Analyzing short time series data from periodically fluctuating rodent populations by threshold models: A nearest block bootstrap approach

The study of the rodent fluctuations of the North was initiated in its modern form with Elton’s pioneering work. Many scientific studies have been designed to collect yearly rodent abundance data, but the resulting time series are generally subject to at least two “problems”: being short and non-linear. We explore the use of the continuous threshold autoregressive (TAR) models for analyzing such data. In the simplest case, the continuous TAR models are additive autoregressive models, being piecewise linear in one lag, and linear in all other lags. The location of the slope change is called the threshold parameter. The continuous TAR models for rodent abundance data can be derived from a general prey-predator model under some simplifying assumptions. The lag in which the threshold is located sheds important insights on the structure of the prey-predator system. We propose to assess the uncertainty on the location of the threshold via a new bootstrap called the nearest block bootstrap (NBB) which combines the methods of moving block bootstrap and the nearest neighbor bootstrap. The NBB assumes an underlying finite-order time-homogeneous Markov process. Essentially, the NBB bootstraps blocks of random block sizes, with each block being drawn from a non-parametric estimate of the future distribution given the realized past bootstrap series. We illustrate the methods by simulations and on a particular rodent abundance time series from Kilpisjärvi, Northern Finland.     

Interpolation solution in generalized stochastic exponential population growth model

In this paper, first we consider model of exponential population growth, then we assume that the growth rate at time t is not completely definite and it depends on some random environment effects. For this case the stochastic exponential population growth model is introduced. Also we assume that the growth rate at time t depends on many different random environment effect, for this case the generalized stochastic exponential population growth model is introduced. The expectations and variances of solutions are obtained. For a case study, we consider the population growth of Iran and obtain the output of models for this data and predict the population individuals in each year.     

Probability maximization models for portfolio selection under ambiguity

This paper considers several probability maximization models for multi-scenario portfolio selection problems in the case that future returns in possible scenarios are multi-dimensional random variables. In order to consider occurrence probabilities and decision makers’ predictions with respect to all scenarios, a portfolio selection problem setting a weight with flexibility to each scenario is proposed. Furthermore, by introducing aspiration levels to occurrence probabilities or future target profit and maximizing the minimum aspiration level, a robust portfolio selection problem is considered. Since these problems are formulated as stochastic programming problems due to the inclusion of random variables, they are transformed into deterministic equivalent problems introducing chance constraints based on the stochastic programming approach. Then, using a relation between the variance and absolute deviation of random variables, our proposed models are transformed into linear programming problems and efficient solution methods are developed to obtain the global optimal solution. Furthermore, a numerical example of a portfolio selection problem is provided to compare our proposed models with the basic model.     

Discrete-time host–parasitoid models with Allee effects: Density dependence versus parasitism

We present two general discrete-time host–parasitoid models with Allee effects on the host. In the first model, it is assumed that parasitism occurs prior to density dependence, while in the second model we assume that density dependence operates first followed by parasitism. It is shown that both models have similar asymptotic behaviour. The parasitoid population will definitely go extinct if the maximal growth rate of the host population is less than or equal to one, independent of whether density dependence or parasitism occurs first. The fate of the population is initial condition dependent if this maximal growth rate exceeds one. In particular, there exists a host population threshold, the Allee threshold, below which the host population goes extinct and so does the parasitoid. This threshold is the same for both models. Numerical examples with different functions are simulated to illustrate our analytical results.     

Simultaneous life testing in a random environment

Examined here is a class of multivariate lifetime distributions generated by a physical model in which a group of like devices is simultaneously exposed to a random wear or damage environment. This random wear is represented by a nonnegative stochastic process with independent increments. Associated with each device is a random threshold and the device fails when the wear attains this threshold. It is shown that tied failure times occur with positive probability. Algorithms are developed to obtain the probabilistic properties of various random variables associated with the joint failure time vector. In particular, these algorithms are used to find the probability of obtaining a specific tie configuration and the large sample behavior of the number of distinct failure times.     

