Routing and location on a network with hazardous threats

The aim of this paper is to minimise the impact of hazards located on or near a network. Two situations are considered: (1) where a hazard is located on a network and affects off network sites and (2) an off-network hazard which can affect traffic on the network. Eight models aimed at optimising different objectives are developed and solved, including finding a route between two nodes on a network which minimises the hazard along it and finding a location on a network where the hazard is minimised.     

Relative ageing for two parallel systems and related problems

Kalashnikov and Rachev [1] have proposed a partial ordering of two life distributions which is equivalent to an increasing hazard (failure rate) ratio, when the ratio exists. The phenomenon of crossing hazards has received considerable attention in recent years. Recently, Sengupta and Deshpande [2) have studied this and two other models of relative ageing. In this paper, we consider the relative ageing properties of two parallel systems with identical but different number of components. We also compare the variances of the two life distributions having the same mean but with increasing hazard ratio. Several examples are given to illustrate the results.     

Modelling the media: the transmission of risk information in the British quality press

Behavioural reactions to potential hazards are likely to bedetermined by psychologically derived factors (risk perceptions)as well as rational estimates of risk .Research has shown thatrisk perception is a complex psychological construct, wheredifferent potential hazards are associated with different hazardcharacteristics. One of the most important determinants of risk perception isknowledge about a given hazard. It is probable that reactionswill be determined not only by what is conveyed in the contentof risk information but also by other characteristics of theinformation source. A predominant source of risk informationfor the public is the media, although there are at present noadequate models available to describe how such risk informationis conveyed. If communication with the public on issues of riskis to be effective, then it must take due account of the roleand effect of the media. Given that the public is suddenly exposedto a problem by the media in the context of a 'crisis', it ispossible that polarization of views among the public and scientificcommunity (and other interested parties, such as pressure groups)will take place, often on the basis of inadequate data. This research examines how the British press communicate riskinformation. The focus is food-related hazards: food biotechnology,microbiological hazards, food irradiation, chemicals and pesticideresidues, and food additives. The development of a content-analysisschedule, and its subsequent application to 182 newspaper reportsrefemng to food-related hazards, is presented. Correspondenceanalysis of the resulting data indicated that substantial differencesin risk reporting occurs for hazards with different perceptualcharacteristics. Methods of testing derived models of mediatransmission of risk information against formation of attitudesare discussed.     

Generalized mixtures in reliability modelling: Applications to the construction of bathtub shaped hazard models and the study of systems

In this paper, we obtain and discuss some general properties of hazard rate (HR) functions constructed via generalized mixtures of two members. These results are applied to determine the shape of generalized mixtures of an increasing hazard rate (IHR) model and an exponential model. In addition, we note that these kind of generalized mixtures can be used to construct bathtub‐shaped HR models. As examples, we study in detail two cases: when the IHR model chosen is a linear HR function and when the IHR model is the extended exponential‐geometric distribution. Finally, we apply the results and show the utility of generalized mixtures in determining the shape of the HR function of different systems, such as mixed systems or consecutive k‐out‐of‐n systems. Copyright © 2008 John Wiley & Sons, Ltd.     

OPTIMAL INVESTMENT IN THE PROTECTION OF A VULNERABLE BIOLOGICAL RESOURCE

It is assumed that the probability of destruction of a biological asset by natural hazards can be reduced through investment in protection. Specifically a model, in which the hazard rate depends on both the age of the asset and the accumulated invested protection capital, is assumed. The protection capital depreciates through time and its effectiveness in reducing the hazard rate is subject to diminishing returns. It is shown how the investment schedule to maximize the expected net present value of the asset can be determined using the methods of deterministic optimal control, with the survival probability regarded as a state variable. The optimal investment pattern involves “bang-bang-singular” control. A numerical scheme for determining jointly the optimal investment policy and the optimal harvest (or replacement) age is outlined and a numerical example involving forest fire protection is given.     

