Semiparametric Inference for Integer-Valued Time Series

离散时间序列出现在各种各样的设置中。能源预报员可能希望了解每月平均刮风天数随时间的变化规律，作为预测风力的一种手段。水文学家可能希望根据对平均阴雨天数的估计来预测未来的河峰。公共卫生官员可能希望评估，在大规模免疫接种等重大医疗干预措施后，流感病例数量是否会随着时间的推移出现下降趋势。时间序列动力学的统计建模的一个目标是预测该序列的未来值。根据估计的平均下雨天数来预测河峰是后一种目标的一个例子。然而，另一个目标是了解时间序列的典型值与一组解释变量之间的关系，同时调整数据的依赖性。估计干预后疾病平均病例数的趋势是前一个目标的一个例子；这种方法被称为回归建模。本研究计划主要关注离散数据的时间序列回归建模。研究了通常的时间序列回归建模问题以及与“大数据”相关的回归问题。当收集到大量的预测因子时，并不是所有的变量都能完全解释平均计数。我将研究一种被称为“收缩估计”的“大数据”技术，它结合了来自部分相关变量的信息，以提高回归模型的预测能力。典型的计数时间序列数据统计模型对产生数据的数据生成机制进行了假设；这些假设在实践中很难得到证实。在我的研究计划中，我放松了关于离散值时间序列数据如何产生的假设，并研究了这种回归建模方法的优点和缺点。我采用估计函数理论建模框架进行模型参数估计，因为它为离散值时间序列估计提供了统一的方法。估计函数理论框架包含了许多用于解释整数值时间序列动力学的统计模型。此外，与需要对数据生成机制进行更严格假设的某些方法相比，该方法易于实现。这项研究计划的结果将使研究人员能够比较和对比从限制较少的方法和现有方法中得出的估计。如果两种方法的结果非常不同，那么该方法的最终用户将能够使用该研究计划的结果作为对其统计结论有信心的终点。**

Discrete time series appear in a wide variety of settings. Energy forecasters may wish to understand patterns in the average number of windy days per month over time as a means to predicting wind power. Hydrologists may wish to forecast future river crests based on estimates of average number of rainy days. Public health officials may want to assess whether there is a declining trend in the number of cases of influenza over time after a major medical intervention such as mass immunizations. ***One objective of statistical modeling of the dynamics of a time series is to predict future values of the series. Forecasting of river crests based on estimates of mean number of rainy days is an example of the latter objective. Yet another objective is to understand the relationship between typical values of the time series and a set of explanatory variables while adjusting for the dependent nature of the data. Estimating the trend in the average number of cases of a disease after an intervention is an example of the former objective; this approach is known as regression modeling. This research program is primarily concerned with time series regression modeling for discrete data. Usual time series regression modeling problems are studied as well as regression problems pertaining to "big data". When a large number of predictors are collected, not all variables will be fully relevant in explaining the mean counts. I will examine a "big data" technique known as "shrinkage estimation" that incorporates information from variables with partial relevance with a view towards improving prediction power of the regression model.***Typical statistical models for count time series data make assumptions about the data generating mechanism giving rise to the data; these assumptions are in practice difficult to verify. In my research program, I relax assumptions about how discrete-valued time series data arise and study the advantages as well as the disadvantages from such a regression modeling approach. I adopt an estimating function theory modeling framework for model parameter estimation as it provides a unifying approach to discrete-valued time series estimation. The estimating function theory framework encompasses many of the statistical models proposed for explaining the dynamics for integer-valued time series. Furthermore, the methodology is easy to implement in contrast to some of the methods that require more stringent assumptions about the data generating mechanism. The findings from this research program will allow researchers to compare and contrast estimates derived from both the less restrictive methodology with that of existing methodology. If the results from both methodologies are very different, then the end users of the methodology will be able to use the results from this research program as an endpoint with confidence in their statistical conclusions. **