基于整合治愈率模型的信贷违约时点预测

传统信用评分方法主要利用统计分类方法,只能预测借款人是否会发生违约,但不能预测违约发生的时点。治愈率模型是二分类和生存分析的混合模型,不仅可以预测是否会发生违约,而且可以预测违约发生的时点,比传统二分类方法可以提供更多的信息。另外,随着大数据的发展,数据源越来越多,针对相同或者相似任务,可以收集到多个数据集,本文提出了融合多源数据的整合治愈率模型,可以对多个数据集同时建模和估计参数,通过复合惩罚函数进行组间和组内双层变量选择,并通过促进两个子模型回归系数符号相同,提高模型的可解释性。通过数值模拟发现,所提方法在变量选择和参数估计上均有明显优势。最后,将所提方法应用于信用贷款的违约时点预测中,模型表现良好。     

Variable selection in proportional hazards cure model with time-varying covariates,application to US bank failures*

From a survival analysis perspective, bank failure data are often characterized by small default rates and heavy censoring. This empirical evidence can be explained by the existence of a subpopulation of banks likely immune from bankruptcy. In this regard, we use a mixture cure model to separate the factors with an influence on the susceptibility to default from the ones affecting the survival time of susceptible banks. In this paper, we extend a semi-parametric proportional hazards cure model to time-varying covariates and we propose a variable selection technique based on its penalized likelihood. By means of a simulation study, we show how this technique performs reasonably well. Finally, we illustrate an application to commercial bank failures in the United States over the period 2006–2016.     

On bootstrap testing inference in cure rate models

Survival models deal with the time until the occurrence of an event of interest. However, in some situations the event may not occur in part of the studied population. The fraction of the population that will never experience the event of interest is generally called cure rate. Models that consider this fact (cure rate models) have been extensively studied in the literature. Hypothesis testing on the parameters of these models can be performed based on likelihood ratio, gradient, score or Wald statistics. Critical values of these tests are obtained through approximations that are valid in large samples and may result in size distortion in small or moderate sample sizes. In this sense, this paper proposes bootstrap corrections to the four mentioned tests and bootstrap Bartlett correction for the likelihood ratio statistic in the Weibull promotion time model. Besides, we present an algorithm for bootstrap resampling when the data presents cure fraction and right censoring time (random and non-informative). Simulation studies are conducted to compare the finite sample performances of the corrected tests. The numerical evidence favours the corrected tests we propose. We also present an application in an actual data set.     

The negative binomial–beta Weibull regression model to predict the cure of prostate cancer

In this article, for the first time, we propose the negative binomial–beta Weibull (BW) regression model for studying the recurrence of prostate cancer and to predict the cure fraction for patients with clinically localized prostate cancer treated by open radical prostatectomy. The cure model considers that a fraction of the survivors are cured of the disease. The survival function for the population of patients can be modeled by a cure parametric model using the BW distribution. We derive an explicit expansion for the moments of the recurrence time distribution for the uncured individuals. The proposed distribution can be used to model survival data when the hazard rate function is increasing, decreasing, unimodal and bathtub shaped. Another advantage is that the proposed model includes as special sub-models some of the well-known cure rate models discussed in the literature. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes. We analyze a real data set for localized prostate cancer patients after open radical prostatectomy.     

Cure rate models: A unified approach

The authors propose a novel class of cure rate models for right‐censored failure time data. The class is formulated through a transformation on the unknown population survival function. It includes the mixture cure model and the promotion time cure model as two special cases. The authors propose a general form of the covariate structure which automatically satisfies an inherent parameter constraint and includes the corresponding binomial and exponential covariate structures in the two main formulations of cure models. The proposed class provides a natural link between the mixture and the promotion time cure models, and it offers a wide variety of new modelling structures as well. Within the Bayesian paradigm, a Markov chain Monte Carlo computational scheme is implemented for sampling from the full conditional distributions of the parameters. Model selection is based on the conditional predictive ordinate criterion. The use of the new class of models is illustrated with a set of real data involving a melanoma clinical trial.     

