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，以此作为预测风力发电能力的一种手段。水文学家可能希望根据对平均雨天数的估计来预测未来的河峰。公共卫生官员可能想要评估在大规模免疫等重大医疗干预之后，流感病例数量是否随着时间的推移而呈下降趋势。对时间序列动态进行统计建模的一个目标是预测该序列的未来值。根据对平均雨天天数的估计对河峰进行预报就是后一目标的一个例子。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，同时根据数据的相依性质进行调整。估计干预后疾病平均病例数的趋势就是前一个目标的一个例子；这种方法被称为回归建模。本研究项目主要涉及离散数据的时间序列回归建模。研究了常见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值时，并不是所有的变量在解释平均值时都是完全相关的。我将研究一种称为“收缩估计”的“大数据”技术，它结合了部分相关变量的信息，以期提高回归模型的预测能力。*统计时间序列数据的典型统计模型对产生数据的数据生成机制做出假设；这些假设在实践中很难验证。在我的研究计划中，我放松了对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的建模框架进行模型参数估计，因为它为离散时间序列估计提供了一种统一的方法。估计函数理论框架包含了许多为解释整数值时间序列的动力学而提出的统计模型。此外，与对数据生成机制要求更严格的假设的一些方法相比，该方法很容易实现。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性较低的方法学和现有方法学得出的估计。如果两种方法的结果非常不同，那么方法的最终用户将能够使用该研究计划的结果作为终点，对他们的统计结论充满信心。

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. **