerated Failure-Time Mover-Stayer Regression Models for the Analysis of Last-Episode Data

This article introduces new models for the analysis of last-episode data—that is, data about the timing of the last occurrence of a repeatable event. The new models are elaborations and extensions of a particular model for backward recurrence times from a renewal process introduced by Allison (1985), and they are accelerated failure-time mover-stayer regression models that permit the separation of covariate effects on event timing from those on event occurrence. Using the data of last residential move from the 1990 Census of Population and Housing, an application of the models, which are found to fit the data adequately, shows distinct patterns in the effects of home ownership, gender, race/ethnicity, education, marital experience, and age group on residential mobility. The patterns differ regarding the relative importance and direction of influence between the effects on the duration of residence in the same house or apartment and the effects on the lifetime probability of not experiencing any residential move.

Estimating time to pregnancy from current durations in a cross-sectional sample.

A new design for estimating the distribution of time to pregnancy is proposed and investigated. The design is based on recording current durations in a cross-sectional sample of women, leading to statistical problems similar to estimating renewal time distributions from backward recurrence times. Non-parametric estimation is studied in some detail and a parametric approach is indicated. The results are illustrated on Monte Carlo simulations and on data from a recent European collaborative study. The role and applicability of this approach is discussed.

On rare and extreme events

In many fields of applied physics, the space-time phenomena to be studied may be described in the following way: events of random amplitudes occur randomly in time. We investigate some statistical properties of this model, with special emphasis on situations where the model for the waiting time between consecutive events or the amplitude of individual events are fractal (power-law) distributions with infinite mean value (the rareness or extreme event hypothesis). Limit laws for cumulative partial sums and the extremal process are characterized. Using asymptotical results on backward and forward recurrence times, limit laws are investigated for the physically realistic situation when the cumulative process is only observed starting from some non-zero observational time.

Cox Regression in a Markov Renewal Model: An Application to the Analysis of Bone Marrow Transplant Data

Abstract We discuss estimation and prediction in a Markov renewal model where intensities of the one-step transitions follow a Cox regression model with baseline hazards and covariates depending on the backward recurrence time. A modification of the Andersen and Gill results is given to extend the asymptotic properties of the partial likelihood estimates to this setting. We further consider prediction of the realization of the process in a single individual based on the follow-up information. Data of the International Bone Marrow Transplant Registry are used for illustration purposes.

The correlation of the backward and forward recurrence times in a renewal process

The joint complementary distribution function is used to obtain the covariance function of the backward and forward recurrence times in an ordinary renewal process for both the time dependent and the steady state cases. Hence, a closed form expression for the steady state correlation of the backward and forward recurrence times is obtained and special cases are investigated

Sample-path analysis of stochastic discrete-event systems

This paper presents a unified sample-path approach for deriving distribution-free relations between performance measures for stochastic discrete-event systems extending previous results for discrete-state processes to processes with a general state space. A unique feature of our approach is that all our results are shown to follow from a single fundamental theorem: the sample-path version of the renewal-reward theorem (Y=λX). As an elementary consequence of this theorem, we derive a version of the rate-conservation law under conditions more general than previously given in the literature. We then focus on relations between continuous-time state frequencies and frequencies at the points of an imbedded point process, giving necessary and sufficient conditions for theASTA (Arrivals See Time Averages), conditionalASTA, and reversedASTA properties. In addition, we provide a unified approach for proving various relations involving forward and backward recurrence times. Finally, we give sufficient conditions for rate stability of an input-output system and apply these results to obtain an elementary proof of the relation between the workload and attained-waiting-time processes in aG/G/l queue.

Kin and Nursing Home Lengths of Stay: A Backward Recurrence Time Approach

There are numerous hypotheses about the role played by kin in decreasing the length of elderly persons' nursing home stays, yet few studies have explicitly examined the relationship between kin and episode length. Here we use backward recurrence time data from the Current Resident File of the 1985 National Nursing Home Survey to estimate models in which the hazard of leaving a nursing home depends upon covariates . Women admitted at age 65 or older stay, on average, 26 months in an institution, while men have an average stay of 19 months. Controlling for demographic factors, insurance status, and various chronic and morbid conditions, we find that, on average, having a living spouse decreases the length of stay by four months for men and three months for women. Having a living child decreases the mean length of stay for women by three months, but has no effect on the mean length of stay for men. Findings underscore the importance of family members in shortening the lengths of time spent by older persons in nursing homes.

The use of multiple integrals in the study of the backward and forward recurrence times for the ordinary renewal process

Using a result in multiple integrals, the joint distribution function of the backward and forward recurrence times for the time dependent ordinary renewal process is derived. A similar methodology is used to find the distribution of their sums. The well known limiting behavior of these distributions is recovered

Joint simulation of backward and forward recurrence times in a superposition of independent renewal processes

It is shown that, in a superposition of finitely many independent renewal processes, an observation from the limiting (when t →∞) joint distribution of backward and forward recurrence times at t can be simulated by simulating an observation of the pair (UW, (1 – U)W), where U and Ware independent random variables with U ~ uniform(0, 1) and W distributed according to the limiting total life distribution of the superposition process.

Joint simulation of backward and forward recurrence times in a superposition of independent renewal processes

It is shown that, in a superposition of finitely many independent renewal processes, an observation from the limiting (when t →∞) joint distribution of backward and forward recurrence times at t can be simulated by simulating an observation of the pair (UW, (1 – U)W), where U and Ware independent random variables with U ~ uniform(0, 1) and W distributed according to the limiting total life distribution of the superposition process.