The pricing of defaultable bonds under a regime-switching jump-diffusion model with stochastic default barrier

In this paper, we investigate the price for the zero-coupon defaultable bond under a structural form credit risk with regime switching. We model the value of a firm and the default threshold by two dependent regime-switching jump-diffusion processes, in which the Markov chain represents the states of an economy. The price is associated with the Laplace transform of the first passage time and the expected discounted ratio of the firm value to the default threshold at default. Closed-form results used for calculating the price are derived when the jump sizes follow a regime-switching double exponential distribution. We present some numerical results for the price of the zero-coupon defaultable bond via Gaver-Stehfest algorithm.     

Assessing the prediction accuracy of cure in the Cox proportional hazards cure model: an application to breast cancer data

A cure rate model is a survival model incorporating the cure rate with the assumption that the population contains both uncured and cured individuals. It is a powerful statistical tool for prognostic studies, especially in cancer. The cure rate is important for making treatment decisions in clinical practice. The proportional hazards (PH) cure model can predict the cure rate for each patient. This contains a logistic regression component for the cure rate and a Cox regression component to estimate the hazard for uncured patients. A measure for quantifying the predictive accuracy of the cure rate estimated by the Cox PH cure model is required, as there has been a lack of previous research in this area. We used the Cox PH cure model for the breast cancer data; however, the area under the receiver operating characteristic curve (AUC) could not be estimated because many patients were censored. In this study, we used imputation‐based AUCs to assess the predictive accuracy of the cure rate from the PH cure model. We examined the precision of these AUCs using simulation studies. The results demonstrated that the imputation‐based AUCs were estimable and their biases were negligibly small in many cases, although ordinary AUC could not be estimated. Additionally, we introduced the bias‐correction method of imputation‐based AUCs and found that the bias‐corrected estimate successfully compensated the overestimation in the simulation studies. We also illustrated the estimation of the imputation‐based AUCs using breast cancer data. Copyright © 2014 John Wiley & Sons, Ltd.     

Bayesian Semiparametric Cure Rate Model with an Unknown Threshold

Abstract.  We propose a Bayesian semiparametric model for survival data with a cure fraction. We explicitly consider a finite cure time in the model, which allows us to separate the cured and the uncured populations. We take a mixture prior of a Markov gamma process and a point mass at zero to model the baseline hazard rate function of the entire population. We focus on estimating the cure threshold after which subjects are considered cured. We can incorporate covariates through a structure similar to the proportional hazards model and allow the cure threshold also to depend on the covariates. For illustration, we undertake simulation studies and a full Bayesian analysis of a bone marrow transplant data set.     

Establishing decision tree-based short-term default credit risk assessment models

ABSTRACT

Traditional credit risk assessment models do not consider the time factor; they only think of whether a customer will default, but not the when to default. The result cannot provide a manager to make the profit-maximum decision. Actually, even if a customer defaults, the financial institution still can gain profit in some conditions. Nowadays, most research applied the Cox proportional hazards model into their credit scoring models, predicting the time when a customer is most likely to default, to solve the credit risk assessment problem. However, in order to fully utilize the fully dynamic capability of the Cox proportional hazards model, time-varying macroeconomic variables are required which involve more advanced data collection. Since short-term default cases are the ones that bring a great loss for a financial institution, instead of predicting when a loan will default, a loan manager is more interested in identifying those applications which may default within a short period of time when approving loan applications. This paper proposes a decision tree-based short-term default credit risk assessment model to assess the credit risk. The goal is to use the decision tree to filter the short-term default to produce a highly accurate model that could distinguish default lending. This paper integrates bootstrap aggregating (Bagging) with a synthetic minority over-sampling technique (SMOTE) into the credit risk model to improve the decision tree stability and its performance on unbalanced data. Finally, a real case of small and medium enterprise loan data that has been drawn from a local financial institution located in Taiwan is presented to further illustrate the proposed approach. After comparing the result that was obtained from the proposed approach with the logistic regression and Cox proportional hazards models, it was found that the classifying recall rate and precision rate of the proposed model was obviously superior to the logistic regression and Cox proportional hazards models.     

Accelerated hazards mixture cure model

We propose a new cure model for survival data with a surviving or cure fraction. The new model is a mixture cure model where the covariate effects on the proportion of cure and the distribution of the failure time of uncured patients are separately modeled. Unlike the existing mixture cure models, the new model allows covariate effects on the failure time distribution of uncured patients to be negligible at time zero and to increase as time goes by. Such a model is particularly useful in some cancer treatments when the treat effect increases gradually from zero, and the existing models usually cannot handle this situation properly. We develop a rank based semiparametric estimation method to obtain the maximum likelihood estimates of the parameters in the model. We compare it with existing models and methods via a simulation study, and apply the model to a breast cancer data set. The numerical studies show that the new model provides a useful addition to the cure model literature.     

