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.     

A general transformation class of semiparametric cure rate frailty models

We consider a class of cure rate frailty models for multivariate failure time data with a survival fraction. This class is formulated through a transformation on the unknown population survival function. It incorporates random effects to account for the underlying correlation, and includes the mixture cure model and the proportional hazards cure model as two special cases. We develop efficient likelihood-based estimation and inference procedures. We show that the nonparametric maximum likelihood estimators for the parameters of these models are consistent and asymptotically normal, and that the limiting variances achieve the semiparametric efficiency bounds. Simulation studies demonstrate that the proposed methods perform well in finite samples. We provide an application of the proposed methods to the data of the age at onset of alcohol dependence, from the Collaborative Study on the Genetics of Alcoholism.     

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??Recurrent event data usually occur in long-term studies which concern
recurrence rates of the disease. In studies of medical sciences, patients who have infected
with the disease, like cancer, were conventionally regarded as impossible to be cured. However,
with the development of medical sciences, recently those patients were found to be possibly
recovered from the disease. The recurrence rate of the events, which is of primary interest,
may be affected by the cure rate that may exist. Therefore, we proposed semiparametric
statistical analysis for recurrent event data with subjects possibly being cured. In our
approach, we present a proportional rate model for recurrence rate with the cure rate adjusted
through a Logistic regression model, and develop some estimating equations for estimation of
the regression parameters, with their large sample properties, including consistency and
asymptotic normality established. Numerical studies under different settings were conducted
for assessing the proposed methodology and the results suggest that they work well for
practical situations. The approach is applied to a bladder cancer dataset which motivated our
study.     

The analysis of an epidemic model on networks

The paper consider an epidemic model with birth and death on networks. We derive the epidemic threshold R₀ dependent on birth rate b, death rate d (natural death) and μ from the infectious disease and natural death, and cure rate γ. And the stability of the equilibriums (the disease-free equilibrium and endemic equilibrium) are analysed. Finally, the effects of various immunization schemes are studied and compared. We show that both targeted, and acquaintance immunization strategies compare favorably to a proportional scheme in terms of effectiveness. For active immunization, the threshold is easier to apply practically. To illustrate our theoretical analysis, some numerical simulations are also included.     

含有治愈部分的可加危险模型的估计

最近可加危险(AH)模型被广泛地应用于生存分析数据,模型的协变量可以假设为时间独立或时间相关的.基于混合治愈模型,有界累计危险治愈模型和"不正确"的比例危险模型.本文将上述的可乘危险模型延伸到可加的危险模型,这里的模型可以允许含治愈部分的生存数据的存在."不正确"的AH模型的识别和参数估计也将在本文给出讨论.     

Maximum likelihood estimation in the proportional hazards cure model

The proportional hazards cure model generalizes Cox’s proportional hazards model which allows that a proportion of study subjects may never experience the event of interest. Here nonparametric maximum likelihood approach is proposed to estimating the cumulative hazard and the regression parameters. The asymptotic properties of the resulting estimators are established using the modern empirical process theory. And the estimators for the regression parameters are shown to be semiparametric efficient.     

A new joint model for longitudinal and survival data with a cure fraction

We develop a new joint cure rate model for longitudinal and survival data. The model allows for multiple longitudinal markers as well as a cure structure for the survival component based on the promotion time cure rate model, as described in Ibrahim et al. (Bayesian Survival Analysis, Springer, New York, 2001). Several characteristics and properties of the new model are discussed and examined. A real dataset from a melanoma clinical trial is given to demonstrate the methodology.     

Joint modeling of longitudinal proportional measurements and survival time with a cure fraction

In cancer clinical trials and other medical studies, both longitudinal measurements and data on a time to an event (survival time) are often collected from the same patients. Joint analyses of these data would improve the efficiency of the statistical inferences. We propose a new joint model for the longitudinal proportional measurements which are restricted in a finite interval and survival times with a potential cure fraction. A penalized joint likelihood is derived based on the Laplace approximation and a semiparametric procedure based on this likelihood is developed to estimate the parameters in the joint model. A simulation study is performed to evaluate the statistical properties of the proposed procedures. The proposed model is applied to data from a clinical trial on early breast cancer.     

