Neural networks offer a versatile, flexible and accurate approach to loss reserving. However, such applications have focused primarily on the (important) problem of fitting accurate central estimates of the outstanding claims. In practice, properties regarding the variability of outstanding claims are equally important (e.g., quantiles for regulatory purposes). In this paper we fill this gap by applying a Mixture Density Network ("MDN") to loss reserving. The approach combines a neural network architecture with a mixture Gaussian distribution to achieve simultaneously an accurate central estimate along with flexible distributional choice. Model fitting is done using a rolling-origin approach. Our approach consistently outperforms the classical over-dispersed model both for central estimates and quantiles of interest, when applied to a wide range of simulated environments of various complexity and specifications. We further extend the MDN approach by proposing two extensions. Firstly, we present a hybrid GLM-MDN approach called "ResMDN". This hybrid approach balances the tractability and ease of understanding of a traditional GLM model on one hand, with the additional accuracy and distributional flexibility provided by the MDN on the other. We show that it can successfully improve the errors of the baseline ccODP, although there is generally a loss of performance when compared to the MDN in the examples we considered. Secondly, we allow for explicit projection constraints, so that actuarial judgement can be directly incorporated in the modelling process. Throughout, we focus on aggregate loss triangles, and show that our methodologies are tractable, and that they out-perform traditional approaches even with relatively limited amounts of data. We use both simulated data -- to validate properties, and real data -- to illustrate and ascertain practicality of the approaches.

翻译：然而,这些应用主要侧重于对未决索赔进行准确的中央估计的(重要的)问题。在实践中,与未决索赔的变异性有关的属性同样重要(例如,用于监管目的的量化);在本文件中,我们通过采用混合密度网络(“MDN”)来填补这一差距,以保留损失。该方法将神经网络结构与混合的Gausian分布结合起来,以同时实现准确的中央估计和灵活的分配选择。模型安装是采用滚动式方法完成的。我们的方法一贯优于传统的超分散式模型,既适用于中央估计,也适用于各种复杂和规格的多种模拟环境。我们进一步扩展了MDN方法以弥补这一差距。首先,我们提出了一种混合的GLMN-MDN方法,称为“ResMDN”。这种混合方法可以平衡传统的GLM模型的可感应力和易懂度,同时采用滚动式方法。我们的方法一贯优于传统的超分散式模型,同时采用传统的超分散式模型,同时在模拟性能上显示我们准确性和分布性能比下的数据。我们一般地将模拟数据比下,我们更精确地展示了精算值数据。