A Two-Parameter Distribution Obtained by Compounding the Generalized Exponential and Exponential Distributions

We introduce a new two-parameter lifetime distribution obtained by compounding the generalized exponential and exponential distributions. We assume that the shape parameter of the generalized exponential distribution is a random variable having the exponential distribution. The shapes of the density and hazard rate functions are derived. The model parameters are estimated by maximum likelihood, and an application of the proposed distribution is presented.

     

Estimation for a class of generalized state-space time series models

State-space models with exponential and conjugate exponential family densities are introduced. Examples include Poisson–Gamma, Binomial–Beta, Gamma–Gamma and Normal–Normal processes. Maximum likelihood and quasilikelihood estimators and their properties are discussed. Results from a simulation study for the Poisson–Gamma model are reported.     

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.     

The beta generalized Gompertz distribution

A new five-parameter continuous model called the beta generalized Gompertz distribution is introduced and studied. This distribution contains the Gompertz, generalized Gompertz, beta Gompertz, generalized exponential, beta generalized exponential, exponential and beta exponential distributions as special sub-models. Some mathematical properties of the new model are derived. We show that the density function of the new distribution can be expressed as a linear combination of Gompertz densities. We obtain explicit expressions for the moments, moment generating function, quantile function, density function of the order statistics and their moments, mean deviations, Bonferroni and Lorenz curves and Rényi entropy. The model parameters are estimated by using the maximum likelihood method of estimation and the observed information matrix is determined. Finally, an application to real data set is given to illustrate the usefulness of the proposed model.     

Random fuzzy shock models and bivariate random fuzzy exponential distribution

In this paper, a random fuzzy shock model and a random fuzzy fatal shock model are proposed. Then bivariate random fuzzy exponential distribution is derived from the random fuzzy fatal shock model. Furthermore, some properties of the bivariate random fuzzy exponential distribution are proposed. Finally, an example is given to show the application of the bivariate random fuzzy exponential distribution.     

Multiway Dependence in Exponential Family Conditional Distributions

Conditionally specified statistical models are frequently constructed from one-parameter exponential family conditional distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field (MRF). A common assumption on the Gibbsian form of the MRF model is that dependence is expressed only through pairs of random variables, which we refer to as the “pairwise-only dependence” assumption. Based on this assumption, J. Besag (1974, J. Roy. Statist. Soc. Ser. B36, 192–225) formulated exponential family “auto-models” and showed the form that one-parameter exponential family conditional densities must take in such models. We extend these results by relaxing the pairwise-only dependence assumption, and we give a necessary form that one-parameter exponential family conditional densities must take under more general conditions of multiway dependence. Data on the spatial distribution of the European corn borer larvae are fitted using a model with Bernoulli conditional distributions and several dependence structures, including pairwise-only, three-way, and four-way dependencies.     

A general model for consecutive-k-out-of-n: F repairable system with exponential distribution and (k−1)-step Markov dependence

In this paper, a general model for consecutive-k-out-of-n: F repairable system with exponential distribution and (k−1)-step Markov dependence is introduced. The lifetime of a component is an exponential random variable, its parameter depends on the number of consecutive failed components that precede the component. The repair time is also an exponential random variable. A priority repair rule on the basis of the system failure risk is adopted. Then the transition density matrix of the system is determined. Some reliability indices, including the system availability, rate of occurrence of failures and reliability are evaluated accordingly. For the demonstration of the model and methodology, a linear system example and a circular system example are investigated.     

单调序列的非齐次GM(1,1)模型

GM(1,1)模型的白化解为齐次指数形式,而一般数据呈非齐次指数形式,存在形式上的差异.本文运用非齐次级比与非齐次指数函数的对应关系,对原始序列中相邻数据做差处理,得到新的序列,将非齐次指数序列转换为齐次指数序列,再建立GM(1,1)模型.实例表明,运用初值优化和非齐次化能提高GM(1,1)模型的精度.     

Dynamics of a hysteretic neuron model

A mathematical model of a single isolated artificial neuron with hysterisis is formulated by means of a neutral delay differential equation. The asymptotic and exponential stability of such a model are investigated. Sufficient conditions for the exponential stability of a linear integral difference inequality are obtained. In the absence of hysterisis effect, our model reduces to a known model of a single neuron. Usually asymptotic stability of neutral delay differential equations is studied by means of degenerate Lyapunov–Kravsovskii functionals. In this article, perhaps for the first time exponential stability of a class of neutral differential equations are studied by means of the exponential stability of an affiliated difference inequality. While generalization to Hopfield type hysteretic neural networks is possible, such a generalization is not considered in this article.     

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.     

Stability analysis of a stochastic Gilpin–Ayala model driven by Lévy noise

A stochastic one-dimensional Gilpin–Ayala model driven by Lévy noise is presented in this paper. Firstly, we show that this model has a unique global positive solution under certain conditions. Then sufficient conditions for the almost sure exponential stability and moment exponential stability of the trivial solution are established. Results show that the jump noise can make the trivial solution stable under some conditions. Numerical example is introduced to illustrate the results.     