Random effects logistic regression model for default prediction of technology credit guarantee fund

Korean government has been funding the small and medium enterprises (SME) with superior technology based on scorecard. However high default rate of funded SMEs has been reported. In order to effectively manage such governmental fund, it is important to develop accurate scoring model for SMEs. In this paper, we provide a random effects logistic regression model to predict the default of funded SMEs based on both financial and non-financial factors. Advantage of such a random effects model lies in the ability of accommodating not only the individual characteristics of each SME but also the uncertainty that cannot be explained by such individual factors. It is expected that our study can contribute to effective management of government funds by proposing the prediction models for defaults of funded SMEs.     

A semiparametric Bayesian generalized linear mixed model for the reliability of Kevlar fibers

We analyze the reliability of NASA composite pressure vessels by using a new Bayesian semiparametric model. The data set consists of lifetimes of pressure vessels, wrapped with a Kevlar fiber, grouped by spool, subject to different stress levels; 10% of the data are right censored. The model that we consider is a regression on the log‐scale for the lifetimes, with fixed (stress) and random (spool) effects. The prior of the spool parameters is nonparametric, namely they are a sample from a normalized generalized gamma process, which encompasses the well‐known Dirichlet process. The nonparametric prior is assumed to robustify inferences to misspecification of the parametric prior. Here, this choice of likelihood and prior yields a new Bayesian model in reliability analysis. Via a Bayesian hierarchical approach, it is easy to analyze the reliability of the Kevlar fiber by predicting quantiles of the failure time when a new spool is selected at random from the population of spools. Moreover, for comparative purposes, we review the most interesting frequentist and Bayesian models analyzing this data set. Our credibility intervals of the quantiles of interest for a new random spool are narrower than those derived by previous Bayesian parametric literature, although the predictive goodness‐of‐fit performances are similar. Finally, as an original feature of our model, by means of the discreteness of the random‐effects distribution, we are able to cluster the spools into three different groups. Copyright © 2012 John Wiley & Sons, Ltd.     

A mixture model-based approach to the clustering of exponential repeated data

The analysis of finite mixture models for exponential repeated data is considered. The mixture components correspond to different unknown groups of the statistical units. Dependency and variability of repeated data are taken into account through random effects. For each component, an exponential mixed model is thus defined. When considering parameter estimation in this mixture of exponential mixed models, the EM-algorithm cannot be directly used since the marginal distribution of each mixture component cannot be analytically derived. In this paper, we propose two parameter estimation methods. The first one uses a linearisation specific to the exponential distribution hypothesis within each component. The second approach uses a Metropolis–Hastings algorithm as a building block of a general MCEM-algorithm.     

Predicting bank loan recovery rates with a mixed continuous‐discrete model

To represent the high concentration of recovery rates at the boundaries, we propose to consider the recovery rate as a mixed random variable, obtained as the mixture of a Bernoulli random variable and a beta random variable. We suggest to estimate the mixture weights and the Bernoulli parameter by two logistic regression models. For the recovery rates belonging to the interval (0,1), we model, jointly, the mean and the dispersion by using two link functions, so we propose the joint beta regression model that accommodates skewness and heteroscedastic errors. This methodological proposal is applied to a comprehensive survey on loan recovery process of Italian banks. In the regression model, we include some macroeconomic variables because they are relevant to explain the recovery rate and allow to estimate it in downturn conditions, as Basel II requires. Copyright © 2012 John Wiley & Sons, Ltd.     

Stochastic models for first-order kinetics of biochemical oxygen demand with random initial conditions, inputs, and coefficients