Mixture cure models in credit scoring: If and when borrowers default

Mixture cure models were originally proposed in medical statistics to model long-term survival of cancer patients in terms of two distinct subpopulations - those that are cured of the event of interest and will never relapse, along with those that are uncured and are susceptible to the event. In the present paper, we introduce mixture cure models to the area of credit scoring, where, similarly to the medical setting, a large proportion of the dataset may not experience the event of interest during the loan term, i.e. default. We estimate a mixture cure model predicting (time to) default on a UK personal loan portfolio, and compare its performance to the Cox proportional hazards method and standard logistic regression. Results for credit scoring at an account level and prediction of the number of defaults at a portfolio level are presented; model performance is evaluated through cross validation on discrimination and calibration measures. Discrimination performance for all three approaches was found to be high and competitive. Calibration performance for the survival approaches was found to be superior to logistic regression for intermediate time intervals and useful for fixed 12 month time horizon estimates, reinforcing the flexibility of survival analysis as both a risk ranking tool and for providing robust estimates of probability of default over time. Furthermore, the mixture cure model’s ability to distinguish between two subpopulations can offer additional insights by estimating the parameters that determine susceptibility to default in addition to parameters that influence time to default of a borrower.     

Optimal investment to minimize the probability of drawdown

We determine the optimal investment strategy in a Black–Scholes financial market to minimize the so-called probability of drawdown, namely, the probability that the value of an investment portfolio reaches some fixed proportion of its maximum value to date. We assume that the portfolio is subject to a payout that is a deterministic function of its value, as might be the case for an endowment fund paying at a specified rate, for example, at a constant rate or at a rate that is proportional to the fund’s value.     

An algorithm to solve the proportional network flow problem

The proportional network flow problem is a generalization of the equal flow problem on a generalized network in which the flow on arcs in given sets must all be proportional. This problem appears in several natural contexts, including processing networks and manufacturing networks. This paper describes a transformation on the underlying network that reduces the problem to the equal flow problem; this transformation is used to show that algorithms that solve the equal flow problem can be directly applied to the proportional network flow problem as well, with no increase in asymptotic running time. Additionally, computational results are presented for the proportional network flow problem demonstrating equivalent performance to the same algorithm for the equal flow problem.     

Likelihood inference for the destructive exponentially weighted Poisson cure rate model with Weibull lifetime and an application to melanoma data

In this paper, we develop the steps of the expectation maximization algorithm (EM algorithm) for the determination of the maximum likelihood estimates (MLEs) of the parameters of the destructive exponentially weighted Poisson cure rate model in which the lifetimes are assumed to be Weibull. This model is more flexible than the promotion time cure rate model as it provides an interesting and realistic interpretation of the biological mechanism of the occurrence of an event of interest by including a destructive process of the initial number of causes in a competitive scenario. The standard errors of the MLEs are obtained by inverting the observed information matrix. An extensive Monte Carlo simulation study is carried out to evaluate the performance of the developed method of estimation. Finally, a known melanoma data are analyzed to illustrate the method of inference developed here. With these data, a comparison is also made with the scenario when the destructive mechanism is not included in the analysis.     

Stage-structured Harvest Models

We formulate a stage-structured population model where the population is divided to two classes, the juveniles and the adults. Then, we include harvest in the model and assume that the harvesting is only on adults. The cases where the harvesting rate is constant, proportional to the amount of adults, or of Holling-II type are studied. While the model dynamics are relatively simple when the harvesting rate is proportional, the model system with a constant or a Holling-II type harvesting rate can have multiple positive equilibria. We explore the existence of all possible equilibria and investigate their stability. We also give numerical examples to confirm our findings.     

人民币汇率参考一篮子货币的结构性变动实证研究

分别以瑞郎和特别提款权为基准货币,基于2005年"汇改"后至2012年7年期间人民币与央行公布的11种参考货币的汇率数据,在邹检验确定结构性变动点的基础上,运用对数线性模型分时段建模实证.结果表明:1)"汇改"以来,人民币汇率已开始参考一篮子货币进行调节,并分别在2006年4季度,2009年1季度和2010年3季度发生了结构性变动;2)人民币汇率制度本质上仍以钉住美元为主,篮子货币的币种选择和权重设定是一个动态调整的过程.     

