Models as Approximations II: A Model-Free Theory of Parametric Regression
Models as Approximations II: A Model-Free Theory of Parametric Regression
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DOI：
10.1214/18-sts694
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发表时间：
2019-11-01
影响因子：
5.7
通讯作者：
Zhao, Linda
中科院分区：
文献类型：
--
作者：
Buja, Andreas;Brown, Lawrence;Zhao, Linda
我们开发了一个无模型理论的一般类型的参数回归独立同分布。意见。该理论将参数模型的参数替换为统计泛函，称为“回归泛函”，定义在联合x-y分布的大型非参数类上，而无需假设正确的模型。参数模型被简化为逻辑学，以提出合理的目标函数。回归函数的一个例子是线性方程的斜率向量，通过OLS拟合到很大程度上任意的x-y分布，而不假设线性模型（见第一部分）。更一般地说，回归泛函可以通过最小化目标函数，求解估计方程或特别构造来定义。在这一框架内，有可能实现以下目标：（1）定义回归泛函的“良好规范”的概念，其取代模型的正确规范的概念，（2）提出基于重新加权分布和数据的回归泛函的良好规范诊断，（3）将回归泛函的采样可变性分解为两个源，一个是由于条件响应分布，另一个是由于回归分布与误设相互作用，都是N-1/2阶，（4）将标准误差的插入/三明治估计作为x-y自助估计的极限情况，和（5）提供理论分析，以表明x-y自助标准误差通常可能优于三明治估计。
We develop a model-free theory of general types of parametric regression for i.i.d. observations. The theory replaces the parameters of parametric models with statistical functionals, to be called "regression functionals," defined on large nonparametric classes of joint x-y distributions, without assuming a correct model. Parametric models are reduced to heuristics to suggest plausible objective functions. An example of a regression functional is the vector of slopes of linear equations fitted by OLS to largely arbitrary x-y distributions, without assuming a linear model (see Part I). More generally, regression functionals can be defined by minimizing objective functions, solving estimating equations, or with ad hoc constructions. In this framework, it is possible to achieve the following: (1) define a notion of "wellspecification" for regression functionals that replaces the notion of correct specification of models, (2) propose a well-specification diagnostic for regression functionals based on reweighting distributions and data, (3) decompose sampling variability of regression functionals into two sources, one due to the conditional response distribution and another due to the regressor distribution interacting with misspecification, both of order N-1/2, (4) exhibit plug-in/sandwich estimators of standard error as limit cases of x-y bootstrap estimators, and (5) provide theoretical heuristics to indicate that x-y bootstrap standard errors may generally be preferred over sandwich estimators.