Model checking for additive hazards model with multivariate survival data

Multivariate failure time data often arise in biomedical studies due to natural or artificial clustering. With appropriate adjustment for the underlying correlation, the marginal additive hazards model characterizes the hazard difference via a linear link function between the hazard and covariates. We propose a class of graphical and numerical methods to assess the overall fitting adequacy of the marginal additive hazards model. The test statistics are based on the supremum of the stochastic processes derived from the cumulative sum of the martingale-based residuals over time and/or covariates. The distribution of the stochastic process can be approximated through a simulation technique. The proposed tests examine how unusual the observed stochastic process is, compared to a large number of realizations from the approximated process. This class of tests is very general and suitable for various purposes of model fitting evaluation. Simulation studies are conducted to examine the finite sample performance, and the model-checking methods are illustrated with data from an otitis media study.     

Empirical likelihood inference for censored median regression with weighted empirical hazard functions

In recent years, median regression models have been shown to be useful for analyzing a variety of censored survival data in clinical trials. For inference on the regression parameter, there have been a variety of semiparametric procedures. However, the accuracy of such procedures in terms of coverage probability can be quite low when the censoring rate is heavy. In this paper, based on weighted empirical hazard functions, we apply an empirical likelihood (EL) ratio method to the median regression model with censoring data and derive the limiting distribution of EL ratio. Confidence region for the regression parameter can then be obtained accordingly. Furthermore, we compared the proposed method with the standard method through extensive simulation studies. The proposed method almost always outperformed the existing method.     

Robust optimization of contaminant sensor placement for community water systems

We present a series of related robust optimization models for placing sensors in municipal water networks to detect contaminants that are maliciously or accidentally injected. We formulate sensor placement problems as mixed-integer programs, for which the objective coefficients are not known with certainty. We consider a restricted absolute robustness criteria that is motivated by natural restrictions on the uncertain data, and we define three robust optimization models that differ in how the coefficients in the objective vary. Under one set of assumptions there exists a sensor placement that is optimal for all admissible realizations of the coefficients. Under other assumptions, we can apply sorting to solve each worst-case realization efficiently, or we can apply duality to integrate the worst-case outcome and have one integer program. The most difficult case is where the objective parameters are bilinear, and we prove its complexity is NP-hard even under simplifying assumptions. We consider a relaxation that provides an approximation, giving an overall guarantee of near-optimality when used with branch-and-bound search. We present preliminary computational experiments that illustrate the computational complexity of solving these robust formulations on sensor placement applications.     

Regression analysis of right-censored failure time data with missing censoring indicators

This paper discusses regression analysis of right-censored failure time data when censoring indicators are missing for some subjects. Several methods have been developed for the analysis under different situations and especially, Goetghebeur and Ryan considered the situation where both the failure time and the censoring time follow the proportional hazards models marginally and developed an estimating equation approach. One limitation of their approach is that the two baseline hazard functions were assumed to be proportional to each other. We consider the same problem and present an efficient estimation procedure for regression parameters that does not require the proportionality assumption. An EM algorithm is developed and the method is evaluated by a simulation study, which indicates that the proposed methodology performs well for practical situations. An illustrative example is provided.     

Optimal Exercise Strategies for Operational Risk Insurance via Multiple Stopping Times

In this paper we demonstrate how to develop analytic closed form solutions to optimal multiple stopping time problems arising in the setting in which the value function acts on a compound process that is modified by the actions taken at the stopping times. This class of problem is particularly relevant in insurance and risk management settings and we demonstrate this on an important application domain based on insurance strategies in Operational Risk management for financial institutions. In this area of risk management the most prevalent class of loss process models is the Loss Distribution Approach (LDA) framework which involves modelling annual losses via a compound process. Given an LDA model framework, we consider Operational Risk insurance products that mitigate the risk for such loss processes and may reduce capital requirements. In particular, we consider insurance products that grant the policy holder the right to insure k of its annual Operational losses in a horizon of T years. We consider two insurance product structures and two general model settings, the first are families of relevant LDA loss models that we can obtain closed form optimal stopping rules for under each generic insurance mitigation structure and then secondly classes of LDA models for which we can develop closed form approximations of the optimal stopping rules. In particular, for losses following a compound Poisson process with jump size given by an Inverse-Gaussian distribution and two generic types of insurance mitigation, we are able to derive analytic expressions for the loss process modified by the insurance application, as well as closed form solutions for the optimal multiple stopping rules in discrete time (annually). When the combination of insurance mitigation and jump size distribution does not lead to tractable stopping rules we develop a principled class of closed form approximations to the optimal decision rule. These approximations are developed based on a class of orthogonal Askey polynomial series basis expansion representations of the annual loss compound process distribution and functions of this annual loss.     

