Bell numbers meet credit risk: new cure rate models capture borrowers who never default

Every survival model carries a quiet assumption: if you follow a subject long enough, the event of interest will eventually happen. In medicine, that means every patient’s disease will recur. In finance, it means every borrower will eventually default. A team of statisticians from Kwame Nkrumah University of Science and Technology in Ghana, the University of Technology and Applied Sciences in Navrongo, and Howard University in Washington, DC, argues that this assumption quietly distorts credit risk analysis, because a substantial share of borrowers will never default at all, no matter how long the loan is observed. In a study published in Frontiers in Applied Mathematics and Statistics, they propose a new family of mixture cure rate models built on the Bell-generated class of distributions, and show that it outperforms established competitors when applied to real loan default data.

Cure rate models were born in oncology. Boag introduced the idea in 1949 to estimate the proportion of cancer patients cured by therapy, and Berkson and Gage formalized the framework in 1952 by adding the fraction of non-susceptible subjects directly to the survival function. The mixture cure model splits the population into two groups: a susceptible group whose default times follow a conventional lifetime distribution, and a cured group that is not merely right-censored but genuinely immune to the event. In credit risk, this structure is natural. Some loans are repaid in full and the borrowers are, for modeling purposes, long-term survivors who will never trigger a default event during the life of the contract.

The innovation in the new study lies in the choice of the baseline distribution for the susceptible group. Rather than relying on classical lifetime distributions or their conventional extensions, the researchers wrapped four well-known distributions—the Weibull, Burr XII, log-logistic, and lognormal—inside the Bell-generated family, a construction introduced by Alsadat and colleagues in 2023. The Bell distribution, named after the mathematician Eric Temple Bell, whose 1934 paper on exponential numbers gave the world the Bell numbers, is particularly good at handling overdispersion, a common feature of real-world data in which variance exceeds what a standard model predicts. The Bell-G mechanism adds extra shape parameters to the baseline distribution, giving the resulting models the flexibility to represent highly skewed data, heavy tails, and non-monotonic hazard functions that conventional models often miss.

Technically, the construction works by compounding. If G(t) and g(t) denote the cumulative distribution and probability density functions of any continuous baseline random variable, the Bell-G family compounds them through the Bell numbers, producing new survival, density, and hazard rate functions with an additional rate parameter. The researchers derived four special submodels: the Bell-Weibull mixture cure rate model, the Bell-Burr XII mixture cure rate model, the Bell-log-logistic mixture cure rate model, and the Bell-lognormal mixture cure rate model. Plots of these functions across different parameter values show survival curves that plateau above zero, confirming the presence of a cured proportion, while the density functions display right-skewed, left-skewed, and approximately symmetric shapes. The hazard rate functions exhibit increasing, decreasing, and modified upside-down bathtub patterns, indicating that the models can accommodate heterogeneous and time-varying risk dynamics.

The team also proved two important theoretical properties. First, the tail behavior of the mixture cure survival function converges to the cure parameter as time increases beyond bounds, which means the long-run survival probability is governed by the fraction of non-susceptible subjects. Second, the Bell-G family is identifiable: different parameter values always produce different distributions, so the model cannot be confused with itself under reparameterization. These properties matter for practical estimation, because they ensure that the maximum likelihood procedure targets a unique distribution and that the estimated cure fraction has a clear interpretation.

Parameter estimation was carried out using the maximum likelihood method, with the log-likelihood function constructed from observed pairs of default times and censoring indicators. Covariate effects on the cure fraction were incorporated through a logistic link function, allowing borrower characteristics to influence the probability of never defaulting. To assess the finite-sample performance of the estimators, the researchers ran Monte Carlo simulations with sample sizes ranging from 50 to 800, repeated 1,000 times for cure percentages of 10 and 8 percent. The results showed that all parameters were consistent: absolute bias and root-mean-squared error decreased as sample size increased, and the average estimates converged to the true parameter values. For the Bell-Weibull model with true cure parameter 0.1, the absolute bias of the cure estimate fell from 0.035 at n=50 to 0.009 at n=800.

The empirical analysis used two real credit default datasets drawn from a lending dataset spanning January 2011 to December 2015, covering loans of $5,000 or more with 60-month repayment periods. A total of 1,476 medical care loans and 53,820 credit card loans were analyzed. The summary statistics revealed heavy right-skewness and heavy tails, with skewness of 1.28 and 1.29 and kurtosis of 3.99 and 5.19 for the medical and credit card datasets respectively. Default rates were 8.06 percent for medical loans and 4.64 percent for credit card loans, while about 91 and 95 percent of observations were right-censored. Kaplan-Meier curves for both datasets reached a plateau well above zero, providing visual evidence of a non-susceptible fraction of borrowers.

Model selection was judged by the Akaike information criterion, the Bayesian information criterion, and visual comparison of fitted survival curves against Kaplan-Meier curves. The proposed Bell-G cure rate models were compared against four established competitors: the Odds Burr XII-Logistic mixture cure rate model, the Weibull Exponentiated Exponential mixture cure rate model, the Log-exponentiated-Weibull mixture cure rate model, and the Exponentiated Weibull Exponential mixture cure rate model. For the medical loan dataset, the Bell-lognormal model achieved the lowest AIC at 1672.020, a reduction of 4.84 units compared to the best competitor, while the other three proposed models each reduced AIC by more than 10 units relative to their competitors. For the credit card dataset, the Bell-Weibull model gave the lowest criteria values, with an AIC of 37291.04, and the proposed family reduced AIC by more than 10 units on average compared to the alternatives.

The researchers also fitted regression versions of the models to the medical loan data, using interest rate, loan amount, and annual income as covariates. Across all fitted regression models, only interest rate was statistically significant at the 5 percent level, with loan amount highly insignificant and annual income marginally so. The coefficients for interest rate and loan amount were negative, meaning that a one-unit increase in either reduces the log odds of the probability of non-default, while the coefficient for annual income was positive. The authors interpret this pattern by noting that interest rate acts as a composite borrower risk signal, set by lenders based on loan amount, income, and overall creditworthiness, so its marginal explanatory power absorbs the effects of the other covariates. The Bell-lognormal regression model provided the most accurate description of the medical loan data, recording the lowest AIC and BIC among all fitted regression models.

The study concludes that the Bell-generated family offers a competitive and flexible alternative for modeling survival data with a cured fraction, with practical implications for lenders who can better identify long-term non-defaulters and improve credit scoring and risk-adjusted pricing. The authors acknowledge limitations: the additional parameters increase computational complexity, maximum likelihood estimation may encounter numerical difficulties in highly censored datasets, and the empirical validation relied on only two datasets. Future research directions include incorporating additional covariates such as borrower age and gender, exploring Bayesian estimation procedures, and developing diagnostic tools for model adequacy, with potential applications extending beyond credit risk to reliability analysis and biomedical survival studies where long-term survivors are commonly observed.

Subject of Research: Statistics and Probability

Article Title: Bell-generated family of mixture cure rate models with application to credit default

Article References: Asabre, E., Nasiru, S., Frempong, N. K., Barley, K., & Appiah, S. K. (2026). Bell-generated family of mixture cure rate models with application to credit default. Frontiers in Applied Mathematics and Statistics, 12, Article 1873298. https://doi.org/10.3389/fams.2026.1873298

Image Credits: AI Generated

DOI: 10.3389/fams.2026.1873298

Keywords: mixture cure rate models, Bell-generated distributions, credit risk, survival analysis, maximum likelihood estimation, time-to-default, Monte Carlo simulation, logistic link function, Bell-generated, family, mixture, cure