Time-varying latent model for longitudinal data with informative observation and terminal event times

Longitudinal data often occur in follow-up studies, and in many situations, there may exist informative observation times and a dependent terminal event such as death that stops the follow-up. We propose a semiparametric mixed effect model with time-varying latent effects in the analysis of longitudinal data with informative observation times and a dependent terminal event. Estimating equation approaches are developed for parameter estimation, and asymptotic properties of the resulting estimators are established. The finite sample behavior of the proposed estimators is evaluated through simulation studies, and an application to a bladder cancer study is provided.     

Control and recovery from rare congestion events in a large multi-server system

We develop deterministic fluid approximations to describe the recovery from rare congestion events in a large multi-server system in which customer holding times have a general distribution. There are two cases, depending on whether or not we exploit the age distribution (the distribution of elapsed holding times of customers in service). If we do not exploit the age distribution, then the rare congestion event is a large number of customers present. If we do exploit the age distribution, then the rare event is an unusual age distribution, possibly accompanied by a large number of customers present. As an approximation, we represent the large multi-server system as an M/G/∞ model. We prove that, under regularity conditions, the fluid approximations are asymptotically correct as the arrival rate increases. The fluid approximations show the impact upon the recovery time of the holding-time distribution beyond its mean. The recovery time may or not be affected by the holding-time distribution having a long tail, depending on the precise definition of recovery. The fluid approximations can be used to analyze various overload control schemes, such as reducing the arrival rate or interrupting services in progress. We also establish large deviations principles to show that the two kinds of rare events have the same exponentially small order. We give numerical examples showing the effect of the holding-time distribution and the age distribution, focusing especially on the consequences of long-tail distributions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Markov processes with creation of mass. II

Summary A theory of Markov processes for which the sample paths have random starting and terminal times was first considered by Hunt [4, 5]. Such processes require that the classical notion of a probability measure space be modified so as to allow -finite probability measures. Such an extension necessitates only minor adjustments of the language of probability theory; e.g., the phrase absolute probability of an event E does not admit the usual interpretation and is not used. Only conditional probabilities are used in the extended theory and do admit the usual interpretation. The purpose of this paper is to establish regularity properties of Markov processes having sample paths with random creation and terminal times. An existence theorem for such processes appears in [3].This research was supported by the National Science Foundation.     

Estimating household purchase rates for consumer nondurable goods

Understanding the purchase rates of households for frequently purchased packaged goods is an important element in developing effective marketing strategies. Previous researchers have attempted to estimate these rates by assuming that the time between purchases is a random variable that follows some common parametric probability distribution such as the exponential or Weibull distribution. Recent research has shown that for many frequently purchased packaged goods, the interpurchase times cannot be adequately described by these commonly used probability distributions. In this study we demonstrate how household purchase rates can be estimated in a robust manner using a generalized semiparametric approach that obviates the need for specifying a parametric form for the distribution of interpurchase times. The motivation being that often there is no theory of household purchase behaviour that specifies a priori the probability distribution underlying the interpurchase times. Our empirical results indicate that, for the data analysed, the household purchase rates exhibit a regular pattern that cannot be recovered by probability distributions often used in previous research. Further, marketing actions taken by sellers do indeed influence household purchase behaviour.     

System availability in a shock model under preventive repair and phase‐type distributions

A device that can fail by shocks or ageing under policy N of maintenance is presented. The interarrival times between shocks follow phase‐type distributions depending on the number of cumulated shocks. The successive shocks deteriorate the system, and some of them can be fatal. After a prefixed number k of nonfatal shocks, the device is preventively repaired. After a fatal shock the device is correctively repaired. Repairs are as good as new, and follow phase‐type distributions. The system is governed by a Markov process whose infinitesimal generator, stationary probability vector, and availability are calculated, obtaining well‐structured expressions due to the use of phase‐type distributions. The availability is optimized in terms of the number k of preventive repairs. Copyright © 2009 John Wiley & Sons, Ltd.     

