Collaborative Research: On Semi-Parametric Mixed Poisson Regression Models

在科学数据的统计分析中，一个基本问题是为所谓的计数数据的分析开发模型，计数数据是指在给定的时间内发生的某种类型的事件的数量。这方面的例子包括在一段特定时期内涉及特定类型船舶损坏的事件数量，职业罪犯一生中每年犯下的罪行数量，以及在一个经济体中每年成立的特定类型公司数量。为了解释增加或减少此类事件发生率的因素，通常会制定泊松或泊松家族的回归模型（其中后者由泊松回归的混合物组成）。最常用的混合泊松回归模型——它提供了比基本泊松模型更大的灵活性——是基于参数混合分布的规范，如伽马分布。一种新的混合泊松回归方法——基于混合分布的非参数规范和模型参数的半参数最大似然估计——已经被主要研究者定义和探索。该项目将系统地完善、重新检查，并更充分地开发估算、测试和模型比较的统计程序，用于这类新模型。将进行计算机模拟研究，以检查相对较小样本的统计估计器的性质。将分析来自不同主题的四个原型数据集，以演示技术并与现有分析进行比较。将进一步开发用于这些技术的经验应用的计算机软件。由于这些新的统计模型在建模事件计数数据方面比传统模型提供了更大的灵活性，因此将要开发的方法可以改进跨各种科学学科的数据分析。

A basic problem in the statistical analysis of scientific data consists of developing models for the analysis of so-called count data -- data consisting of the numbers of events of some type occuring over a given period of time. Examples include the number of incidents involving damage to ships of a specified type over a given time period, the number of crimes committed per year in the life course of a career criminal, and the number of firms of a given type founded per year in an economy. To account for factors that increase or decrease the rates of occurrence of such events, regression models of the Poisson or Poisson family (where the latter consists of mixtures of Poisson regressions) typically are formulated. The most commonly used mixed Poisson regression models -- which offer considerably more flexibility than the basic Poisson model -- are based on the specification of a parametric mixing distribution such as the gamma distribution. A new methodology for mixed Poisson regressions -- based on a nonparametric specification of the mixing distribution and the semiparametric maximum likelihood estimation of the model parameters -- has been defined and explored by the Principal Investigators. This project will systematically refine, reexamine, and more fully develop statistical procedures of estimation, testing, and model comparison for this new class of models. A computer simulation study will be conducted to examine the properties of the statistical estimtors for relatively small samples. Four prototypical datasets from diverse subject matters will be analyzed to demonstrate the techniques and to make comparisons with existing analyses. Computer software for the empirical application of the techniques will be further developed. Because these new statistical models provide more flexibility in modeling event count data than conventional models, the methods to be developed could improve data analyses across a variety of scientific disciplines.