Nonparametric versus Parametric Estimation of the Cure Fraction Using Interval Censored Data

This article discusses estimation of the cure rate by means of the bounded cumulative hazard (BCH) model using interval censored data. The parametric and nonparametric estimation methods within the framework of the EM algorithm were employed for cure rate estimation and their results compared. The Turnbull estimator was used in the nonparametric estimation while in parametric method both the exponential and Weibull distributions were considered. We show via simulation that the nonparametric method is a viable alternative to the parametric one when the censoring rate is rapidly increasing.     

Latent cure rate model under repair system and threshold effect

In this paper, we formulate a simple latent cure rate model with repair mechanism for a cell exposed to radiation. This latent approach is a flexible alternative to the models proposed by Klebanov et al. [A stochastic model of radiation carcinogenesis: latent time distributions and their properties. Math Biosci. 1993;18:51–75], Kim et al. [A new threshold regression model for survival data with a cure fraction. Lifetime Data Anal. 2011;17:101–122], and is along the lines of the destructive cure rate model formulated recently by Rodrigues et al. [Destructive weighted Poisson cure rate model. Lifetime Data Anal. 2011b;17:333–346]. A new version of the modified Gompertz model and the promotion cure rate model that takes into account the first passage time of reaching the critical point are discussed, and the estimation of tumor size at detection is then addressed from the Bayesian viewpoint. In addition, a simulation study and an application to real data set illustrate the usefulness of the proposed cure rate model.     

A Bayesian approach to mixture cure models with spatial frailties for population‐based cancer relative survival data

As the treatments of cancer progress, a certain number of cancers are curable if diagnosed early. In population‐based cancer survival studies, cure is said to occur when mortality rate of the cancer patients returns to the same level as that expected for the general cancer‐free population. The estimates of cure fraction are of interest to both cancer patients and health policy makers. Mixture cure models have been widely used because the model is easy to interpret by separating the patients into two distinct groups. Usually parametric models are assumed for the latent distribution for the uncured patients. The estimation of cure fraction from the mixture cure model may be sensitive to misspecification of latent distribution. We propose a Bayesian approach to mixture cure model for population‐based cancer survival data, which can be extended to county‐level cancer survival data. Instead of modeling the latent distribution by a fixed parametric distribution, we use a finite mixture of the union of the lognormal, loglogistic, and Weibull distributions. The parameters are estimated using the Markov chain Monte Carlo method. Simulation study shows that the Bayesian method using a finite mixture latent distribution provides robust inference of parameter estimates. The proposed Bayesian method is applied to relative survival data for colon cancer patients from the Surveillance, Epidemiology, and End Results (SEER) Program to estimate the cure fractions. The Canadian Journal of Statistics 40: 40–54; 2012 © 2012 Statistical Society of Canada     

Standard Exponential Cure Rate Model with Noninformative or Informative Uniform-Exponential Censoring

In survival data analysis it is frequent the occurrence of a significant amount of censoring to the right indicating that there may be a proportion of individuals in the study for which the event of interest will never happen. This fact is not considered by the ordinary survival theory. Consequently, the survival models with a cure fraction have been receiving a lot of attention in the recent years. In this article, we consider the standard mixture cure rate model where a fraction p ₀ of the population is of individuals cured or immune and the remaining 1 ? p ₀ are not cured. We assume an exponential distribution for the survival time and an uniform-exponential for the censoring time. In a simulation study, the impact caused by the informative uniform-exponential censoring on the coverage probabilities and lengths of asymptotic confidence intervals is analyzed by using the Fisher information and observed information matrices.     

The Poisson Generalized Linear Failure Rate Model

We formulate a new cure rate survival model by assuming that the number of competing causes of the event of interest has the Poisson distribution, and the time to this event has the generalized linear failure rate distribution. A new distribution to analyze lifetime data is defined from the proposed cure rate model, and its quantile function as well as a general expansion for the moments is derived. We estimate the parameters of the model with cure rate in the presence of covariates for censored observations using maximum likelihood and derive the observed information matrix. We obtain 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 proposed cure rate survival model is illustrated in an application to real data.     

