General proportional mean residual life model

By considering a covariate random variable in the ordinary proportional mean residual life (PMRL) model, we introduce and study a general model, taking more situations into account with respect to the ordinary PMRL model. We investigate how stochastic structures of the proposed model are affected by the stochastic properties of the baseline and the mixing variables in the model. Several characterizations and preservation properties of the new model under different stochastic orders and aging classes are provided. In addition, to illustrate different properties of the model, some examples are presented.     

Semiparametric additive intensity model with frailty for recurrent events

The seminal Cox’s proportional intensity model with multiplicative frailty is a popular approach to analyzing the frequently encountered recurrent event data in scientific studies. In the case of violating the proportional intensity assumption, the additive intensity model is a useful alternative. Both the additive and proportional intensity models provide two principal frameworks for studying the association between the risk factors and the disease recurrences. However, methodology development on the additive intensity model with frailty is lacking, although would be valuable. In this paper, we propose an additive intensity model with additive frailty to formulate the effects of possibly time-dependent covariates on recurrent events as well as to evaluate the intra-class dependence within recurrent events which is captured by the frailty variable. The asymptotic properties for both the regression parameters and the association parameters in frailty distribution are established. Furthermore, we also investigate the large-sample properties of the estimator for the cumulative baseline intensity function.     

Additive Positive Stable Frailty Models

In this article, we describe an additive stable frailty model for multivariate times to events data using a flexible baseline hazard, and assuming that the frailty component for each individual is described by additive functions of independent positive stable random variables with possibly different stability indices. Dependence properties of this frailty model are investigated. To carry out inference, the likelihood function is derived by replacing high-dimensional integration by Monte Carlo simulation. Markov chain Monte Carlo algorithms enable estimation and model checking in the Bayesian framework.      

Enhanced annuities and the impact of individual underwriting on an insurer’s profit situation

We analyze the effect of enhanced annuities on an insurer engaging in individual underwriting. We use a frailty model for heterogeneity of the insured population and model individual underwriting by a random variable that positively correlates with the corresponding frailty factor. For a given annuity portfolio, we analyze the effect of the quality of the underwriting on the insurer’s profit/loss situation and the impact of adverse selection effects.     

Bayesian estimation of generalized gamma shared frailty model

Multivariate survival analysis comprises of event times that are generally grouped together in clusters. Observations in each of these clusters relate to data belonging to the same individual or individuals with a common factor. Frailty models can be used when there is unaccounted association between survival times of a cluster. The frailty variable describes the heterogeneity in the data caused by unknown covariates or randomness in the data. In this article, we use the generalized gamma distribution to describe the frailty variable and discuss the Bayesian method of estimation for the parameters of the model. The baseline hazard function is assumed to follow the two parameter Weibull distribution. Data is simulated from the given model and the Metropolis–Hastings MCMC algorithm is used to obtain parameter estimates. It is shown that increasing the size of the dataset improves estimates. It is also shown that high heterogeneity within clusters does not affect the estimates of treatment effects significantly. The model is also applied to a real life dataset.     

Gamma shared frailty model based on reversed hazard rate for bivariate survival data

The unknown or unobservable risk factors in the survival analysis cause heterogeneity between the individuals. Frailty models are used in the survival analysis to account for the unobserved heterogeneity in the individual risks to disease and death. In this paper, we suggest the shared gamma frailty model with the reversed hazard rate. We introduce the Bayesian estimation procedure using MCMC technique to estimate the parameters involved in the model and compare the frailty model with the baseline model. We apply the proposed models to Australian twin data set and suggest a better model.     

Compound negative binomial shared frailty models for bivariate survival data

In this paper, we introduce a new shared frailty model called the compound negative binomial shared frailty model with three different baseline distributions namely, Weibull, generalized exponential and exponential power distribution. To estimate the parameters involved in these models we adopt Markov Chain Monte Carlo (MCMC) approach. Also we apply these three models to a real life bivariate survival data set of McGrilchrist and Aisbett (1991) related to kidney infection and suggest a better model for the data.     

Stochastic and deterministic simulations of a delayed genetic oscillation model: Investigating the validity of reductions

Quasi-stationary approximations are commonly used in order to simplify and reduce the number of equations of genetic circuit models. Protein/protein and protein/DNA binding reactions are considered to occur on much shorter time scale than protein production and degradation processes and often tacitly assumed at a quasi-equilibrium. Taking a biologically inspired, typical, small, abstract, negative feedback, genetic circuit model as study case, we investigate in this paper how different quasi-stationary approximations change the system behaviour both in deterministic and stochastic frameworks. We investigate the consistence between the deterministic and stochastic behaviours of our time-delayed negative feedback genetic circuit model with different implementations of quasi-stationary approximations. Quantitative and qualitative differences are observed among the various reduction schemes and with the underlying microscopic model, for biologically reasonable ranges and combinations of the microscopic model kinetic rates. The different reductions do not behave in the same way: correlations and amplitudes of the stochastic oscillations are not equally captured and the population behaviour is not always in consistence with the deterministic curves.     