Picosecond dynamics in amorphous systems with local order

We discuss here the rapid time-dependent dynamics of the intermolecular potential in a varied class of “glassy” systems through the evaluation of the vibrational amplitude correlation, i.e. the numerical Fourier transform of the isotropic Raman profile, of an isolated and appropriate system oscillator (Linear Response Theory). We consider the stochastic fluctuations of the intermolecular potential coupling to the oscillator coordinate to represent an equilibrium renewal process that switches, “at random”, the transition frequency of the oscillator about its equilibrium value (“Vibrational Dephasing with Motional Narrowing”). We assign the stretched exponential, ▪ to give a useful and problem-related limiting description of the probability density function of the “backward recurrence time”, i.e. the time interval from time t back to the previous frequency switch, and the “forward recurrence time”, i.e. the time from t the next instant of an oscillator frequency switch. The assumed F̃(t) causes a series of switches (“events’) that are clustered at early observation times, modelling the particular, rapid dynamics of many glass-like systems in terms of “catastrophic” relaxation steps in the medium. Examples are comformational rearrangement of alkyl chains in liquid crystals, jump motion of holes in fused inorganic salts, and jump motion of water-shell molecules between ions in concentrated electrolytes. We discuss here the first and third example.

The role of weighted distributions in stochastic modeling

C. R. Rao pointed out that “The role of statistical methodology is to extract the relevant information from a given sample to answer specific questions about the parent population” and raised the question “What population does a sample represent”? Wrong specification can lead to invalid inference giving rise to a third kind of error. Rao introduced the concept of weighted distributions as a method of adjustment applicable to many situations. In this paper, we study the relationship between the weighted distributions and the parent distributions in the context of reliability and life testing. These relationships depend on the nature of the weight function and give rise to interesting connections between the different ageing criteria of the two distributions. As special cases, the length biased distribution, the equilibrium distribution of the backward and forward recurrence times and the residual life distribution, which frequently arise in practice, are studied and their relationships with the original di...

UNEMPLOYMENT DURATION: MODELLING AND ESTIMATION WITH PARTICULAR REFERENCE TO THE AUSTRALIAN LONGITUDINAL SURVEY

SummarySome of the issues in the analysis of durations of unemployment spells are considered, including, in particular, notions of duration dependence and the effect of the sampling scheme on the correct incorporation of incomplete or censored spells. Attention is restricted to analysis of spells overlapping a particular time point, and consideration is given to the situations where one has data only on the backward recurrence times, and where additional follow‐up information on the forward recurrence times is also available. In the former case a quasi‐likelihood approach is suggested as a possible alternative to full parametric modelling under some special conditions, while in the latter case arguments are forwarded in favour of analyses based on the conditional distribution of the forward recurrence times given the backward recurrence times. The methods are briefly illustrated using samples extracted from the Australian Longitudinal Survey.

Joint simulation of backward and forward recurrence times in a renewal process

It is shown that, in a renewal process with inter-arrival distributionF,an observation from the asymptotic (whent→∞) joint distribution of backward and forward recurrence times attcan be simulated by simulating an observation of the pair (UW, (1 –U)W), whereUandWare independent random variables withU~ uniform(0, 1) andWdistributed according to the length-biased version ofF.

Joint simulation of backward and forward recurrence times in a renewal process

It is shown that, in a renewal process with inter-arrival distribution F, an observation from the asymptotic (when t →∞) joint distribution of backward and forward recurrence times at t can be simulated by simulating an observation of the pair (UW, (1 – U)W), where U and W are independent random variables with U ~ uniform(0, 1) and W distributed according to the length-biased version of F.

A Method for Analyzing Backward Recurrence Time Data on Residential Mobility

Data on duration of current residence are used to estimate models of duration-dependent residential mobility rates. "We use the theory of backward recurrence times in renewal processes to infer the hazard rate of residential mobility from the density of the duration of current residence. We demonstrate that the inferences made using the asymptotic solution of the density of the backward recurrence times will be severely biased if the renewal process has not reached stability. We present an unbiased finite time solution and estimate univariate and multivariate models of residential mobility rates using 1980 U.S. census data on duration of current residence. These models include mover-stayer models represented by mixture densities. The multivariate analysis shows that the most important covariates of residential mobility are age, homeownership, race, and presence of school-aged children."

Some characterizations of renewal densities with emphasis in reliability

Consider a renewal proess on the nonnegative real line with distribution function F(x). Then the backward or forward recurrence times Ut and Vt are , in general, non–independent r.v.s. and are indenpendence iff F(.) is exponential. Several authors have studied the characterizations of the exponential distribution and hence of the Poisson process by certain properties of the distributions of Ut and Vt.

Backward and forward recurrence times, and their interrelationships in Bellman-Harris branching processes

Random variables are constructed for a Bellman-Harris branching process which are the extensions of backward and forward recurrence times from an ordinary renewal process. The sum of the two extended random variables is also constructed, and then the distribution functions for all three random variables are developed. The asymptotic limits for all three time-dependent distributions are also obtained.

Bounds, Inequalities, and Monotonicity Properties for Some Specialized Renewal Processes

Renewal processes with increasing mean residual life and decreasing failure rate interarrival time distributions are investigated. Various two-sided bounds are obtained for $M(t)$, the expected number of renewals in $\lbrack 0, t\rbrack$. It is shown that if the interarrival time distribution has increasing mean residual life with mean $\mu$, then the expected forward recurrence time is increasing in $t \geqslant 0$, as is $M(t) - t/\mu$. If the interarrival time distribution has decreasing failure rate then $M(t)$ is concave, and the forward and backward recurrence time distributions are stochastically increasing in $t \geqslant 0$.

Two-dimensional histograms of forward and backward recurrence time intervals for two pulse series analysis

Detection of time relations between two pulse series in the span of individual interpulse intervals is described. The method is based on application of the scatter diagram of forward and backward recurrence time intervals.