The Asymptotic Behavior of INAR (p) Models

An integer-valued autoregressive model with random time delay under random environment is presented. The geometric ergodicity of the iterative sequence determined by this new model is discussed. Moreover, sufficient conditions for stationarity and β-mixing property with exponential decay for the INAR model with random time delay under random environment are developed.     

Generalized Linear Mixed Models Based on Latent Markov Heterogeneity Structures

We describe a generalized linear mixed model in which all random effects may evolve over time. Random effects have a discrete support and follow a first‐order Markov chain. Constraints control the size of the parameter space and possibly yield blocks of time‐constant random effects. We illustrate with an application to the relationship between health education and depression in a panel of adolescents, where the random effects are highly dimensional and separately evolve over time.     

Some results on residual life and inactivity time at random time

In this article, we enhance the study of residual life at random time (RLRT) and inactivity time at random time (ITRT). To this aim, first we provide some stochastic orderings results among ITRT in two-sample problems when they failed at two different random times. Then, we develop some sufficient conditions which lead to the stochastic comparisons of RLRT and ITRT based on variance residual life order. The results are expected to be useful in reliability theory, forensic science, queue theory, and actuarial science.     

A characterization of probability distributions similar to the exponential

A concept of the lack-of-memory property at a given time point c > 0 is introduced. It is equivalent to the concept of the almost-lack-of-memory (ALM) property of the random variables. A representation theorem is given for the cumulative distribution function of such random variables as well as for corresponding decompositions in terms of independent random variables. It is shown that a periodic failure rate for a random variable is equivalent to the ALM property. In addition some properties of the service time of an unreliable server are observed.     

A class of neutral to the right priors induced by superposition of beta processes

A random distribution function on the positive real line which belongs to the class of neutral to the right priors is defined. It corresponds to the superposition of independent beta processes at the cumulative hazard level. The definition is constructive and starts with a discrete time process with random probability masses obtained from suitably defined products of independent beta random variables. The continuous time version is derived as the corresponding infinitesimal weak limit and is described in terms of completely random measures. It takes the interpretation of the survival distribution resulting from independent competing failure times. We discuss prior specification and illustrate posterior inference on a real data example.     

On k-match problems

Several waiting time random variables for a duplication within a memory window of size k in a sequence of {1,2,…,m}-valued random variables are investigated. The exact distributions of the waiting time random variables are derived by the method of conditional probability generating functions. In particular, the exact distribution of the waiting time for the first k-match is obtained when the underlying sequence is generated by higher order Markov dependent trials. Examples for numerical calculations are also given.     

Discrete time cold standby repairable system: Combinatorial analysis

ABSTRACT

In this article, we obtain exact expression for the distribution of the time to failure of discrete time cold standby repairable system under the classical assumptions that both working time and repair time of components are geometric. Our method is based on alternative representation of lifetime as a waiting time random variable on a binary sequence, and combinatorial arguments. Such an exact expression for the time to failure distribution is new in the literature. Furthermore, we obtain the probability generating function and the first two moments of the lifetime random variable.     

SURVIVAL ANALYSIS WITH TIME DEPENDENT FRAILTY USING A LONGITUDINAL MODEL

A method of estimation for generalised mixed models is applied to the estimation of regression parameters in a proportional hazards model with time dependent frailty. A parameter representing change over time is introduced and is modelled in turn into a fixed effect, a normally distributed random effect and a longitudinal effect in which the random component relates to the patient characteristics. Both maximum likelihood and residual maximum likelihood estimators are given.     

The repair alert models with fixed and random effects

This article extends a random preventive maintenance scheme, called repair alert model, when there exist environmental variables that effect on system lifetimes. It can be used for implementing age-dependent maintenance policies on engineering devices. In other words, consider a device that works for a job and is subject to failure at a random time X, and the maintenance crew can avoid the failure by a possible replacement at some random time Z. The new model is flexible to including covariates with both fixed and random effects. The problem of estimating parameters is also investigated in details. Here, the observations are in the form of random signs censoring data (RSCD) with covariates. Therefore, this article generalizes derived statistical inferences on the basis of RSCD albeit without covariates in past literature. To do this, it is assumed that the system lifetime distribution belongs to the log-location-scale family of distributions. A real dataset is also analyzed on basis of the results obtained.     

