Multi-state Systems with Graduate Failure and Equal Transition Intensities

We consider unrecoverable homogeneous multi-state systems with graduate failures, where each component can work at M + 1 linearly ordered levels of performance. The underlying process of failure for each component is a homogeneous Markov process such that the level of performance of one component can change only for one level lower than the observed one, and the failures are independent for different components. We derive the probability distribution of the random vector X, representing the state of the system at the moment of failure and use it for testing the hypothesis of equal transition intensities. Under the assumption that these intensities are equal, we derive the method of moments estimators for probabilities of failure in a given state vector and the intensity of failure. At the end we calculate the reliability function for such systems. Received: May 18, 2007., Revised: July 8, 2008., Accepted: September 29, 2008.     

Quick response policy with Bayesian information updates

In this paper we investigate the quick response (QR) policy with different Bayesian models. Under QR policy, a retailer can collect market information from the sales of a pre-seasonal product whose demand is closely related to a seasonal product’s demand. This information is then used to update the distribution for the seasonal product’s demand by a Bayesian approach. We study two information update models: one with the revision of an unknown mean, and the other with the revision of both an unknown mean and an unknown variance. The impacts of the information updates under both models are compared and discussed. We also identify the features of the pre-seasonal product which can bring more significant profit improvement. We conclude that an effective QR policy depends on a precise information update model as well as a selection of an appropriate pre-seasonal product as the observation target.     

Pricing electricity derivatives within a Markov regime-switching model: a risk premium approach

In this paper we derive analytic formulas for electricity derivatives under assumption that electricity spot prices follow a 3-regime Markov regime-switching model with independent spikes and drops and periodic transition matrix. Since the classical derivatives pricing methodology cannot be used in the case of non-storable commodities, we employ the concept of the risk premium. The obtained theoretical results are then used for the European Energy Exchange data analysis. We calculate the risk premium in the case of the calibrated 3-regime MRS model. We find a time varying structure of the risk premium and an evidence for a negative risk premium (or positive forward premium), especially at short times before delivery. Finally, we use the obtained risk premium to calculate prices of European options written on spot, as well as, forward prices.     

Stochastic Breakeven Analysis

This paper presents a practical approach for studying the profit probability distribution defined in stochastic breakeven analysis. After presenting a procedure for calculating the mean and the second to fourth central moments of the profit distribution, the several important uses of these four moments are discussed. A major application of the four moments is in the fitting of a Pearson's curve. Using a numerical example in conjunction with a set of published tables, we then demonstrate the simplicity of our proposed approach as well as the resultant high accuracy in estimating probabilities of various profit levels.     

Risk-Based Integration of Strategic and Tactical Capacity Planning

We consider capacity management with a long-term strategic choice, such as the number of production lines to install before demand is known, and short-term tactical decisions relating to production, inventory, and subcontracting (recourse actions made after demand is known). We present an integrated scenario-based mathematical modeling and solution framework. For a single-product environment, we examine properties of total profit as a function of demand and the long-term capacity z. We investigate two measures of risk (profit variance and mean downside risk) and their corresponding profit-risk frontiers. Computational experiments are used to illustrate parameter sensitivity results obtained from the model.     

相对损失函数下的倍度保费

传统的倍度保费公式利用均方损失函数估计特定保人的风险. 然而, 索取保费与真实保费之间的比例比它们差的绝对值更适合于衡量保费的公平性. 基于这一点, 我们提出了两种计算保费的损失函数: 均方相对损失函数和熵相对损失函数, 并且给出了倍度因子的估计公式及它们的性质.     

多利率下的寿险费率测算模型

为了能够在多利率条件下测算人寿保险的费率,本文建立了一个线性规划模型.根据该模型,能够合理安排保费资金的投资期限以达到最大的保险利益,从而为费率和红利的测算提供了依据.列出了两个典型寿险产品的计算数据,结果表明,寿险费率的测算主要取决于长期利率.对于储蓄型寿险,资金的运用应该以长期投资为主,分红水平可以由长期利率与预定利率之差来确定.     

The Moments of the M/M/s Queue Length Process

A representation for the moments of the number of customers in a M/M/s queueing system is deduced from the Karlin and McGregor representation for the transition probabilities. This representation allows us to study the limit behavior of the moments as time tends to infinity. We study some consequences of the representation for the mean.     

