Abstract
The identification of Hawkes-like processes can pose significant challenges. Despite substantial amounts of data, standard estimation methods show significant bias or fail to converge. To overcome these issues, we propose an alternative approach based on an expectation-maximization algorithm, which instrumentalizes the internal branching structure of the process, thus improving convergence behavior. Furthermore, we show that our method provides a tight lower bound for maximum-likelihood estimates. The approach is discussed in the context of a practical application, namely the collection of outstanding unsecured consumer debt.
Highlights
In contrast to the exogenous intensity of an inhomogeneous Poisson point process, the intensity of a Hawkes process is self-exciting: it depends endogenously on the arrival history [1,2]
Any arrival event induces an intensity jump which dissipates through a memory kernel, and this in turn influences the probability of the arrival event
We focus on the identification of a class of linear controlled marked Hawkes processes, where the arrival events include scalar marks and the arrival intensity is regulated by an impulse control
Summary
In contrast to the exogenous intensity of an inhomogeneous Poisson point process, the intensity of a Hawkes process is self-exciting: it depends endogenously on the arrival history [1,2]. We focus on the identification of a class of linear controlled marked Hawkes processes, where the arrival events include scalar marks and the arrival intensity is regulated by an impulse control This class of controlled self-exciting processes was considered by Chehrazi and Weber [7] to predict the repayment behavior of unsecured loans placed in credit collections. Estimation (MLE), may well be asymptotically consistent, the corresponding estimators tend to exhibit a significant bias as soon as the amount of available data is sparse This is the case in many practical applications such as credit collections where a delinquent account over the collection history usually features only a few repayment events. To ameliorate convergence behavior and estimation performance of standard MLE methods, we propose a robust estimation method based on an expectation-maximization (EM) algorithm The latter exploits the branching structure of the process, featuring a primal-dual type approximation. Using a fairly generic setup (in the context of credit collections, to fix ideas), we show that the EM algorithm achieves substantial improvements in convergence behavior and an increased robustness with respect to a broad range of starting values for the parameter vector
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