The effects of service disciplines on the performance of a nonreliable terminal system

In this paper, we consider a stochastic queueing model for the performance evaluaton of a real-life computer system consisting of n terminals connected with a CPU. A user at terminal i has thinking and processing time depending on the index i. Let us suppose that the operational system is subject to random breakdowns, which may be software or hardware ones, stopping the service both at the terminals and at the CPU. The failure-free operation times of the system and the restoration times are random variables. Busy terminals are also subject to random breakdowns not affecting the system operation. The failure-free operation times and the repair times of a busy terminal i are random variables with distribution function depending on index i. The breakdowns are serviced by a single repairman providing preemptive priority to the system's failure, while the restoration at the terminals are carried out according to the FIFO rule. We assume that each user generates only one job at a time, and he waits at the CPU before he starts thinking again, that is, the terminal is inactive while waiting at the CPU, and it cannot break down. All random variables involved in the model construction are assumed to be exponentially distributed and independent of each other. The aim of this paper is to investigate the effect of different service disciplines, such as FIFO, processor sharing, priority processor sharing, and polling, on the main performance measures, such as utilizations, response times, throughput, and mean queue length. Supported by the Hungarian National Foundation for Scientific Research (grant No. ORKA T014974/95). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló. Hungary, 1997, Part II.     

Asymptotics of stochastic networks with subexponential service times

We analyse the tail behaviour of stationary response times in the class of open stochastic networks with renewal input admitting a representation as (max,+)-linear systems. For a K-station tandem network of single server queues with infinite buffer capacity, which is one of the simplest models in this class, we first show that if the tail of the service time distribution of one server, say server i ₀ ∈ {1,...,K}, is subexponential and heavier than those of the other servers, then the stationary distribution of the response time until the completion of service at server j ⩾ i ₀ asymptotically behaves like the stationary response time distribution in an isolated single-server queue with server i ₀. Similar asymptotics are given in the case when several service time distributions are subexponential and asymptotically tail-equivalent. This result is then extended to the asymptotics of general (max,+)-linear systems associated with i.i.d. driving matrices having one (or more) dominant diagonal entry in the subexponential class. In the irreducible case, the asymptotics are surprisingly simple, in comparison with results of the same kind in the Cramér case: the asymptotics only involve the excess distribution of the dominant diagonal entry, the mean value of this entry, the intensity of the arrival process, and the Lyapunov exponent of the sequence of driving matrices. In the reducible case, asymptotics of the same kind, though somewhat more complex, are also obtained. As a direct application, we give the asymptotics of stationary response times in a class of stochastic Petri nets called event graphs. This is based on the assumption that the firing times are independent and that the tail of the firing times of one of the transitions is subexponential and heavier than those of the others. An extension of these results to nonrenewal input processes is discussed. Asymptotics of queue size processes are also considered. This revised version was published online in June 2006 with corrections to the Cover Date.     

Waiting time distributions for closed M/M/N processor sharing queues

We consider a multiple server processor sharing model with a finite number of terminals (customers). Each terminal can submit at most one job for service at any time. The think times of the terminals and the service time demands are independently exponentially distributed. We focus our attention on the exact detailed analysis of the waiting time distribution of a tagged job. We give the Laplace-Stieltjes transform of the waiting time distribution conditioned on the job's service time demand and the state of the other terminals and show that these transforms can be efficiently evaluated and inverted. Further results include the representation of conditioned waiting times as mixtures of a constant and several exponentially distributed components. The numerical precision of our results is being compared with results from a discrete approximation of the waiting time distributions.The main part of this research was carried out at the Institut für Mathematische Stochastik of the Technische UniversitÄt Braunschweig.     