两阶段免费增值商业模式中的定价优化与时序研究

免费增值商业模式在信息产品和服务当中被广泛采用。针对企业首先推出免费产品再推出付费产品以充分利用两种产品推出的时间差来提升消费者学习效应、降低消费者使用成本的情形,本文首先建立两阶段模型,求解给定系统参数情况下的付费版产品最优定价问题并给出解析解,然后通过与企业应用单阶段免费增值模式时的利润进行比较,解析地得到企业选择两阶段模式可获得较高利润的条件,最后在数值计算基础上讨论了学习效应强度对企业利润的影响,和优化学习效应强度以拓展两阶段模式适用范围的问题。本文的研究成果为拟采用免费增值商业模式的企业提供了关于产品最优定价和模式选择的决策支持。     

Infinitely dimensional control Markov branching chains in random environments

First of all we introduce the concepts of infinitely dimensional control Markov branching chains in random environments (β-MBCRE) and prove the existence of such chains, then we introduce the concepts of conditional generating functionals and random Markov transition functions of such chains and investigate their branching property. Base on these concepts we calculate the moments of the β-MBCRE and obtain the main results of this paper such as extinction probabilities, polarization and proliferation rate. Finally we discuss the classification of β-MBCRE according to the different standards.     

An investigation of phase-distribution moment-matching algorithms for use in queueing models

Algorithms for matching moments to phase-type distributions are evaluated on the basis of their performance in their intended application, queueing models. The moment-matching algorithms under consideration match two moments to a hyperexponential distribution with balanced means and three moments to a mixture of two Erlang distributions of common order. These algorithms are used to approximate an interarrival-time distribution for a queueing model, and the accuracy of associated performance-measure approximations is then used to evaluate the moment-matching algorithms. Three performance measures are considered, and attention is focussed on the steady-state mean queue length (number in system) of theGI/M/1 queue. Performance-measure approximations are compared to three-moment bounds and performance-measure values arising from hypothetical approximated distributions.     

A χ2 goodness-of-fit test for Markov renewal processesgoodness-of-fit test for Markov renewal processes

A Markov Renewal Process (M.R.P.) is a process similar to a Markov chain, except that the time required to move from one state to another is not fixed, but is a random variable whose distribution may depend on the two states between which the transition is made. For an M.R.P. ofm (<∞) states we derive a goodness-of-fit test for a hypothetical matrix of transition probabilities. This test is similar to the test Bartlett has derived for Markov chains. We calculate the first two moments of the test statistic and modify it to fit the moments of a standard χ². Finally, we illustrate the above procedure numeerically for a particular case of a two-state M.R.P. Dwight B. Brock is mathematical statistican, Office of Statistical Methods, National Center for Health Statistics, Rockville, Maryland. A. M. Kshisagar is Associate Professor, Department of Statistics, Southern Methodist University. This research was partially supported by Office of Naval Research Contract No. N000 14-68-A-0515, and by NIH Training Grant GM-951, both with Southern Methodist University. This article is partially based on Dwight B. Brock's Ph.D. dissertation accepted by Southern Methodist University.     

自回归相依结构下累积索赔的矩及应用

在不同时期的索赔具有自回归相依结构的条件下,给出了累积索赔的前两阶矩,并进一步讨论了在保费计算中的应用.数值算例揭示了相依参数对风险矩以及保险费的影响,表明了保险公司适时开发新业务、淘汰旧业务的必要性.     

Multifractal products of cylindrical pulses

 New multiplicative and statistically self-similar measures μ are defined on ℝ as limits of measure-valued martingales. Those martingales are constructed by multiplying random functions attached to the points of a statistically self-similar Poisson point process defined in a strip of the plane. Several fundamental problems are solved, including the non-degeneracy and the multifractal analysis of μ. On a bounded interval, the positive and negative moments of diverge under broad conditions. First received: 14 September 1999 / Resubmited: 27 June 2001 / Revised version: 30 May 2002 / Published online: 30 September 2002 Mathematics Subject Classification (2002): 28A80, 60G18, 60G44, 60G55, 60G57 Key words or phrases: Random measures – Multifractal analysis – Continuous time martingales – Statistically self-similar Poisson point processes     

Sensitivity analysis of discrete Markov chains via matrix calculus

Markov chains are often used as mathematical models of natural phenomena, with transition probabilities defined in terms of parameters that are of interest in the scientific question at hand. Sensitivity analysis is an important way to quantify the effects of changes in these parameters on the behavior of the chain. Many properties of Markov chains can be written as simple matrix expressions, and hence matrix calculus is a powerful approach to sensitivity analysis. Using matrix calculus, we derive the sensitivity and elasticity of a variety of properties of absorbing and ergodic finite-state chains. For absorbing chains, we present the sensitivities of the moments of the number of visits to each transient state, the moments of the time to absorption, the mean number of states visited before absorption, the quasistationary distribution, and the probabilities of absorption in each of several absorbing states. For ergodic chains, we present the sensitivity of the stationary distribution, the mean first passage time matrix, the fundamental matrix, and the Kemeny constant. We include two examples of application of the results to demographic and ecological problems.     

