Control charts for monitoring correlated counts with a finite range

Correlated count data processes with a finite range can be adequately described by a first‐order binomial autoregressive model. However, in several practical applications, these data demonstrate extra‐binomial variation, and a more appropriate choice is the first‐order beta‐binomial autoregressive model. In this paper, we propose and study control charts that can be used for the monitoring of these 2 processes. Practical guidelines concerning their statistical design are provided, whereas the effect of the extra‐binomial variation is investigated as well. Finally, the practical application of the proposed schemes is illustrated via a real‐data example.     

The maximum surplus in a finite‐time interval for a discrete‐time risk model with exchangeable,dependent claim occurrences

This paper investigates a discrete‐time risk model that involves exchangeable dependent loss generating claim occurrences and compound binomially distributed aggregate loss amounts. First, a general framework is presented to derive the distribution of a surplus sequence using the model. This framework is then applied to obtain the distribution of any function of a surplus sequence in a finite‐time interval. Specifically, the distribution of the maximum surplus is obtained under nonruin conditions. Based on this distribution, the computation of the minimum surplus distribution is given. Asset and risk management–oriented implications are discussed for the obtained distributions based on numerical evaluations. In addition, comparisons are made involving the corresponding results of the classical discrete‐time compound binomial risk model, for which claim occurrences are independent and identically distributed.     

Computation methods for combinatorial sums and Euler‐type numbers related to new families of numbers

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well‐known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright © 2016 John Wiley & Sons, Ltd.     

Generalized discrete autoregressive moving‐average models

This article proposes the generalized discrete autoregressive moving‐average (GDARMA) model as a parsimonious and universally applicable approach for stationary univariate or multivariate time series. The GDARMA model can be applied to any type of quantitative time series. It allows to compute moment properties in a unique way, and it exhibits the autocorrelation structure of the traditional ARMA model. This great flexibility is obtained by using data‐specific variation operators, which is illustrated for the most common types of time series data, such as counts, integers, reals, and compositional data. The practical potential of the GDARMA approach is demonstrated by considering a time series of integers regarding votes for a change of the interest rate, and a time series of compositional data regarding television market shares.     

A bivariate INAR(1) time series model with geometric marginals

In this paper we introduce a simple bivariate integer-valued time series model with positively correlated geometric marginals based on the negative binomial thinning mechanism. Some properties of the model are considered. The unknown parameters of the model are estimated using the modified conditional least squares method.     

具有常红利边界的复合马尔可夫二项模型

主要讨论复合马尔可夫二项模型.在模型中引进一个常数红利边界策略,得到了Gerber-Shiu罚金函数所满足的线性方程组,且证明该方程组存在唯一解.最后,作为罚金函数的一些应用实例给出了一些具体风险量的计算公式.     

Information initiatives of mobile retailers: a regression analysis of zero‐truncated count data with underdispersion

Our paper presents an empirical analysis of the association between firm attributes in electronic retailing and the adoption of information initiatives in mobile retailing. In our attempt to analyze the collected data, we find that the count of information initiatives exhibits underdispersion. Also, zero‐truncation arises from our study design. To tackle the two issues, we test four zero‐truncated (ZT) count data models—binomial, Poisson, Conway–Maxwell–Poisson, and Consul's generalized Poisson. We observe that the ZT Poisson model has a much inferior fit when compared with the other three models. Interestingly, even though the ZT binomial distribution is the only model that explicitly takes into account the finite range of our count variable, it is still outperformed by the other two Poisson mixtures that turn out to be good approximations. Further, despite the rising popularity of the Conway–Maxwell–Poisson distribution in recent literature, the ZT Consul's generalized Poisson distribution shows the best fit among all candidate models and suggests support for one hypothesis. Because underdispersion is rarely addressed in IT and electronic commerce research, our study aims to encourage empirical researchers to adopt a flexible regression model in order to make a robust assessment on the impact of explanatory variables. Copyright © 2014 John Wiley & Sons, Ltd.     

The Poisson random effect model for experience ratemaking: Limitations and alternative solutions

Poisson random effect models with a shared random effect have been widely used in actuarial science for analyzing the number of claims. In particular, the random effect is a key factor in a posteriori risk classification. However, the necessity of the random effect may not be properly assessed due to the dual role of the random effect; it affects both the marginal distribution of the number of claims and the dependence among the numbers of claims obtained from an individual over time. We first show that the score test for the nullity of the variance of the shared random effect can falsely indicate significant dependence among the numbers of claims even though they are independent. To mitigate this problem, we propose to separate the dual role of the random effect by introducing additional random effects to capture the overdispersion part, which are called saturated random effects. In order to circumvent heavy computational issues by the saturated random effects, we choose a gamma distribution for the saturated random effects because it gives the closed form of marginal distribution. In fact, this choice leads to the negative binomial random effect model that has been widely used for the analysis of frequency data. We show that safer conclusions about the a posteriori risk classification can be made based on the negative binomial mixed model under various situations. We also derive the score test as a sufficient condition for the existence of the a posteriori risk classification based on the proposed model.     

