Estimation and Inference for Generalized Geoadditive Models

In many application areas, data are collected on a count or binary response with spatial covariate information. In this article, we introduce a new class of generalized geoadditive models (GGAMs) for spatial data distributed over complex domains. Through a link function, the proposed GGAM assumes that the mean of the discrete response variable depends on additive univariate functions of explanatory variables and a bivariate function to adjust for the spatial effect. We propose a two-stage approach for estimating and making inferences of the components in the GGAM. In the first stage, the univariate components and the geographical component in the model are approximated via univariate polynomial splines and bivariate penalized splines over triangulation, respectively. In the second stage, local polynomial smoothing is applied to the cleaned univariate data to average out the variation of the first-stage estimators. We investigate the consistency of the proposed estimators and the asymptotic normality of the univariate components. We also establish the simultaneous confidence band for each of the univariate components. The performance of the proposed method is evaluated by two simulation studies. We apply the proposed method to analyze the crash counts data in the Tampa-St. Petersburg urbanized area in Florida. Supplementary materials for this article are available online.

在诸多应用场景中，研究人员常会采集带有空间协变量信息的计数型或二分类响应数据。本文提出一类面向复杂区域空间数据的广义地理可加模型（Generalized Geoadditive Models, GGAMs）。该模型通过连接函数，设定离散响应变量的均值由解释变量的可加单变量函数，与用于校正空间效应的双变量函数共同决定。针对该模型各分量的估计与统计推断问题，本文提出两阶段估计方法：第一阶段中，模型的单变量分量与地理分量分别通过单变量多项式样条与三角剖分下的双变量惩罚样条进行近似；第二阶段则对经清洗的单变量数据施加局部多项式平滑，以平均消除第一阶段估计量的随机变异。本文推导并证明了所提估计量的相合性，以及单变量分量的渐近正态性；此外，还为各单变量分量构建了联合置信带。本文通过两项模拟实验评估了所提方法的性能表现，并将其应用于美国佛罗里达州坦帕-圣彼得堡都市区的碰撞计数数据开展实证分析。本文的补充材料可在线获取。