Semiparametric estimation based on parametric modeling of the cause-specific hazard ratios in competing risks

This paper is intended as an investigation of estimating cause-specific cumulative hazard and cumulative incidence functions in a competing risks model. The proportional model in which ratios of the cause-specific hazards to the overall hazard are assumed to be constant (independent of time) is a well-known semiparametric model. We are here concerned with relaxation of the proportionality assumption. The set C of all causes are decomposed into two disjoint subsets of causes as C=C₁∪C₂. The relative risk of cause A in the sub-causes C₁ can be represented as a function defined by ratio of the cause-specific hazard of cause A to the sum of cause-specific hazards in the sub-causes C₁. We call this function the risk pattern function of cause A in C₁, and consider a semiparametric model in which risk pattern functions in C₁ are not constant (independent of time) but those functional forms, except for finite-dimensional parameters, are known. Based on this model, semiparametric estimators are obtained, and estimated variances of them are derived by delta methods. We investigate asymptotic properties of the semiparametric estimators and compare them with the nonparametric estimators. The semiparametric procedure is illustrated with the radiation-exposed mice data set, which represents lifetimes and causes of death of mice exposed to radiation in two different environments.     

Optimizing Preventive Maintenance Models

We deal with the problem of scheduling preventive maintenance (PM) for a system so that, over its operating life, we minimize a performance function which reflects repair and replacement costs as well as the costs of the PM itself. It is assumed that a hazard rate model is known which predicts the frequency of system failure as a function of age. It is also assumed that each PM produces a step reduction in the effective age of the system. We consider some variations and extensions of a PM scheduling approach proposed by Lin et al. [6]. In particular we consider numerical algorithms which may be more appropriate for hazard rate models which are less simple than those used in [6] and we introduce some constraints into the problem in order to avoid the possibility of spurious solutions. We also discuss the use of automatic differentiation (AD) as a convenient tool for computing the gradients and Hessians that are needed by numerical optimization methods. The main contribution of the paper is a new problem formulation which allows the optimal number of occurrences of PM to be determined along with their optimal timings. This formulation involves the global minimization of a non-smooth performance function. In our numerical tests this is done via the algorithm DIRECT proposed by Jones et al. [19]. We show results for a number of examples, involving different hazard rate models, to give an indication of how PM schedules can vary in response to changes in relative costs of maintenance, repair and replacement. Part of this work was carried out while the first author was a Visiting Professor in the Department of Mechanical Engineering at the University of Alberta in December 2003.     

Calculation of changes in life expectancy based on proportional hazards model of an intervention

Mortality projections are of great interest to the pension and insurance industry and with an ageing population, the projections need to cover a longer period. A significant question is how to incorporate in mortality projections the longevity risk due to medical advances and uptake of health interventions. We show how hazard ratios obtained from medical studies in combination with the baseline hazards described by Gompertz or Weibull survival distributions, can be translated into changes in individual and population period life expectancy. The impact of medical advances and health interventions can differ among groups of people, such as by sex, age, and deprivation. Changes in life expectancy depend on the composition of the population and these attributes. These calculations are illustrated by a case study on statins, a drug that can significantly improve life expectancy. An R program implementing our methodology is provided in the Appendix.     

Optimal selection of obsolescence mitigation strategies using a restless bandit model

Obsolescence of embedded parts is a serious concern for managers of complex systems where the design life of the system typically exceeds 20 years. Capital asset management teams have been exploring several strategies to mitigate risks associated with Diminishing Manufacturing Sources (DMS) and repeated life extensions of complex systems. Asset management cost and the performance of a system depend heavily on the obsolescence mitigation strategy chosen by the decision maker. We have developed mathematical models that can be used to calculate the impact of various obsolescence mitigation strategies on the Total Cost of Ownership (TCO) of a system. We have used classical multi-arm bandit (MAB) and restless bandit models to identify the best strategy for managing obsolescence in such instances wherein organizations have to deal with continuous technological evolution under uncertainty. The results of dynamic programming and greedy heuristic are compared with Gittins index solution.     