Crisis cycles in the slightly unstable block random model

ABSTRACT

The paper proposes a new approach for studying the time to time appearing breakdowns in economy. Block random model can describe stability of large complicated systems with variable number of participants. Theoretical background of the model is given by a theorem about the eigenvalues of block random matrices [Juhász F. On the characteristic values of non-symmetric block random matrices. J Theoret Probab. 1990;67:199–205; On the structural eigenvalues of block random matrices. Linear Algebra Appl. 1996;246:225–231]. The model takes into account not only effects of participants but of groups formed from them as well. Slight instability means group level stability and participant level instability [Juhász F. On the turbulence of slightly unstable block random systems. In: Taylor C, et al., editors. Numerical methods for laminar and turbulent flow. Atlanta; 1995. p. 113–121]. Lability index of block random systems is introduced for measuring instability. It is showed that lability index of a slightly unstable block random model is growing while number of participants increases. Alteration in the number of participants makes it possible to describe crisis cycles.     

A guided walk Metropolis algorithm

The random walk Metropolis algorithm is a simple Markov chain Monte Carlo scheme which is frequently used in Bayesian statistical problems. We propose a guided walk Metropolis algorithm which suppresses some of the random walk behavior in the Markov chain. This alternative algorithm is no harder to implement than the random walk Metropolis algorithm, but empirical studies show that it performs better in terms of efficiency and convergence time.     

Sampling efficiency for an alternating poisson process

The efficiency of schemes for sampling an alternating Poisson process (0,1 observations) is evaluated by the inverse ratio of the variance of the proportion estimate, p, to the binomial variance. The variance ratio presented by D.R. Cox (in Renewal Theory) for fixed interval sampling is generalized to accommodate random sampling and random sampling after a time delay equal to a fixed proportion, γ , of the mean time between observations, δ. The result is a sampling design tool that provides quantifications for the effect of various spacings between observations and of fixed vs. random sampling. Direct application is made to thes field of work sampling.     

Preservation of generalized ageing classes on the residual life at random time

In this paper, we provide some new preservation properties of generalized ageing classes (s-IFR) on the residual life at random time, where s is a nonnegative integer. We also obtain bounds of the residual life at exponential random time. Results are expected to be useful in the reliability, queue theory and actuarial science.     

A class of dynamic piecewise exponential models with random time grid

A novel fully Bayesian approach for modeling survival data with explanatory variables using the Piecewise Exponential Model (PEM) with random time grid is proposed. We consider a class of correlated Gamma prior distributions for the failure rates. Such prior specification is obtained via the dynamic generalized modeling approach jointly with a random time grid for the PEM. A product distribution is considered for modeling the prior uncertainty about the random time grid, turning possible the use of the structure of the Product Partition Model (PPM) to handle the problem. A unifying notation for the construction of the likelihood function of the PEM, suitable for both static and dynamic modeling approaches, is considered. Procedures to evaluate the performance of the proposed model are provided. Two case studies are presented in order to exemplify the methodology. For comparison purposes, the data sets are also fitted using the dynamic model with fixed time grid established in the literature. The results show the superiority of the proposed model.     

SOJOURN TIMES IN NON-HOMOGENEOUS QBD PROCESSES WITH PROCESSOR SHARING

We study sojourn times of customers in a processor sharing queue with a service rate that varies over time, depending on the number of customers and on the state of a random environment. An explicit expression is derived for the Laplace–Stieltjes transform of the sojourn time conditional on the state upon arrival and the amount of work brought into the system. Particular attention is paid to the conditional mean sojourn time of a customer as a function of his required amount of work, and we establish the existence of an asymptote as the amount of work tends to infinity. The method of random time change is then extended to include the possibility of a varying service rate. By means of this method, we explain the well-established proportionality between the conditional mean sojourn time and required amount of work in processor sharing queues without random environment. Based on numerical experiments, we propose an approximation for the conditional mean sojourn time. Although first presented for exponentially distributed service requirements, the analysis is shown to extend to phase-type services. The service discipline of discriminatory processor sharing is also shown to fall within the framework.     

