Propriety of posterior distribution for dichotomous quantal response models

In this article, we investigate the property of posterior distribution for dichotomous quantal response models using a uniform prior distribution on the regression parameters. Sufficient and necessary conditions for the propriety of the posterior distribution with a general link function are established. In addition, the sufficient conditions for the existence of the posterior moments and the posterior moment generating function are also obtained. Finally, the relationship between the propriety of posterior distribution and the existence of the maximum likelihood estimate is examined.

     

Existence of MLE and posteriors for a recognition-memory model

Necessary and sufficient conditions are developed for the existence of the maximum likelihood estimate (MLE) for a recognition-memory model. The propriety of posteriors is shown for a class of bounded priors. Under a constant prior, an easy-to-implement Gibbs sampler is developed and illustrated via a real data set.     

Stationary distribution and extinction of a stochastic SIRI epidemic model with relapse

In this paper, we study the dynamics of a stochastic Susceptible-Infective-Removed-Infective (SIRI) epidemic model with relapse. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of an ergodic stationary distribution to the model. Moreover, sufficient conditions for extinction of the disease are also obtained.     

具有标准发生率的随机SIR传染病模型的建立及平稳分布与疾病灭绝研究

研究了一类具有标准发生率以及考虑随机扰动与系统变量成正比的随机SIR传染病模型.首先,对于任意的正的初值,系统存在唯一的全局正解以及通过构造合适的随机李雅普诺夫函数,得到了模型遍历平稳分布存在的充分条件.其次,给出了疾病灭绝的充分条件,并与模型遍历平稳分布存在的充分条件作对比,得出了在特定条件下随机SIR模型的阈值.最后通过数值模拟验证了结果的正确性.     

Dynamical behavior of a stochastic predator-prey model with stage structure for prey

Abstract

In the present paper, we focus on a stochastic predator-prey model with stage structure for prey. Firstly, by using the stochastic Lyapunov function method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the predator population in two cases. Some examples and numerical simulations are carried out to validate our analytical findings.     

Stationary distribution of a stochastic food chain chemostat model with general response functions

In this paper, we focus on a food chain chemostat model with general response functions, perturbed by white noise. Under appropriate assumptions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution by using stochastic Lyapunov analysis method. Our main effort is to construct the suitable Lyapunov function.     

Stationary distribution and extinction of a stochastic predator–prey model with distributed delay

In this paper, we focus on a stochastic predator–prey model with distributed delay. We first obtain the existence of a stationary distribution to the positive solutions by stochastic Lyapunov function method. Then we establish sufficient conditions for extinction of the predator population, that is, the prey population is survival and the predator population is extinct.     

On posterior consistency in nonparametric regression problems

We provide sufficient conditions to establish posterior consistency in nonparametric regression problems with Gaussian errors when suitable prior distributions are used for the unknown regression function and the noise variance. When the prior under consideration satisfies certain properties, the crucial condition for posterior consistency is to construct tests that separate from the outside of the suitable neighborhoods of the parameter. Under appropriate conditions on the regression function, we show there exist tests, of which the type I error and the type II error probabilities are exponentially small for distinguishing the true parameter from the complements of the suitable neighborhoods of the parameter. These sufficient conditions enable us to establish almost sure consistency based on the appropriate metrics with multi-dimensional covariate values fixed in advance or sampled from a probability distribution. We consider several examples of nonparametric regression problems.     

Stationary distribution and extinction of a stochastic one-prey two-predator model with Holling type II functional response

In this article, we investigate a stochastic one-prey two-predator model with Holling type II functional response. We first establish sufficient conditions for persistence and extinction of prey and predator populations, then by constructing a suitable stochastic Lyapunov function, we establish sharp sufficient criteria for the existence of a unique ergodic stationary distribution of the positive solutions to the model. The results show that the smaller white noise can ensure the persistence of prey and predator populations while the larger white noise can lead to the extinction of prey and predator populations.     

Dynamics of a Stochastic Predator–Prey Model with Stage Structure for Predator and Holling Type II Functional Response

In this paper, we develop and study a stochastic predator–prey model with stage structure for predator and Holling type II functional response. First of all, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then, we obtain sufficient conditions for extinction of the predator populations in two cases, that is, the first case is that the prey population survival and the predator populations extinction; the second case is that all the prey and predator populations extinction. The existence of a stationary distribution implies stochastic weak stability. Numerical simulations are carried out to demonstrate the analytical results.     

Posterior propriety in nonparametric mixed effects model

It is well known that spline smoothing estimator relates to the Bayesian estimate under partially informative normal prior.In this paper,we derive the conditions for the propriety of the posterior in the nonparametric mixed efects model under this class of partially informative normal prior for fxed efect with inverse gamma priors on the variance components and hierarchical priors for covariance matrix of random efect,then we explore the Gibbs sampling procedure.     