Analyzing right-censored length-biased data with additive hazards model

Length-biased data are often encountered in observational studies, when the survival times are left-truncated and right-censored and the truncation times follow a uniform distribution. In this article, we propose to analyze such data with the additive hazards model, which specifies that the hazard function is the sum of an arbitrary baseline hazard function and a regression function of covariates. We develop estimating equation approaches to estimate the regression parameters. The resultant estimators are shown to be consistent and asymptotically normal. Some simulation studies and a real data example are used to evaluate the finite sample properties of the proposed estimators.     

Impact of insurance for operational risk: Is it worthwhile to insure or be insured for severe losses?

Under the Basel II standards, the Operational Risk (OpRisk) advanced measurement approach allows a provision for reduction of capital as a result of insurance mitigation of up to 20%. This paper studies different insurance policies in the context of capital reduction for a range of extreme loss models and insurance policy scenarios in a multi-period, multiple risk setting. A Loss Distributional Approach (LDA) for modeling of the annual loss process, involving homogeneous compound Poisson processes for the annual losses, with heavy-tailed severity models comprised of α-stable severities is considered. There has been little analysis of such models to date and it is believed insurance models will play more of a role in OpRisk mitigation and capital reduction in future. The first question of interest is when would it be equitable for a bank or financial institution to purchase insurance for heavy-tailed OpRisk losses under different insurance policy scenarios? The second question pertains to Solvency II and addresses quantification of insurer capital for such operational risk scenarios. Considering fundamental insurance policies available, in several two risk scenarios, we can provide both analytic results and extensive simulation studies of insurance mitigation for important basic policies, the intention being to address questions related to VaR reduction under Basel II, SCR under Solvency II and fair insurance premiums in OpRisk for different extreme loss scenarios. In the process we provide closed-form solutions for the distribution of loss processes and claims processes in an LDA structure as well as closed-form analytic solutions for the Expected Shortfall, SCR and MCR under Basel II and Solvency II. We also provide closed-form analytic solutions for the annual loss distribution of multiple risks including insurance mitigation.     

Time to default in credit scoring using survival analysis: a benchmark study

We investigate the performance of various survival analysis techniques applied to ten actual credit data sets from Belgian and UK financial institutions. In the comparison we consider classical survival analysis techniques, namely the accelerated failure time models and Cox proportional hazards regression models, as well as Cox proportional hazards regression models with splines in the hazard function. Mixture cure models for single and multiple events were more recently introduced in the credit risk context. The performance of these models is evaluated using both a statistical evaluation and an economic approach through the use of annuity theory. It is found that spline-based methods and the single event mixture cure model perform well in the credit risk context.     

Additive–multiplicative hazards model with current status data

The additive–multiplicative hazards (AMH) regression model specifies an additive and multiplicative form on the hazard function for the counting process associated with a multidimensional covariate process, which contains the Cox proportional hazards model and the additive hazards model as its special cases. In this paper, we study the AMH model with current status data, where the cumulative hazard hazard function is assumed to be nonparametric and is estimated using B-splines with monotonicity constraint on the functional, while a simultaneous sieve maximum likelihood estimation is proposed to estimate regression parameters. The proposed estimator for the parameter vector is shown to be asymptotically normal and semiparametric efficient. The B-splines estimator of the functional of the cumulative hazard function is shown to achieve the optimal nonparametric rate of convergence. A simulation study is conducted to examine the finite sample performance of the proposed estimators and algorithm, and a real data example is presented for illustration.     

Additive hazards models for gap time data with multiple causes

Recurrent event data with multiple causes are often observed in biomedical studies. The additive hazards model describes a different aspect of the association between covariates and the failure time than does the proportional hazards model. In this paper, we introduce additive hazards models for the analysis of gap time data of recurrent events with multiple causes. We estimate the regression parameter vector and cumulative baseline cause specific hazard rate function using counting process approach. Asymptotic properties of the estimators are studied. The proposed model is applied to the kidney dialysis data given in Lawless (2003). A simulation study is carried out to assess the performance of the estimates.