投资收益下的两类双二项风险模型的破产概率及比较

本文讨论了含投资因素的双二项风险模型,得到了破产概率表达式,并对几类相关的双二项风险模型的调节系数及破产概率上界进行了比较.     

Random dynamics and finance: constructing implied binomial trees from a predetermined stationary density

We introduce a general binomial model for asset prices based on the concept of random maps. The asymptotic stationary distribution for such model is studied using techniques from dynamical systems. In particular, we present a technique to construct a general binomial model with a predetermined stationary distribution. This technique is independent of the chosen distribution making our model potentially useful in financial applications. We briefly explore the suitability of our construction as an implied binomial tree. Copyright © 2006 John Wiley & Sons, Ltd.     

Pell identities via a quadratic field

In this article, we derive a number of identities involving both Pell numbers and binomial coefficients. We also consider briefly the potential for obtaining Pell identities involving trinomial coefficients and beyond. A key point is the simplicity of the derivations, and indeed this work can lead on to a number of interesting explorations for first-year undergraduates.     

Estimating retail demand with Poisson mixtures and out‐of‐sample likelihood

Estimation of retail demand is critical to decisions about procuring, shipping, and shelving. The idea of Poisson demand process is central to retail inventory management and numerous studies suggest that negative binomial (NB) distribution characterize retail demand well. In this study, we reassess the adequacy of estimating retail demand with the NB distribution. We propose two Poisson mixtures—the Poisson–Tweedie family (PTF) and the Conway–Maxwell–Poisson distribution—as generic alternatives to the NB distribution. On the basis of the principle of likelihood and information theory, we adopt out‐of‐sample likelihood as a metric for model selection. We test the procedure on consumer demand for 580 stock‐keeping unit store sales datasets. Overall the PTF and the Conway–Maxwell–Poisson distribution outperform the NB distribution for 70% of the tested samples. As a general case of the NB model, the PTF has particularly strong performance for datasets with relatively small means and high dispersion. Our finding carries useful implications for researchers and practitioners who seek for flexible alternatives to the oft‐used NB distribution in characterizing retail demand. Copyright © 2013 John Wiley & Sons, Ltd.     

光滑树图期权定价模型的叉熵分析法

为了克服CRR模型收敛的波动性,以及强调历史信息的预测作用的情况,提出了一个新奇的光滑收敛的树图模型.新模型基于历史信息,运用最小叉熵原理
来推导树图的关键参数p,u,d, 然后使用倒推法推断期权的价格.显然,新模型所得的期权的价格隐含着历史信息.由于最小叉熵原理是一个凸规划问题,能求得唯一的最优解,所以,新模型也适用于不完全金融市场期权定价.最后,数值算例表明,相比于CRR模型,新模型收敛光滑平稳且有更高的计算精度;对上涨(下跌)的二元期权、欧式期权,新模型都能光滑收敛于B-S公式.     

Risk models based on time series for count random variables

In this paper, we generalize the classical discrete time risk model by introducing a dependence relationship in time between the claim frequencies. The models used are the Poisson autoregressive model and the Poisson moving average model. In particular, the aggregate claim amount and related quantities such as the stop-loss premium, value at risk and tail value at risk are discussed within this framework.     

Bayesian modeling and forecasting of Value‐at‐Risk via threshold realized volatility

This study proposes a threshold realized generalized autoregressive conditional heteroscedastic (GARCH) model that jointly models daily returns and realized volatility, thereby taking into account the bias and asymmetry of realized volatility. We incorporate this threshold realized GARCH model with skew Student‐t innovations as the observation equation, view this model as a sharp transition model, and treat the realized volatility as a proxy for volatility under this nonlinear structure. Through the Bayesian Markov chain Monte Carlo method, the model can jointly estimate the parameters in the return equation, the volatility equation, and the measurement equation. As an illustration, we conduct a simulation study and apply the proposed method to the US and Japan stock markets. Based on quantile forecasting and volatility estimation, we find that the threshold heteroskedastic framework with realized volatility successfully models the asymmetric dynamic structure. We also investigate the predictive ability of volatility by comparing the proposed model with the traditional GARCH model as well as some popular asymmetric GARCH and realized GARCH models. This threshold realized GARCH model with skew Student‐t innovations outperforms the competing risk models in out‐of‐sample volatility and Value‐at‐Risk forecasting.     

