In many applications, the success probability 𝜋 of a Bernoulli distributed variable 𝑌 is influenced by another variable 𝑋. For example for loans granted, it is necessary to rate debtors in different rating classes, where the probability of default (PD) 𝜋 of 𝑌 is assumed to be homogeneous within and heterogeneous between the rating classes. The PD of a debtor is largely influenced by macroeconomic and individual variables (𝑋). In this work, we study a Bernoulli mixture model for 𝑌, where the success probability of 𝑌 changes systematically at the breakpoints. We focus on cross-sectional data and our main objective is to estimate all 𝑘 breakpoints with 𝑘 either known or unknown and their corresponding success probabilities between each pair of neighbouring breakpoints. To the best of our knowledge, an estimator for estimating multiple breakpoints has not yet been developed in this context. Thus, we develop an approach with a view to closing this research gap. We show that our estimator works for independent and identically distributed (i.i.d.) 𝑋 as well as for a linear one-factor model for 𝑋. A theoretical foundation for this estimator is also presented. In practice, the number of breakpoints 𝑘 is often unknown a priori. As the multiple estimator is based on an iterative procedure, we propose stopping criteria for estimating 𝑘 correctly. We conduct a simulation study in the context of credit rating to demonstrate the performance of the developed estimator. Furthermore, we apply the new estimator on credit risk data from the Sächsische Aufbaubank, the Development Bank of Saxony. To simplify the use of the new estimator, we also develop an R package called MultipleBreakpoints.

Objective: We consider the following problem from credit risk modeling: Our sample (Xi; Yi), 1 < i < n, consists of pairs of variables. The first variable Xi measures the creditworthiness of individual i. The second variable Yi is the default indicator of individual i. It has two states: Yi = 1 indicates a default, Yi = 0 a non-default. A default occurs, if individual i cannot meet its contractual credit obligations, i. e. it cannot pay back its outstandings regularly. In afirst step, our objective is to estimate the threshold between good and bad creditworthiness in the sense of dividing the range of Xi into two rating classes: One class with good creditworthiness and a low probability of default and another class with bad creditworthiness and a high probability of default. Methods: Given observations of individual creditworthiness Xi and defaults Yi, the field of change point analysis provides a natural way to estimate the breakpoint between the rating classes. In order to account for dependency between the observations, the literature proposes a combination of three model classes: These are a breakpoint model, a linear one-factor model for the creditworthiness Xi, and a Bernoulli mixture model for the defaults Yi. We generalize the dependency structure further and use a generalized link between systematic factor and idiosyncratic factor of creditworthiness. So the systematic factor cannot only change the location, but also the form of the distribution of creditworthiness. Results: For the case of two rating classes, we propose several estimators for the breakpoint and for the default probabilities within the rating classes. We prove the strong consistency of these estimators in the given non-i.i.d. framework. The theoretical results are illustrated by a simulation study. Finally, we give an overview of research opportunities.

Objective: We consider the following problem from credit risk modeling: Our sample (Xi; Yi), 1 < i < n, consists of pairs of variables. The first variable Xi measures the creditworthiness of individual i. The second variable Yi is the default indicator of individual i. It has two states: Yi = 1 indicates a default, Yi = 0 a non-default. A default occurs, if individual i cannot meet its contractual credit obligations, i. e. it cannot pay back its outstandings regularly. In afirst step, our objective is to estimate the threshold between good and bad creditworthiness in the sense of dividing the range of Xi into two rating classes: One class with good creditworthiness and a low probability of default and another class with bad creditworthiness and a high probability of default. Methods: Given observations of individual creditworthiness Xi and defaults Yi, the field of change point analysis provides a natural way to estimate the breakpoint between the rating classes. In order to account for dependency between the observations, the literature proposes a combination of three model classes: These are a breakpoint model, a linear one-factor model for the creditworthiness Xi, and a Bernoulli mixture model for the defaults Yi. We generalize the dependency structure further and use a generalized link between systematic factor and idiosyncratic factor of creditworthiness. So the systematic factor cannot only change the location, but also the form of the distribution of creditworthiness. Results: For the case of two rating classes, we propose several estimators for the breakpoint and for the default probabilities within the rating classes. We prove the strong consistency of these estimators in the given non-i.i.d. framework. The theoretical results are illustrated by a simulation study. Finally, we give an overview of research opportunities.