STUDY ON THE THREE-SPECIES ALMOST PERIODIC MODEL WITH THE TYPE II FUNCTIONAL RESPONSE

By using Lyapunov function and differential inequality, sufficient conditions for the existence of global asymptotical stable almost periodic solution of the three-species almost periodic modei with the type II functional response are obtained.     

On the posterior median estimators of possibly sparse sequences

We adopt the Bayesian paradigm and discuss certain properties of posterior median estimators of possibly sparse sequences. The prior distribution considered is a mixture of an atom of probability at zero and a symmetric unimodal distribution, and the noise distribution is taken as another symmetric unimodal distribution. We derive an explicit form of the corresponding posterior median and show that it is an antisymmetric function and, under some conditions, a shrinkage and a thresholding rule. Furthermore we show that, as long as the tails of the nonzero part of the prior distribution are heavier than the tails of the noise distribution, the posterior median, under some constraints on the involved parameters, has the bounded shrinkage property, extending thus recent results to larger families of prior and noise distributions. Expressions of posterior distributions and posterior medians in particular cases of interest are obtained. The asymptotes of the derived posterior medians, which provide valuable information of how the corresponding estimators treat large coefficients, are also given. These results could be particularly useful for studying frequentist optimality properties and developing statistical techniques of the resulting posterior median estimators of possibly sparse sequences for a wider set of prior and noise distributions.     

Dynamical behavior of a stochastic HBV infection model with logistic hepatocyte growth

This paper is concerned with a stochastic HBV infection model with logistic growth. First, by constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence of ergodic stationary distribution of the solution to the HBV infection model. Then we obtain sufficient conditions for extinction of the disease. The stationary distribution shows that the disease can become persistent in vivo.     

The threshold for a stochastic HIV-1 infection model with Beddington-DeAngelis incidence rate

In this paper, we consider a stochastic HIV-1 infection model with Beddington-DeAngelis incidence rate. Before exploring its long-time behavior we show that there is a global positive solution of this model. Then sufficient conditions for extinction of the disease are established. Moreover, we give sufficient conditions for the existence of a stationary distribution of the model through constructing a suitable stochastic Lyapunov function. The stationary distribution implies that the disease is persistent in the mean. Therefore, a threshold value for the disease to disappear or prevail is obtained. Finally, some numerical examples are illustrated to support our theoretical results.     

Synchronized stationary distribution of stochastic coupled systems based on graph theory

This paper deals with the synchronized stationary distribution of stochastic coupled systems. The response system is constructed to help achieve a synchronized stationary distribution. Firstly, an error system obtained by the drive system and the response system is given and an appropriate Lyapunov function for the error system is constructed. On the basis of the graph theory and the Lyapunov method, some sufficient conditions are proposed to guarantee the existence of a stationary distribution for the error system, which reflects the coupling structure has a close relationship with synchronized stationary distribution. Then, an application to stochastic coupled oscillators is presented and sufficient conditions are obtained to illustrate the feasibility of the theoretical results. Finally, a numerical example is provided to demonstrate the effectiveness of theoretical results.     

Dynamics of a stochastic chemostat competition model with plasmid-bearing and plasmid-free organisms

In this paper, we consider a chemostat model of competition between plasmid-bearing and plasmid-free organisms, perturbed by white noise. Firstly, we prove the existence and uniqueness of the global positive solution. Then by constructing suitable Lyapunov functions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution. Furthermore, conditions for extinction of plasmid-bearing organisms are obtained. Theoretical analysis indicates that large noise intensity $\sigma_{2}^{2}$ is detrimental to the survival of plasmid-bearing organisms and is not conducive to the commercial production of genetically altered organisms. Finally, numerical simulations are presented to illustrate the results.     

The Dynamics of Stochastic Predator-prey Models with Non-constant Mortality Rate and General Nonlinear Functional Response

In this paper, we investigate the dynamics of stochastic predator- prey models with non-constant mortality rate and general nonlinear functional response. For the stochastic system, we firstly prove the existence of the global unique positive solution. Secondly, we establish sufficient conditions for the extinction and persistence in the mean of autonomous stochastic model and obtain a critical value between them. Then by constructing a appropriate Lyapunov function, we prove that there exists a unique stationary distribution and it has ergodicity in the case of persistence. Finally, numerical simulations are introduced to illustrate our theoretical results.     

On periodic phenomena and stationary distribution of queueing system M/G/1 with group arrivals and batch service

In this paper several models of queueing system M/G/1 with group arrivals and batch service are considered, and the following fundamental questions are considered: (1) what is the structure of the phase space of the imbedded Markov chain? (2) what are the necessary and sufficient conditions causing the imbedded Markov chain to be reducible or irreducible, and periodic or aperiodic? (3) what are the necessary and sufficient conditions of the existence of stationary distribution? The generating function of stationary distribution is obtained.     

Markov调制的随机系统平稳分布的存在与唯一性(英文)

本文讨论Markov调制的随机系统平稳分布的存在与唯一性. 利用Lyapunov方法与耦合方法得到这类系统存在唯一平稳分布的一些充分条件, 这些条件易于验证具可操作性.