Social forecasting for company planning: Brian C. TWISS (Ed.) Macmillan,London, 1982, xxv + 310 pages, £25.00

The transportation system considered in this paper has a number of vehicles with no capacity constraint, which take passengers from a source terminal to various destinations and return to the terminal. The trip times are considered to be independent and identically distributed random variables with a common exponential distribution. Passengers arrive at the terminal in accordance with a Poisson process. The system is operated under the following policy: when a vehicle is available and there are at least α passengers waiting for service, then a vehicle is dispatched immediately. The passenger queue length and waiting time distributions are obtained under steady-state conditions. System performance measures such as average passenger queue length and waiting time are then derived. A minimum average cost criterion is then used to determine the optimal fleet size and dispatching policy. This is a generalization of the results of Weiss for a single-vehicle system.     

On a loss system with a given number of customers

A one-server loss system with Poisson arrival stream and deterministic service times is considered conditional on the number of customers who appeared up to a givenT. This condition implies that the arrival times form a sample of the uniform distribution on (0,T]. We derive several characteristics of interest, such as the blocking probability at any given timet (0,T], the probability that exactlyi of the customers in (0,T] are served and, as a generalization, the distribution of the number of served customers arriving in any subinterval of (0,T].     

带相依终止事件的复发事件数据的可加可乘比率模型

本文对带相依终止事件的复发事件数据提出了一个联合建模分析方法,用一个带脆弱变量的可加可乘比率模型来刻画复发事件过程,还用带脆弱变量的Cox风险率模型来刻画终止事件过程,而且这两个过程的相依性由脆弱变量来刻画.我们利用估计方程的方法,对模型参数进行了估计,给出了所得估计的渐近性质.同时,通过数值模拟分析验证了估计的渐近性质.最后,利用该方法分析了弗吉尼亚大学慢性心脏病病人医疗诊费数据.     

Stopped Markov chains with stationary occupation times

Summary. Let E be a finite set equipped with a group G of bijective transformations and suppose that X is an irreducible Markov chain on E that is equivariant under the action of G. In particular, if E=G with the corresponding transformations being left or right multiplication, then X is a random walk on G. We show that when X is started at a fixed point there is a stopping time U such that the distribution of the random vector of pre-U occupation times is invariant under the action of G. When G acts transitively (that is, E is a homogeneous space), any non-zero, finite expectation stopping time with this property can occur no earlier than the time S of the first return to the starting point after all states have been visited. We obtain an expression for the joint Laplace transform of the pre-S occupation times for an arbitrary finite chain and show that even for random walk on the group of integers mod r the pre-S occupation times do not generally have a group invariant distribution. This appears to contrast with the Brownian analog, as there is considerable support for the conjecture that the field of local times for Brownian motion on the circle prior to the counterpart of S is stationary under circular shifts. Received: 6 December 1995 / In revised form: 11 June 1997     

A simulation-based genetic algorithm approach for reducing emissions from import container pick-up operation at container terminal

Emissions from idle truck engines are a main source of pollution at container terminals. In this study, we focus on reducing such emission from waiting trucks as well as the related crane operations with a new truck arrival control method that gives individual truck limited time slots for entry. We develop a method to optimize the time slot assignment for individual trucks, aiming at minimizing total emissions from trucks and cranes at import yards. The method applies discrete event simulation to estimate total truck waiting times and crane moving distance, and then applies a genetic algorithm to minimize the generated emissions from these trucks and cranes. The experiment result shows that the truck arrivals should be controlled based on the stacking of import containers, and that such control is necessary for reducing truck idling emissions at a congested container terminal.     

Nonparametric estimation with doubly censored and truncated data

Data from longitudinal studies in which an initiating event and a subsequent event occur in sequence are called doubly censored data if the time of both events is interval-censored. In some cases, the second event also suffers left-truncation. This article is concerned with using doubly censored and truncated data to estimate the distribution function F of the duration time, i.e. the elapsed time between the originating event and the subsequent event. An iterative procedure is proposed to obtain the estimate of F. A simulation study is conducted to investigate the performance of the proposed estimator. A modified data set is used to illustrate the proposed approach.     