A stochastic model for internal HIV dynamics

In this paper we analyse a stochastic model representing HIV internal virus dynamics. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling. We show that the model established in this paper possesses non-negative solutions as this is essential in any population dynamics model. We also carry out analysis on the asymptotic behaviour of the model. We approximate one of the variables by a mean reverting process and find out the mean and variance of this process. Numerical simulations conclude the paper.     

Persistence and extinction of a stochastic Gilpin–Ayala model with jumps

This paper considers a stochastic Gilpin–Ayala model with jumps. First, we show the model that has a unique global positive solution. Then we establish the sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence of the solution. The threshold between weak persistence and extinction is obtained. Finally, we make simulations to conform our analytical results. The results show that the jump process can change the properties of the population model significantly. Copyright © 2014 John Wiley & Sons, Ltd.     

Stochastic comparisons in multivariate mixed model of proportional reversed hazard rate with applications

This paper studies the multivariate mixed proportional reversed hazard rate model having dependent mixing variables. Stochastic comparison as well as aging properties in this model are investigated, and stochastic monotone properties of the population vector with respect to the mixing vector are also discussed. Moreover, MTP₂ dependence among the mixing vectors is proved to imply the increasingness of the reversed hazard rate with respect to the baseline one. Finally, some interesting applications are presented as well.     

A model of an age-structured population in a multipatch environment

We propose a model of an age-structured population divided into N geographical patches. We distinguish two time scales, at the fast time scale we have the migration dynamics and at the slow time scale the demographic dynamics. The demographic process is described using the classical McKendrick-von Foerster model for each patch, and a simple matrix model including the transfer rates between patches depicts the migration process.Assuming that 0 is a simple strictly dominant eigenvalue for the migration matrix, we transform the model (an e.d.p. problem with N state variables) into a classical McKendrick-von Foerster model (scalar e.d.p. problem) for the global variable: total population density. We prove, under certain assumptions, that the semigroup associated to our problem has the property of positive asynchronous exponential growth and so we compare its asymptotic behaviour to that of the transformed scalar model. This type of study can be included in the so-called aggregation methods, where a large scale dynamical system is approximately described by a reduced system. Aggregation methods have been already developed for systems of ordinary differential equations and for discrete time models.An application of the results to the study of the dynamics of the Sole larvae is also provided.     

Vibrations of the Beam with Variable Cross‐Section Caused by Moving Stochastic Load

We present the problem of vibrations of a beam with variable geometry which are caused by stochastic moving load. The approach is based on concepts of the tolerance averaged model [4]. In this way we formulate the averaged equations of the structured beam which describe the length‐scale effect. Using the averaged equations we obtain the probabilistic characteristics of the beam with periodically variable geometry.     

Reliability modelling incorporating load share and frailty

The stochastic behaviour of lifetimes of a two component system is often primarily influenced by the system structure and by the covariates shared by the components. Any meaningful attempt to model the lifetimes must take into consideration the factors affecting their stochastic behaviour. In particular, for a load share system, we describe a reliability model incorporating both the load share dependence and the effect of observed and unobserved covariates. The model includes a bivariate Weibull to characterize load share, a positive stable distribution to describe frailty, and also incorporates effects of observed covariates. We investigate various interesting reliability properties of this model using cross ratio functions and conditional survivor functions. We implement maximum likelihood estimation of the model parameters and discuss model adequacy and selection. We illustrate our approach using a simulation study. For a real data situation, we demonstrate the superiority of the proposed model that incorporates both load share and frailty effects over competing models that incorporate just one of these effects. An attractive and computationally simple cross‐validation technique is introduced to reconfirm the claim. We conclude with a summary and discussion.     

A Stochastic Model for Population and Well-Being Dynamics

This article presents a stochastic dynamic model to study the demographic evolution per sexes and the corresponding well-being of a general human population. The main model variables are population per sexes and well-being. The considered well-being variable is the Gender-Related Development Index (GDI), a United Nations index. The model's objectives are to improve future well-being and to reach a stable population in a country. The application case consists of adapting, validating, and using the model for Spain in the 2000–2006 period. Some instance strategies have been tested in different scenarios for the 2006–2015 period to meet these objectives by calculating the reliability of the results. The optimal strategy is “government invests more in education and maintains the present health investment tendency.”     