Nonlinear stability of Runge-Kutta methods applied to infinite-delay-differential equations

In functional differential equations (FDEs), there is a class of infinite delay-differential equations (IDDEs) with proportional delays, which aries in many scientific fields such as electric mechanics, quantum mechanics, and optics. Ones have found that there exist very different mathematical challenges between FDEs with proportional delays and those with constant delays. Some research on the numerical solutions and the corresponding analysis for the linear FDEs with proportional delays have been presented by several authors. However, up to now, the research for nonlinear case still remains to be done. For this, in the present paper, we deal with nonlinear stability of the Runge-Kutta (RK) methods for a class of IDDEs with proportional delays. It is shown under the suitable conditions that a (k, l)-algebraically stable RK method for this kind of nonlinear IDDE is globally and asymptotically stable.     

Properties of Epoxy Systems with Clay Nanocomposites

Different types of montmorillonite clays are often used for the modification of thermoplastic polymers. In the case of epoxy systems, the presence of the clays can complicate the cure process. Therefore, the influence of montmorillonite concentration and the temperature regime on the cure rate and mechanical properties of the composite material obtained is investigated in this paper.     

Dynamic adjustment: an electoral method for relaxed double proportionality

We describe an electoral system for distributing seats in a parliament. It gives proportionality for the political parties and close to proportionality for constituencies. The system suggested here is a version of the system used in Sweden and other Nordic countries with permanent seats in each constituency and adjustment seats to give proportionality on the national level. In the national election of 2010 the current Swedish system failed to give proportionality between parties. We examine here one possible cure for this unwanted behavior. The main difference compared to the current Swedish system is that the number of adjustment seats is not fixed, but rather dynamically determined to be as low as possible and still insure proportionality between parties.     

The new Neyman type A beta Weibull model with long-term survivors

For the first time, we propose a flexible cure rate survival model by assuming that the number of competing causes of the event of interest follows the Neyman type A distribution and the time to this event has the beta Weibull distribution. This new model can be used to analyze survival data when the hazard rate function is increasing, decreasing, bathtub or unimodal-shaped. It includes some commonly used lifetime distributions and some well-known cure rate models as special cases. Maximum likelihood and non-parametric bootstrap are used to estimate the regression parameters. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. The usefulness of the new model is illustrated by means of an application in the medical area.     

Default risk analysis via a discrete‐time cure rate model

Cure models represent an appealing tool when analyzing default time data where two groups of companies are supposed to coexist: those which could eventually experience a default (uncured) and those which could not develop an endpoint (cured). One of their most interesting properties is the possibility to distinguish among covariates exerting their influence on the probability of belonging to the populations’ uncured fraction, from those affecting the default time distribution. This feature allows a separate analysis of the two dimensions of the default risk: whether the default can occur and when it will occur, given that it can occur. Basing our analysis on a large sample of Italian firms, the probability of being uncured is here estimated with a binary logit regression, whereas a discrete time version of a Cox's proportional hazards approach is used to model the time distribution of defaults. The extension of the cure model as a forecasting framework is then accomplished by replacing the discrete time baseline function with an appropriate time‐varying system level covariate, able to capture the underlying macroeconomic cycle. We propose a holdout sample procedure to test the classification power of the cure model. When compared with a single‐period logit regression and a standard duration analysis approach, the cure model has proven to be more reliable in terms of the overall predictive performance. Copyright © 2013 John Wiley & Sons, Ltd.     

A new criterion to global exponential stability for impulsive neural networks with continuously distributed delays

This paper considers the global exponential stability and exponential convergence rate of impulsive neural networks with continuously distributed delays in which the state variables on the impulses are related to the unbounded distributed delays. By establishing a new impulsive delay differential inequality, a new criterion concerning global exponential stability for these networks is derived, and the estimated exponential convergence rate is also obtained. The result extends and improves on earlier publications. In addition, two numerical examples are given to illustrate the applicability of the result. Copyright © 2010 John Wiley & Sons, Ltd.