Biochemical oxygen demand (BOD) is a parameter of prime importance in surface water pollution studies and in the design and operation of waste-water treatment plants. A general, stochastic analytical model (denoted S1) is developed for the temporal expectation and (heteroscedastic) variance of first-order BOD kinetics. The model is obtained by integrating the moment equation, which is derived from the mathematical theory of stochastic differential equations. This model takes into account random initial conditions, random inputs, and random coefficients, which appear in the model formulation as initial condition (σ_O²), input (σ_l²), and coefficient (σ_c²) variance parameters, respectively. By constraining these three variance parameters to either vanish or to be nonnegative, model S1 is allowed (under appropriate combinations of the constraints) to split into six stochastic “submodels” (denoted S2 to S7), with each of these submodels being a particular case of the general model. Model S1 also degenerates to the deterministic model (denoted D) when each of the variance parameters vanish. The deterministic parameters (i.e., the rate coefficient and the ultimate BOD) and the stochastic variance parameters of the seven models are estimated on sets of replicated BOD data using the maximum likelihood principle. In this study, two (S5 and S7) of these seven stochastic models are found to be appropriate for BOD. The stochastic input (S5) model (i.e., null initial condition and coefficient variance parameters) shows the best prediction capabilities, while the next best is the stochastic initial condition (S7) model (i.e., null input and coefficient variance parameters).     

TFR、TRV和CE模型序加试验下WEIBULL分布产品的失效分布

本针对TFR模型，首次提出将步加试验推广至序加试验，就两参数Weibull分布给出了损伤因子函数，同时给出了产品寿命的残存函数，另外针对TRV模型，在序加试验下就两参数Weibull分布给出了损伤系数，同时给出了产品寿命的残存函数。     

A comprehensive probabilistic analysis of approximate SIR-type epidemiological models via full randomized discrete-time Markov chain formulation with applications

This paper provides a comprehensive probabilistic analysis of a full randomization of approximate SIR-type epidemiological models based on discrete-time Markov chain formulation. The randomization is performed by assuming that all input data (initial conditions, the contagion, and recovering rates involved in the transition matrix) are random variables instead of deterministic constants. In the first part of the paper, we determine explicit expressions for the so called first probability density function of each subpopulation identified as the corresponding states of the Markov chain (susceptible, infected, and recovered) in terms of the probability density function of each input random variable. Afterwards, we obtain the probability density functions of the times until a given proportion of the population remains susceptible, infected, and recovered, respectively. The theoretical analysis is completed by computing explicit expressions of important randomized epidemiological quantities, namely, the basic reproduction number, the effective reproduction number, and the herd immunity threshold. The study is conducted under very general assumptions and taking extensive advantage of the random variable transformation technique. The second part of the paper is devoted to apply our theoretical findings to describe the dynamics of the pandemic influenza in Egypt using simulated data excerpted from the literature. The simulations are complemented with valuable information, which is seldom displayed in epidemiological models. In spite of the nonlinear mathematical nature of SIR epidemiological model, our results show a strong agreement with the approximation via an appropriate randomized Markov chain. A justification in this regard is discussed.     

Resonance in Beverton–Holt population models with periodic and random coefficients

An increase in the mean population density in a fluctuating environment is known as resonance. Resonance has been observed in laboratory experiments and has been studied in discrete-time population models. We investigate this phenomenon in the Beverton–Holt model with either periodic or random variables for two biologically relevant coefficients: the intrinsic growth rate and the carrying capacity. Three types of resonance are defined: arithmetic, geometric and harmonic. Conditions are derived for each type of resonance in the case of period-2 coefficients and some results for period p>2. For period 2, regions in parameter space where each type of resonance occurs are shown to be subsets of each other. For the case of random coefficients with constant intrinsic growth rate, it is shown that the three types of resonance do not occur. Numerical examples illustrate resonance and attenuance (decrease in the mean population density) in the Beverton–Holt model when the coefficients are discrete random variables.     

Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal

In this paper, we investigate the dynamical behavior of two nonlinear models for viral infection with humoral immune response. The first model contains four compartments; uninfected target cells, actively infected cells, free virus particles and B cells. The intrinsic growth rate of uninfected cells, incidence rate of infection, removal rate of infected cells, production rate of viruses, neutralization rate of viruses, activation rate of B cells and removal rate of B cells are given by more general nonlinear functions. The second model is a modification of the first one by including an eclipse stage of infected cells. We assume that the latent-to-active conversion rate is also given by a more general nonlinear function. For each model we derive two threshold parameters and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the models. By using suitable Lyapunov functions and LaSalle’s invariance principle, we prove the global asymptotic stability of the all equilibria of the models. We perform some numerical simulations for the models with specific forms of the general functions and show that the numerical results are consistent with the theoretical results.     