On the Counting Process for a Class of Markovian Arrival Processes with an Application to a Queueing System

In this paper, we consider the counting process for a class of Markovian arrival processes (MAPs). We assume that the representing matrices in such MAPs are expanded in terms of matrix representations of the standard generators in the Lie algebra of the special linear group. The primary purpose of this paper is to construct an explicit solution of the time-dependent distribution and factorial moments of the number of arrival events in (0,t] of the counting process for this class of MAPs. Our construction relies on the Baker–Hausdorff lemma and the specific structure of the representing matrices. To investigate the efficiency of CPU usage with the explicit solution, we have conducted numerical experiments on computing the time-dependent distribution of the counting process through the explicit solution and uniformization-based method. We show that the CPU times required to compute the time-dependent distribution of the number of arrival events in (0,t] through the explicit solution have little sensitivity to t, while the consumption of CPU times with the uniformization-based method becomes greater as t increases. For illustrative purposes, we present a system performance analysis of a queueing system for possible use in automatic call distribution (ACD) systems. As an application of the explicit solution, we use it to express the waiting time distribution of the queueing system. Some numerical examples are also given with comparisons to computer simulations.AMS subject classification: 60K20, 60K25, 68M20This revised version was published online in June 2005 with corrected coverdate     

The population-sample decomposition approach to multivariate estimation methods

In this paper it is argued that all multivariate estimation methods, such as OLS regression, simultaneous linear equations systems and, more widely, what are known as LISREL methods, have merit as geometric approximation methods, even if the observations are not drawn from a multivariate normal parent distribution and consequently cannot be viewed as ML estimators. It is shown that for large samples the asymptotical distribution of any estimator, being a totally differentiable covariance function, may be assessed by the δ method. Finally, we stress that the design of the sample and a priori knowledge about the parent distribution may be incorporated to obtain more specific results. It turns out that some fairly traditional assumptions, such as assuming some variables to be non-random, fixed over repeated samples, or the existence of a parent normal distribution, may have dramatic effects on the assessment of standard deviations and confidence bounds, if such assumptions are not realistic. The method elaborated by us does not make use of such assumptions.     

Asymptotics for non-parametric likelihood estimation with doubly censored multivariate failure times

This paper considers non-parametric estimation of a multivariate failure time distribution function when only doubly censored data are available, which occurs in many situations such as epidemiological studies. In these situations, each of multivariate failure times of interest is defined as the elapsed time between an initial event and a subsequent event and the observations on both events can suffer censoring. As a consequence, the estimation of multivariate distribution is much more complicated than that for multivariate right- or interval-censored failure time data both theoretically and practically. For the problem, although several procedures have been proposed, they are only ad-hoc approaches as the asymptotic properties of the resulting estimates are basically unknown. We investigate both the consistency and the convergence rate of a commonly used non-parametric estimate and show that as the dimension of multivariate failure time increases or the number of censoring intervals of multivariate failure time decreases, the convergence rate for non-parametric estimate decreases, and is slower than that with multivariate singly right-censored or interval-censored data.     

Hidden Markov Filter Estimation of the Occurrence Time of an Event in a Financial Market

Abstract

In this paper, we use filtering techniques to estimate the occurrence time of an event in a financial market. The occurrence time is being viewed as a Markov stopping time with respect to the σ-field generated by a hidden Markov process. We also generalize our result to the Nth occurrence time of that event.     

Limit distributions of smallest gap and largest repeated part in integer partitions

We study two parameters in random integer partitions, namely the first gap and the last repeated part, that have been introduced by Grabner and Knopfmacher in a recent paper (Ramanujan J. 12(3):439–454, 2006). More generally, the first part that occurs at most r times and the last part that occurs at least r times are considered. For both parameters, we determine the limit distribution, which turn out to be the Rayleigh and Gumbel distributions, respectively. This also generalises the well-known result by Erdős and Lehner on the distribution of the largest part in a random integer partition. Furthermore, extensions to general Λ-partitions and results on related parameters, such as the length of the first gap, are provided.