European generic scoring models using survival analysis

Credit scoring discriminates between ‘good’ and ‘bad’ credit risks to assist credit-grantors in making lending decisions. Such discrimination may not be a good indicator of profit, while survival analysis allows profit to be modelled. The paper explores the application of parametric accelerated failure time and proportional hazards models and Cox non-parametric model to the data from the retail card (revolving credit) from three European countries. The predictive performance of three national models is tested for different timescales of default and then compared to that of a single generic model for a timescale of 25 months. It is found that survival analysis national and generic models produce predictive quality, which is very close to the current industry standard—logistic regression. Stratification is investigated as a way of extending Cox non-parametric proportional hazards model to tackle heterogeneous segments in the population.     

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.     

PHAB scores: proportional hazards analysis behavioural scores

Credit scoring is one of the most widely used applications of quantitative analysis in business. Behavioural scoring is a type of credit scoring that is performed on existing customers to assist lenders in decisions like increasing the balance or promoting new products. This paper shows how using survival analysis tools from reliability and maintenance modelling, specifically Cox's proportional hazards regression, allows one to build behavioural scoring models. Their performance is compared with that of logistic regression. Also the advantages of using survival analysis techniques in building scorecards are illustrated by estimating the expected profit from personal loans. This cannot be done using the existing risk behavioural systems.     

A time-dependent proportional hazards survival model for credit risk analysis

In the consumer credit industry, assessment of default risk is critically important for the financial health of both the lender and the borrower. Methods for predicting risk for an applicant using credit bureau and application data, typically based on logistic regression or survival analysis, are universally employed by credit card companies. Because of the manner in which the predictive models are fit using large historical sets of existing customer data that extend over many years, default trends, anomalies, and other temporal phenomena that result from dynamic economic conditions are not brought to light. We introduce a modification of the proportional hazards survival model that includes a time-dependency mechanism for capturing temporal phenomena, and we develop a maximum likelihood algorithm for fitting the model. Using a very large, real data set, we demonstrate that incorporating the time dependency can provide more accurate risk scoring, as well as important insight into dynamic market effects that can inform and enhance related decision making.     

Neural network survival analysis for personal loan data

Traditionally, credit scoring aimed at distinguishing good payers from bad payers at the time of the application. The timing when customers default is also interesting to investigate since it can provide the bank with the ability to do profit scoring. Analysing when customers default is typically tackled using survival analysis. In this paper, we discuss and contrast statistical and neural network approaches for survival analysis. Compared to the proportional hazards model, neural networks may offer an interesting alternative because of their universal approximation property and the fact that no baseline hazard assumption is needed. Several neural network survival analysis models are discussed and evaluated according to their way of dealing with censored observations, time-varying inputs, the monotonicity of the generated survival curves and their scalability. In the experimental part, we contrast the performance of a neural network survival analysis model with that of the proportional hazards model for predicting both loan default and early repayment using data from a UK financial institution.     

Cox模型与BP神经网络在处理非线性数据时的性能比较

本文采用BP神经网络、Cox模型和bootstrap方法,比较BP神经网络与Cox模型在处理非线性资料时的性能。两种方法的预测一致性的均数分别为0.7525和0.7706。对于非线性资料,BP神经网络的预测效果优于Cox模型。     

SICA for Cox’s proportional hazards model with a diverging number of parameters

The smooth integration of counting and absolute deviation （SICA） penalized variable selection procedure for high-dimensional linear regression models is proposed by Lv and Fan （2009）. In this article, we extend their idea to Cox＇s proportional hazards （PH） model by using a penalized log partial likelihood with the SICA penalty. The number of the regression coefficients is allowed to grow with the sample size. Based on an approximation to the inverse of the Hessian matrix, the proposed method can be easily carried out with the smoothing quasi-Newton （SQN） algorithm. Under appropriate sparsity conditions, we show that the resulting estimator of the regression coefficients possesses the oracle property. We perform an extensive simulation study to compare our approach with other methods and illustrate it on a well known PBC data for predicting survival from risk factors.     

Chi-squared tests for general composite hypotheses from censored samples

We give chi-squared goodness-of-fit tests for parametric models including various regression models such as accelerated failure time, proportional hazards, generalized proportional hazards, frailty models, transformation models, models with cross-effects of survival functions. Choice of random grouping intervals as data functions is considered.     

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.     

Empirical likelihood analysis of the Buckley-James estimator

The censored linear regression model, also referred to as the accelerated failure time (AFT) model when the logarithm of the survival time is used as the response variable, is widely seen as an alternative to the popular Cox model when the assumption of proportional hazards is questionable. Buckley and James [Linear regression with censored data, Biometrika 66 (1979) 429-436] extended the least squares estimator to the semiparametric censored linear regression model in which the error distribution is completely unspecified. The Buckley-James estimator performs well in many simulation studies and examples. The direct interpretation of the AFT model is also more attractive than the Cox model, as Cox has pointed out, in practical situations. However, the application of the Buckley-James estimation was limited in practice mainly due to its illusive variance. In this paper, we use the empirical likelihood method to derive a new test and confidence interval based on the Buckley-James estimator of the regression coefficient. A standard chi-square distribution is used to calculate the P-value and the confidence interval. The proposed empirical likelihood method does not involve variance estimation. It also shows much better small sample performance than some existing methods in our simulation studies.     