Risk model with fuzzy random individual claim amount

In this paper, we consider a risk model in which individual claim amount is assumed to be a fuzzy random variable and the claim number process is characterized as a Poisson process. The mean chance of the ultimate ruin is researched. Particularly, the expressions of the mean chance of the ultimate ruin are obtained for zero initial surplus and arbitrary initial surplus if individual claim amount is an exponentially distributed fuzzy random variable. The results obtained in this paper coincide with those in stochastic case when the fuzzy random variables degenerate to random variables. Finally, two numerical examples are presented.     

Nonparametric estimation of education productivity incorporating nondiscretionary inputs with an application to Dutch schools

In this paper we develop a Malmquist productivity index for public sector production characterized by the influence of environmental variables. We extend Johnson and Ruggiero (2011) to the more general case of variable returns to scale to further decompose the Malmquist productivity index into technical, efficiency, scale and environmental change. We apply our model to analyze productivity of Dutch schools using 2002–2007 data. The results indicate that the environment influences the productivity index as well as the technical, efficiency, scale and environmental change components. We see that schools with a moderate classification of environment have the highest productivity numbers. In line with expectations, schools with the worst environment also perform worse and would perform better with an improved environment.     

A Mathematical Framework for Critical Transitions: Normal Forms, Variance and Applications

Critical transitions occur in a wide variety of applications including mathematical biology, climate change, human physiology and economics. Therefore it is highly desirable to find early-warning signs. We show that it is possible to classify critical transitions by using bifurcation theory and normal forms in the singular limit. Based on this elementary classification, we analyze stochastic fluctuations and calculate scaling laws of the variance of stochastic sample paths near critical transitions for fast-subsystem bifurcations up to codimension two. The theory is applied to several models: the Stommel–Cessi box model for the thermohaline circulation from geoscience, an epidemic-spreading model on an adaptive network, an activator–inhibitor switch from systems biology, a predator–prey system from ecology and to the Euler buckling problem from classical mechanics. For the Stommel–Cessi model we compare different detrending techniques to calculate early-warning signs. In the epidemics model we show that link densities could be better variables for prediction than population densities. The activator–inhibitor switch demonstrates effects in three time-scale systems and points out that excitable cells and molecular units have information for subthreshold prediction. In the predator–prey model explosive population growth near a codimension-two bifurcation is investigated and we show that early-warnings from normal forms can be misleading in this context. In the biomechanical model we demonstrate that early-warning signs for buckling depend crucially on the control strategy near the instability which illustrates the effect of multiplicative noise.     

Location-aided routing with uncertainty in mobile ad hoc networks: A stochastic semidefinite programming approach

We study location-aided routing under mobility in wireless ad hoc networks. Due to node mobility, the network topology changes continuously, and clearly there exists an intricate tradeoff between the message passing overhead and the latency in the route discovery process. Aiming to obtain a clear understanding of this tradeoff, we use stochastic semidefinite programming (SSDP), a newly developed optimization model, to deal with the location uncertainty associated with node mobility. In particular, we model both the speed and the direction of node movement by random variables and construct random ellipses accordingly to better capture the location uncertainty and the heterogeneity across different nodes. Based on SSDP, we propose a stochastic location-aided routing (SLAR) strategy to optimize the tradeoff between the message passing overhead and the latency. Our results reveal that in general SLAR can significantly reduce the overall overhead than existing deterministic algorithms, simply because the location uncertainty in the routing problem is better captured by the SSDP model.     

Lagrangian Decomposition for large-scale two-stage stochastic mixed 0-1 problems

In this paper we study solution methods for solving the dual problem corresponding to the Lagrangian Decomposition of two-stage stochastic mixed 0-1 models. We represent the two-stage stochastic mixed 0-1 problem by a splitting variable representation of the deterministic equivalent model, where 0-1 and continuous variables appear at any stage. Lagrangian Decomposition (LD) is proposed for satisfying both the integrality constraints for the 0-1 variables and the non-anticipativity constraints. We compare the performance of four iterative algorithms based on dual Lagrangian Decomposition schemes: the Subgradient Method, the Volume Algorithm, the Progressive Hedging Algorithm, and the Dynamic Constrained Cutting Plane scheme. We test the tightness of the LD bounds in a testbed of medium- and large-scale stochastic instances.