On the limiting distribution of sample central moments

We investigate the limiting behavior of sample central moments, examining the special cases where the limiting (as the sample size tends to infinity) distribution is degenerate. Parent (non-degenerate) distributions with this property are called singular, and we show in this article that the singular distributions contain at most three supporting points. Moreover, using the delta-method, we show that the (second-order) limiting distribution of sample central moments from a singular distribution is either a multiple, or a difference of two multiples of independent Chi-square random variables with one degree of freedom. Finally, we present a new characterization of normality through the asymptotic independence of the sample mean and all sample central moments.

     

A mathematical programming approach to optimise insurance premium pricing within a data mining framework

In this paper we provide evidence of the benefits of an approach which combines data mining and mathematical programming to determining the premium to charge automobile insurance policy holders in order to arrive at an optimal portfolio. An non-linear integer programming formulation is proposed to determine optimal premiums based on the insurer's need to find a balance between profitability and market share. The non-linear integer programming approach to solving this problem is used within a data mining framework which consists of three components: classifying policy holders into homogenous risk groups and predicting the claim cost of each group using k-means clustering; determining the price sensitivity (propensity to pay) of each group using neural networks; and combining the results of the first two components to determine the optimal premium to charge. We have earlier presented the results of the first two components. In this paper we present the results of the third component. Using our approach, we have been able to increase revenue without affecting termination rates and market share.     

Bose-Einstein condensation in the excited band and the energy spectrum of the Bose-Hubbard model

Based on the mean field approximation, we investigate the transition into the Bose-Einstein condensate phase in the Bose-Hubbard model with two local states and boson hopping in only the excited band. In the hard-core boson limit, we study the instability associated with this transition, which appears at excitation energies δ < |t ₀ |, where |t ₀ | is the particle hopping parameter. We discuss the conditions under which the phase transition changes from second to first order and present the corresponding phase diagrams (Θ,μ) and (|t ₀ |, μ), where Θ is the temperature and μ is the chemical potential. Separation into the normal and Bose-Einstein condensate phases is possible at a fixed average concentration of bosons. We calculate the boson Green’s function and one-particle spectral density using the random phase approximation and analyze changes in the spectrum of excitations of the “particle” or “hole” type in the region of transition from the normal to the Bose-Einstein condensate phase.     

On the optimal product mix in life insurance companies using conditional value at risk

This paper proposes a Conditional Value-at-Risk Minimization (CVaRM) approach to optimize an insurer’s product mix. By incorporating the natural hedging strategy of Cox and Lin (2007) and the two-factor stochastic mortality model of Cairns et al. (2006b), we calculate an optimize product mix for insurance companies to hedge against the systematic mortality risk under parameter uncertainty. To reflect the importance of required profit, we further integrate the premium loading of systematic risk. We compare the hedging results to those using the duration match method of Wang et al. (forthcoming), and show that the proposed CVaRM approach has a narrower quantile of loss distribution after hedging—thereby effectively reducing systematic mortality risk for life insurance companies.     

Risk measures, distortion parameters, and their empirical estimation

Risk measures are of considerable current interest. Among other uses, they allow an insurer to calculate a risk-loaded premium for a random loss. However, the premium principle in use by the insurer may be, at least in part, based on considerations other than risk. It is then important to quantify the degree to which the premium compensates the insurer for the risk associated with the loss. This can be done by choosing a suitable risk measure and solving for the parameter that leads to the insurer’s premium. When the loss distribution is unknown, this becomes a statistical estimation problem.In this paper, we investigate the nonparametric estimation of the parameter associated with a distortion-based risk measure. It is assumed that the premium principle is known, but no information is assumed about the loss distribution, and therefore empirical estimators are used. We explore the asymptotic properties of the resulting estimator of the risk measure parameter in general and for three well-known risk measures in particular: the proportional hazards transform, the Wang transform, and the conditional tail expectation.