Using principal components in a proportional hazards model with applications in condition-based maintenance

This paper proposes the application of a principal components proportional hazards regression model in condition-based maintenance (CBM) optimization. The Cox proportional hazards model with time-dependent covariates is considered. Principal component analysis (PCA) can be applied to covariates (measurements) to reduce the number of variables included in the model, as well as to eliminate possible collinearity between the covariates. The main issues and problems in using the proposed methodology are discussed. PCA is applied to a simulated CBM data set and two real data sets obtained from industry: oil analysis data and vibration data. Reasonable results are obtained.     

Modelling credit risk of portfolio of consumer loans

One of the issues that the Basel Accord highlighted was that, though techniques for estimating the probability of default and hence the credit risk of loans to individual consumers are well established, there were no models for the credit risk of portfolios of such loans. Motivated by the reduced form models for credit risk in corporate lending, we seek to exploit the obvious parallels between behavioural scores and the ratings ascribed to corporate bonds to build consumer-lending equivalents. We incorporate both consumer-specific ratings and macroeconomic factors in the framework of Cox Proportional Hazard models. Our results show that default intensities of consumers are significantly influenced by macro factors. Such models then can be used as the basis for simulation approaches to estimate the credit risk of portfolios of consumer loans.     

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 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.     

Jackknifed random weighting for Cox proportional hazards model

The Cox proportional hazards model is the most used statistical model in the analysis of survival time data.Recently,a random weighting method was proposed to approximate the distribution of the maximum partial likelihood estimate for the regression coefficient in the Cox model.This method was shown not as sensitive to heavy censoring as the bootstrap method in simulation studies but it may not be second-order accurate as was shown for the bootstrap approximation.In this paper,we propose an alternative random weighting method based on one-step linear jackknife pseudo values and prove the second accuracy of the proposed method.Monte Carlo simulations are also performed to evaluate the proposed method for fixed sample sizes.     

A comparison of neural networks and linear scoring models in the credit union environment

The purpose of the present paper is to explore the ability of neural networks such as multilayer perceptrons and modular neural networks, and traditional techniques such as linear discriminant analysis and logistic regression, in building credit scoring models in the credit union environment. Also, since funding and small sample size often preclude the use of customized credit scoring models at small credit unions, we investigate the performance of generic models and compare them with customized models. Our results indicate that customized neural networks offer a very promising avenue if the measure of performance is percentage of bad loans correctly classified. However, if the measure of performance is percentage of good and bad loans correctly classified, logistic regression models are comparable to the neural networks approach. The performance of generic models was not as good as the customized models, particularly when it came to correctly classifying bad loans. Although we found significant differences in the results for the three credit unions, our modular neural network could not accommodate these differences, indicating that more innovative architectures might be necessary for building effective generic models.     

Empirical likelihood method for linear transformation models

Empirical likelihood inferential procedure is proposed for right censored survival data under linear transformation models, which include the commonly used proportional hazards model as a special case. A log-empirical likelihood ratio test statistic for the regression coefficients is developed. We show that the proposed log-empirical likelihood ratio test statistic converges to a standard chi-squared distribution. The result can be used to make inference about the entire regression coefficients vector as well as any subset of it. The method is illustrated by extensive simulation studies and a real example.     

Boosting in Cox regression: a comparison between the likelihood-based and the model-based approaches with focus on the R-packages <Emphasis Type="Italic">CoxBoost</Emphasis> and <Emphasis Type="Italic">mboost</Emphasis>

Despite the limitations imposed by the proportional hazards assumption, the Cox model is probably the most popular statistical tool used to analyze survival data, thanks to its flexibility and ease of interpretation. For this reason, novel statistical/machine learning techniques are usually adapted to fit its requirements, including boosting. Boosting is an iterative technique originally developed in the machine learning community to handle classification problems, and later extended to the statistical field, where it is used in many situations, including regression and survival analysis. The popularity of boosting has been further driven by the availability of user-friendly software such as the R packages mboost and CoxBoost, both of which allow the implementation of boosting in conjunction with the Cox model. Despite the common underlying boosting principles, these two packages use different techniques: the former is an adaptation of model-based boosting, while the latter adapts likelihood-based boosting. Here we contrast these two boosting techniques as implemented in the R packages from an analytic point of view; we further examine solutions adopted within these packages to treat mandatory variables, i.e. variables that—for several reasons—must be included in the model. We explore the possibility of extending solutions currently only implemented in one package to the other. A simulation study and a real data example are added for illustration.     

Model combination for credit risk assessment: A stacked generalization approach

The development of credit risk assessment models is often considered within a classification context. Recent studies on the development of classification models have shown that a combination of methods often provides improved classification results compared to a single-method approach. Within this context, this study explores the combination of different classification methods in developing efficient models for credit risk assessment. A variety of methods are considered in the combination, including machine learning approaches and statistical techniques. The results illustrate that combined models can outperform individual models for credit risk analysis. The analysis also covers important issues such as the impact of using different parameters for the combined models, the effect of attribute selection, as well as the effects of combining strong or weak models.     

基于加速失效时间模型模拟生存时间应用于检测生存性状位点

对于考察预指定情形下的统计模型的性能、性质及适应性,模拟研究是非常重要的统计工具.作为生存分析中两个最受欢迎的模型之一,由于加速失效时间模型中的因变量是生存时间的对数,且此模型能够以线性形式回归带有易解释的参数的协变量,从而加速失效模型比COX比例风险模型更便于拟合生存数据.首先提出了关于带有广义F-分布的加速失效模型的模拟研究中生成生存时间的方法,然后给出了描述加速失效时间模型的误差分布和相应的生存时间之间的一般的关系式,并给出了广义F-分布是如何生成生存时间的.最后,为证实所建议模拟技术的性能和有效性,将此方法应用于检测生存性状位点的模型中.