Random time step probabilistic methods for uncertainty quantification in chaotic and geometric numerical integration

A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomise ODE solvers by adding a random forcing term, we show that a probability measure over the numerical solution of ODEs can be obtained by introducing suitable random time steps in a classical time integrator. This intrinsic randomisation allows for the conservation of geometric properties of the underlying deterministic integrator such as mass conservation, symplecticity or conservation of first integrals. Weak and mean square convergence analysis is derived. We also analyse the convergence of the Monte Carlo estimator for the proposed random time step method and show that the measure obtained with repeated sampling converges in the mean square sense independently of the number of samples. Numerical examples including chaotic Hamiltonian systems, chemical reactions and Bayesian inferential problems illustrate the accuracy, robustness and versatility of our probabilistic numerical method.     

Long-term experiments and strip plot designs

In a long-term experiment usually the experimenter needs to know whether the effect of a treatment varies over time. But time usually has both a fixed and a random effects over the output and the difficulty in the analysis depends on the particular design considered and the availability of covariates. Actually, as shown in the paper, the presence of covariates can be very useful to model the random effect of time. In this paper a model to analyze data from a long-term strip plot design with covariates is proposed. Its effectiveness will be tested using both simulated and real data from a crop rotation experiment.     

On the analysis of long-term experiments

Summary.  Long-term experiments are commonly used tools in agronomy, soil science and other disciplines for comparing the effects of different treatment regimes over an extended length of time. Periodic measurements, typically annual, are taken on experimental units and are often analysed by using customary tools and models for repeated measures. These models contain nothing that accounts for the random environmental variations that typically affect all experimental units simultaneously and can alter treatment effects. This added variability can dominate that from all other sources and can adversely influence the results of a statistical analysis and interfere with its interpretation. The effect that this has on the standard repeated measures analysis is quantified by using an alternative model that allows for random variations over time. This model, however, is not useful for analysis because the random effects are confounded with fixed effects that are already in the repeated measures model. Possible solutions are reviewed and recommendations are made for improving statistical analysis and interpretation in the presence of these extra random variations.     

Quasi-Likelihood Approach to Bioavailability and Bioequivalence Analysis

The outcomes AUCT (area-under-curve from time zero to time t) of n individuals randomized to one of two groups TR or RT, where the group name denotes the order in which the subjects receive a test formulation (T) or a reference formulation (R), are used to assess average bioequivalence for the two formulations. The classical method is the mixed model, for example, proc mixed or proc glm with random statement in SAS can be used to analyze this type of data. This is equivalent to the marginal likelihood approach in a normal–normal model. There are some limitations for this approach. It is not appropriate if the random effect is not normally distributed. In this article, we introduce a hierarchical quasi-likelihood approach. Instead of assuming the random effect is normal, we make assumptions only about the mean and the variance function of the random effect. Our method is flexible to model the random effect. Since we can estimate the random effect for each individual, we can check the adequacy of the distribution assumption about the random effect. This method can also be used to handle high-dimensional crossover data. Simulation studies are conducted to check the finite sample performance of the method under various conditions and two real data examples are used for illustration.     

An Efficient Simulation Method for the Computation of a Class of Conditional Expectations

Suppose that the random vector X and the random variable Y are jointly continuous. Also suppose that an observation x of X can be easily simulated and that the probability density function of Y conditional on X = x is known. The paper presents an efficient simulation-based algorithm for estimating E{ g ( X , Y ) | h ( X , Y ) = r } where g and h are real-valued functions. This algorithm is applicable to time series problems in which X = ( X ₁, . . . , X _n−1) and Y = X_n where { x_t } is a discrete time stochastic process for which ( X₁ , . . . , X_n ) is a continuous random vector. A numerical example from time series analysis illustrates the algorithim, for prediction for an ARCH(1) process.