Coherent Forecasting and Risk Analysis of Count Processes

计数时间序列出现在日常生活的许多情况下，特别是在经济背景下。计数时间序列可能显示出完全不同的特征，例如，显著的序列依赖性或边缘分布与过度分散或零通货膨胀。这种时间序列的预测对于以下方面是重要的，例如，需求规划、经济发展评估或为可能的极端情况（风险）做好准备。为了在生成预测时考虑计数的整数特征，连贯的预测方案通过精确计算预测分布并随后使用，例如，其中，但是，由于精确预测分布的计算并不总是显而易见的，因此从业者通常基于实值时间序列的方法来生成计数时间序列预测，例如，用高斯阿尔马模型作为近似。由于以这种方式生成的预测通常不会导致非负值和整数值，因此所产生的预测必须在之后进行截断和舍入。关键问题是：这种对真实预测分布的近似有多好，以这种方式生成的预测有多可靠？预期的研究项目是围绕计数过程的连贯预测。其目的是始终比较基于模型的方案和近似方案的一致性预测。对于基于模型的方案，除了具有不同边际分布的INAR模型（例如，具有过度分散和零通货膨胀），并且有限支持情况下的模型也考虑到趋势和季节性。对于所有这些类型的模型，将分析所产生的点和区间预测的质量，也在估计模型参数的情况下。在极端预测分位数的背景下，关于基于计数时间序列的风险分析的更一般的问题变得相关，这是预期研究项目的第二个主要部分。目标是基于预测分位数和由此导出的风险度量来预测风险，总是基于条件预测分布（也在估计不确定性下）。它的目的是评估良好的风险预测，并回答这个问题，风险措施是最适合的风险分析计数时间序列。

Count time series appear in many circumstances of everyday life, and in particular in situations with an economic context. Count time series may show quite different characteristics, e.g., pronounced serial dependencies or marginal distributions with overdispersion or zero inflation. The forecasting of such time series is important for, e.g., the planning of demand, the assessment of economic development, or to prepare for a possible extreme situation (risk).To account for the integer character of counts when generating forecasts, coherent forecasting schemes lead to discrete predictions in a natural way, by computing the predictive distribution exactly and by then using, e.g., quantiles thereof. But since the calculation of the exact predictive distribution is not always obvious, practitioners often generate their count time series forecasts based on methods for real-valued time series, e.g., by using the Gaussian ARMA models as an approximation. Since a prediction generated in this way does generally not lead to non-negative and integer values, the resulting forecasts have to be truncated and rounded afterwards. The essential question is: How good does such kind of approximation of the true predictive distribution work, and how reliable are the forecasts generated in this way?The intended research project is centered around the coherent forecasting of count processes. The aim is to always compare a model-based and an approximate scheme for coherent forecasting. For the model-based scheme, a broad variety of models will be considered, besides INAR models with diverse marginal distributions (e.g., with overdispersion and zero inflation) and models for the case of a finite support also models allowing for trend and seasonality. For all these types of models, the quality of the resulting point and interval forecasts will be analyzed, also in the case of estimated model parameters.In the context of extreme forecasting quantiles, the more general question about a risk analysis based on count time series becomes relevant, which is the second main part of the intended research project. The aim is the prediction of a risk based on forecasting quantiles and on risk measures derived thereof, always based on conditional forecasting distributions (also under estimation uncertainty). It is intended to evaluate the goodness of the risk forecasts and to answer the question, which risk measure is best suited for the risk analysis of count time series.