The admissible parameter space for exponential smoothing models

We discuss the admissible parameter space for some state space models, including the models that underly exponential smoothing methods. We find that the usual parameter restrictions (requiring all smoothing parameters to lie between 0 and 1) do not always lead to stable models. We also find that all seasonal exponential smoothing methods are unstable as the underlying state space models are neither reachable nor observable. This instability does not affect the forecasts, but does corrupt the state estimates. The problem can be overcome with a simple normalizing procedure. Finally we show that the admissible parameter space of a seasonal exponential smoothing model is much larger than that for a basic structural model, leading to better forecasts from the exponential smoothing model when there is a rapidly changing seasonal pattern.     

Cost analysis of the <Emphasis Type="Italic">R</Emphasis>-unloader queueing system

This paper deals with an unloader queueing model in which N identical trailers are unloaded by one or more unloaders. Theoretical solutions are obtained with the assumption of negative exponential distributions for trip times, unloading times, and finishing times. A cost model is developed to determine the optimal number of trailers. In order to evaluate the model robustness, simulation models are developed for three different distributions, exponential, second-order Erlang, and triangular. We demonstrate, through simulation results, that the unloader queueing model is sufficiently robust to the variations of probability distributions.     

Exponential statistical manifold

We consider the non-parametric statistical model ε(p) of all positive densities q that are connected to a given positive density p by an open exponential arc, i.e. a one-parameter exponential model p(t), t ∈ I, where I is an open interval. On this model there exists a manifold structure modeled on Orlicz spaces, originally introduced in 1995 by Pistone and Sempi. Analytic properties of such a manifold are discussed. Especially, we discuss the regularity of mixture models under this geometry, as such models are related with the notion of e- and m-connections as discussed by Amari and Nagaoka.     

Mixed-Type Duality on Nonsmooth Minimax Fractional Programming Involving Exponential (p,r)-Invexity

Considering a nonsmooth minimax fractional programming problem involving exponential (p, r)-invexity, we construct a mixed-type dual problem, which is performed by an incomplete Lagrangian dual model. This mixed-type dual model involves the Wolfe type dual and Mond-Weir type dual as the special cases under exponential (p, r)-invexity. We establish the mixed-type duality problem with conditions for exponential (p, r)-invexity and prove that the optimal values of the primary problem and the mixed-type duality problem have no duality gap under the framwork of exponential (p, r)-invexity.     

A load share model for non-identical components of a k-out-of-m system

We consider a k-out-of-m load sharing system when the lifetimes of the components are not necessarily identically distributed random variables. For such systems, a model for the load sharing phenomenon through the exponentiated conditional survival functions of ordered failure times is proposed. This model is more general than the load sharing model with identically distributed component lifetimes and leads to a different family of distributions for ordered random variables. A general expression for the reliability of the system is given. The computations of the reliability for a two component parallel load sharing system corresponding to the exponential and Weibull distributions are discussed. For illustrative purpose, we discuss the inference procedures for a two component parallel load sharing system corresponding to the exponential distributions. A simulation study is carried out to assess the proposed estimation and testing procedures. The applicability of the proposed load sharing model is shown through two data sets.     

Consequences of Schanuel's condition for zeros of exponential terms

Assuming “Schanuel's Condition” for a certain class of exponential fields, Sturm's technique for polynomials in real closed fields can be extended to more complicated exponential terms in the corresponding exponential field. Hence for this class of terms the exact number of zeros can be calculated. These results give deeper insights into the model theory of exponential fields. MSC: 03C65, 03C60, 12L12.     

Efficient stochastic model for the point kinetics equations

A new stochastic model for the point kinetics equations with I-delayed neutron precursor groups is presented. In this stochastic model, the point kinetics equations are separated into three terms: prompt neutrons, delayed neutrons and external neutrons source. The matrix form of the efficient stochastic model is solved by a semi-analytical method. The semi-analytical method is based on the exponential function of the coefficient matrix. The eigenvalues of the coefficient matrix and Gaussian elimination are used to calculate this exponential function. The mean and standard deviation of neutron and precursor populations of the efficient stochastic model with step, ramp, and sinusoidal reactivities are computed. The results of the efficient stochastic model are compared with the results of Allen's stochastic model for the point kinetics equations. This comparison confirms that the efficient stochastic model is an accurate model compared with the deterministic point kinetics equations. This stochastic model is efficient to study the natural behavior of neutron and precursor populations in the nuclear reactor dynamics.     

Dimension estimate of the exponential attractor for the chemotaxis‐growth system

We consider a chemotaxis‐growth model which takes into account diffusion, chemotaxis, production of chemical substance, and growth. We present estimates from above and below of the fractal dimension dim?? of the exponential attractor ?? in terms of the coefficients of the system. Comparisons are made between the sizes of the global and exponential attractors. Numerical simulations are presented which confirm the analytical results obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

Razumikhin method to global exponential stability for coupled neutral stochastic delayed systems on networks

Global exponential stability for coupled neutral stochastic delayed systems on networks (CNSDSNs) is investigated in this paper. By means of combining the Razumikhin method with graph theory, some sufficient conditions that can be verified easily are derived to ensure the global exponential stability for CNSDSNs. Finally, a specific model of CNSDSNs is discussed, and numerical test manifests the effectiveness of the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.