Generalized binomial and negative binomial distributions of order<Emphasis Type="Italic">k</Emphasis> by the<Emphasis Type="Italic">l</Emphasis>-overlapping enumeration scheme

In this paper, we investigate the exact distribution of the waiting time for ther-th ℓ-overlapping occurrence of success-runs of a specified length in a sequence of two state Markov dependent trials. The probability generating functions are derived explicitly, and as asymptotic results, relationships of a negative binomial distribution of orderk and an extended Poisson distribution of orderk are discussed. We provide further insights into the run-related problems from the viewpoint of the ℓ-overlapping enumeration scheme. We also study the exact distribution of the number of ℓ-overlapping occurrences of success-runs in a fixed number of trials and derive the probability generating functions. The present work extends several properties of distributions of orderk and leads us a new type of geneses of the discrete distributions.     

供应链零售商预测技术研究—基于牛鞭效应的视角

一阶自回归(AR(1))序列模拟需求过程是传统文献采用的经典模型,然而上述文献关于需求过程参数(如需求自回归系数)对牛鞭效应的影响分析缺乏实践意义,为了更符合企业的实际决策过程,本文建立了需求依赖于价格、而以AR(1)序列模拟价格过程的需求函数模型,分析了最小均方差、移动平均和指数平滑预测下的牛鞭效应,确定了零售商的预测技术选择条件。研究表明:(1)产品市场规模不影响零售商预测技术的选择;(2)当产品价格敏感系数较小或价格自回归系数较小时,零售商应选择最小均方差预测技术;(3)当产品价格敏感系数和价格自回归系数均较大时,零售商应选择移动平均预测技术。     

Two‐level discretization of the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity

The r‐Laplacian has played an important role in the development of computationally efficient models for applications, such as numerical simulation of turbulent flows. In this article, we examine two‐level finite element approximation schemes applied to the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity, where r is the order of the power‐law artificial viscosity term. In the two‐level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single‐step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two‐level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide rigorous numerical analysis of the two‐level approximation scheme and derive scalings which vary based on the coefficient r, coarse mesh size H, fine mesh size h, and filter radius δ. We also investigate the two‐level algorithm in several computational settings, including the 3D numerical simulation of flow past a backward‐facing step at Reynolds number Re = 5100. In all numerical tests, the two‐level algorithm was proven to achieve the same order of accuracy as the standard one‐level algorithm, at a fraction of the computational cost. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Global energy‐tracking identities and global energy‐tracking splitting FDTD schemes for the Drude Models of Maxwell's equations in three‐dimensional metamaterials

In this article, we study the Drude models of Maxwell's equations in three‐dimensional metamaterials. We derive new global energy‐tracking identities for the three dimensional electromagnetic problems in the Drude metamaterials, which describe the invariance of global electromagnetic energy in variation forms. We propose the time second‐order global energy‐tracking splitting FDTD schemes for the Drude model in three dimensions. The significant feature is that the developed schemes are global energy‐preserving, unconditionally stable, second‐order accurate both in time and space, and computationally efficient. We rigorously prove that the new schemes satisfy these energy‐tracking identities in the discrete form and the discrete variation form and are unconditionally stable. We prove that the schemes in metamaterials are second order both in time and space. The superconvergence of the schemes in the discrete H¹ norm is further obtained to be second order both in time and space. Their approximations of divergence‐free are also analyzed to have second‐order accuracy both in time and space. Numerical experiments confirm our theoretical analysis results. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 763–785, 2017     

Second‐order performance analysis of discrete‐time queues fed by DAR(2) sources with a focus on the marginal effect of the additional traffic parameter

Discrete autoregressive process of order 1 (DAR(1)) has been used as a popular stochastic model for correlated traffic sources because it parsimoniously uses a single parameter to capture the desirable correlation structure. In contrast with DAR(1), discrete autoregressive process of order 2 (DAR(2)) uses one more parameter to provide a much richer pattern in the autocorrelation function and is able to capture slower decay rate and longer memory. To investigate how the additional traffic parameter in DAR(2) influences the queueing performance, this paper provides an analysis of the discrete‐time DAR(2)/D/1 queue. The performance measures concerned are the mean and second‐order statistics of queue size, which are both important in the queueing systems seen in telecommunication networks. Under a mild condition, these performance indices are derived in closed form that allows for efficient computing. An approximate version of these results is also developed to relax the condition and cover more general sources, and both versions serve as a simple tool set for performance evaluation. The numerical examples use this tool to demonstrate that the DAR(2) source may cause up to 30% poorer performance than DAR(1) when the traffic is heavy, bursty, and highly correlated. This indicates that the effect from slower decay rate in autocorrelation is not negligible and using the extra parameter is necessary particularly when the queue is heavily loaded with correlated traffic. Copyright © 2012 John Wiley & Sons, Ltd.