Measurement Error Models with Nonconstant Covariance Matrices

In this paper we consider measurement error models when the observed random vectors are independent and have mean vector and covariance matrix changing with each observation. The asymptotic behavior of the sample mean vector and the sample covariance matrix are studied for such models. Using the derived results, we study the case of the elliptical multiplicative error-in-variables models, providing formal justification for the asymptotic distribution of consistent slope parameter estimators. The model considered extends a normal model previously considered in the literature. Asymptotic relative efficiencies comparing several estimators are also reported.     

A fragmentation model with neighborhood interaction

Several models of fragmentation have been studied that suppose random fracture forces. In this article, we did a numerical study on a dynamic model for fragmentation in which the fracture forces are generated by neighboring fragments and are proportional to the size of the common boundary between two fragments. The following assumptions were also considered: the material defects are represented by a random distribution of point flaws; the total mass is conserved; and the iterative fracture of each fragment is randomly stopped by a condition that considers a constant probability and a minimal fragment size. The motivation for this model was to determine under what circumstances a continuous fragmentation model with fracture forces defined by the neighbors’ interaction produces results that are in agreement with the experimental evidence. The main result of this work establishes that the fragment size distribution follows a power-law for fragments of greater area than the minimal fragment size m_fs. The visualizations present complex fracture patterns that resemble real systems.     

一类带有一般接触率和常数输入的流行病模型的全局分析

借助极限系统理论和构造适当的Liapunov函数,对带有一般接触率和常数输入的SIR型和SIRS型传染病模型进行讨论．当无染病者输入时,地方病平衡点存在的阈值被找到A·D2对相应的SIR模型,关于无病平衡点和地方病平衡点的全局渐近稳定性均得到充要条件;对相应的SIRS模型,得到无病平衡点和地方病平衡点全局渐近稳定的充分条件．当有染病者输入时,模型不存在无病平衡点．对相应的SIR模型,地方病平衡点是全局渐近稳定的;对相应的SIRS模型,得到地方病平衡点全局渐近稳定的充分条件．     

Statistical aspects of random fragmentations

Three random fragmentation of an interval processes are investigated. For each of them, there is a splitting probability and a probability not to split at each step of the fragmentation process whose overall effect is to stabilize the global number of splitting events. Some of their statistical features are studied in each case among which fragments’ size distribution, partition function, structure of the underlying random fragmentation tree, occurrence of a phase transition. In the first homogeneous model, splitting probability does not depend on fragments’ size at each step. In the next two fragmentation models, splitting probability is fragments’ length dependent. In the first such models, fragments further split with probability one if their sizes exceed some cutoff value only; in a second model considered, splitting probability of finite-size objects is assumed to increase algebraically with fragments’ size at each step. The impact of these dependencies on statistical properties of the resulting random partitions are studied. Several examples are supplied.     

Local Unit Roots and Global Stationarity of TARMA Models

In this paper the (strict and weak) stationarity of threshold autoregressive moving average models is discussed. After examining the strict stationarity, mainly based on the random coefficient autoregressive representation of the model, we provide sufficient conditions for its weak stationarity that allow to obtain a wider stationarity region with respect to some previous results given in the literature. These conditions are discussed to distinguish between global and local stationarity, whose relation has been considered in detail. The threshold process has been further evaluated to face the problem related to the so called existence of a threshold structure in the data generating process that is strictly related to the stationarity and has significant relevance when the parameters of the model have to be estimated.     

Optimizing flow rates in a queueing network with side constraints

In this paper, modified versions of the classical deterministic maximum flow and minimum cost network flow problems are presented in a stochastic queueing environment. In the maximum flow network model, the throughput rate in the network is maximized such that for each arc of the network the resulting probability of finding congestion along that arc in excess of a desirable threshold does not exceed an acceptable value. In the minimum cost network flow model, the minimum cost routing of a flow of given magnitude is determined under the same type of constraints on the arcs. After proper transformations, these models are solved by Ford and Fulkerson's labeling algorithm and out-of-kilter algorithm, respectively.