Correlated defaults,temporal correlation,expert information and predictability of default rates

ABSTRACT

Dependence among defaults both across assets and over time is an important characteristic of financial risk. A Bayesian approach to default rate estimation is proposed and illustrated using prior distributions assessed from an experienced industry expert. Two extensions of the binomial model are proposed. The first allows correlated defaults yet remains consistent with Basel II’s asymptotic single-factor model. The second adds temporal correlation in default rates through autocorrelation in the systemic factor. Implications for the predictability of default rates are considered. The single-factor model generates more forecast uncertainty than does the parameter uncertainty. A robustness exercise illustrates that the correlation indicated by the data is much smaller than that specified in the Basel II regulations.     

A flexible model for survival data with a cure rate: a Bayesian approach

In this paper we deal with a Bayesian analysis for right-censored survival data suitable for populations with a cure rate. We consider a cure rate model based on the negative binomial distribution, encompassing as a special case the promotion time cure model. Bayesian analysis is based on Markov chain Monte Carlo (MCMC) methods. We also present some discussion on model selection and an illustration with a real data set.     

A general hazard model for lifetime data in the presence of cure rate

Historically, the cure rate model has been used for modeling time-to-event data within which a significant proportion of patients are assumed to be cured of illnesses, including breast cancer, non-Hodgkin lymphoma, leukemia, prostate cancer, melanoma, and head and neck cancer. Perhaps the most popular type of cure rate model is the mixture model introduced by Berkson and Gage [1]. In this model, it is assumed that a certain proportion of the patients are cured, in the sense that they do not present the event of interest during a long period of time and can found to be immune to the cause of failure under study. In this paper, we propose a general hazard model which accommodates comprehensive families of cure rate models as particular cases, including the model proposed by Berkson and Gage. The maximum-likelihood-estimation procedure is discussed. A simulation study analyzes the coverage probabilities of the asymptotic confidence intervals for the parameters. A real data set on children exposed to HIV by vertical transmission illustrates the methodology.     

Integrating phase 2 into phase 3 based on an intermediate endpoint while accounting for a cure proportion—With an application to the design of a clinical trial in acute myeloid leukemia

For a trial with primary endpoint overall survival for a molecule with curative potential, statistical methods that rely on the proportional hazards assumption may underestimate the power and the time to final analysis. We show how a cure proportion model can be used to get the necessary number of events and appropriate timing via simulation. If phase 1 results for the new drug are exceptional and/or the medical need in the target population is high, a phase 3 trial might be initiated after phase 1. Building in a futility interim analysis into such a pivotal trial may mitigate the uncertainty of moving directly to phase 3. However, if cure is possible, overall survival might not be mature enough at the interim to support a futility decision. We propose to base this decision on an intermediate endpoint that is sufficiently associated with survival. Planning for such an interim can be interpreted as making a randomized phase 2 trial a part of the pivotal trial: If stopped at the interim, the trial data would be analyzed, and a decision on a subsequent phase 3 trial would be made. If the trial continues at the interim, then the phase 3 trial is already underway. To select a futility boundary, a mechanistic simulation model that connects the intermediate endpoint and survival is proposed. We illustrate how this approach was used to design a pivotal randomized trial in acute myeloid leukemia and discuss historical data that informed the simulation model and operational challenges when implementing it.     

Bayesian analysis of elapsed times in continuous‐time Markov chains

The authors consider Bayesian analysis for continuous‐time Markov chain models based on a conditional reference prior. For such models, inference of the elapsed time between chain observations depends heavily on the rate of decay of the prior as the elapsed time increases. Moreover, improper priors on the elapsed time may lead to improper posterior distributions. In addition, an infinitesimal rate matrix also characterizes this class of models. Experts often have good prior knowledge about the parameters of this matrix. The authors show that the use of a proper prior for the rate matrix parameters together with the conditional reference prior for the elapsed time yields a proper posterior distribution. The authors also demonstrate that, when compared to analyses based on priors previously proposed in the literature, a Bayesian analysis on the elapsed time based on the conditional reference prior possesses better frequentist properties. The type of prior thus represents a better default prior choice for estimation software.