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无模型框架下线性回归的子模型一致界AI 解读

For the last two decades, high-dimensional data and methods have proliferated throughout the literature. Yet, the classical technique of linear regression has not lost its usefulness in applications. In fact, many high-dimensional estimation techniques can be seen as variable selection that leads to a smaller set of variables (a “submodel”) where classical linear regression applies. We analyze linear regression estimators resulting from model selection by proving estimation…

半监督线性回归AI 解读

We study a regression problem where for some part of the data we observe both the label variable (Y) and the predictors (X), while for other part of the data only the predictors are given. Such a problem arises, for example, when observations of the label variable are costly and may require a skilled human agent. When the conditional expectation E[Y|X] is not exactly linear, one can consider the best linear approximation to the conditional expectation, which can be estimated…

Ogburn 等对《广义线性模型参数的弱假设推断》的讨论AI 解读

Not all conditional associations between outcomes and exposures are of interest. Those that are tend to be directional: up or down. The simplest way to assess directionality is to fit a confounder- adjusted linear exposure term, as the authors propose. We agree with this approach as some of us have argued that linear slopes are meaningful and interpretable even if the directional association is not linear (Buja et al., 2019, section 10). The authors, and Whitney et al.…

Buja 等对“广义线性模型参数的弱假设推断”的讨论AI 解读

We thank the authors for their stimulating article. At the highest level, it agrees with our thinking as expressed in the second of two articles in Statistical Science (Buja et al., 2019). The guiding principle is to focus on quantities of interest that are meaningful statistical functionals, beyond the statistical models that motivate them, even when those models are fully misspecified. In some form, this focus has existed for decades in the semi-parametric literature. We…

Modern data-driven approaches to modeling make extensive use of covariate/model selection. Such selection incurs a cost: it invalidates classical statistical inference. A conservative remedy to the problem was proposed by Berk et al. (Ann. Statist. 41 (2013) 802–837) and further extended by Bachoc, Preinerstorfer and Steinberger (2016). These proposals, labeled “PoSI methods,” provide valid inference after arbitrary model selection. They are computationally NP-hard and have…

有效的模型选择后推断AI 解读

It is common practice in statistical data analysis to perform data-driven variable selection and derive statistical inference from the resulting model. Such inference enjoys none of the guarantees that classical statistical theory provides for tests and confidence intervals when the model has been chosen a priori. We propose to produce valid “post-selection inference” by reducing the problem to one of simultaneous inference and hence suitably widening conventional confidence…

利用双向正则化奇异值分解分析双向函数型数据AI 解读

Two-way functional data consist of a data matrix whose row and column domains are both structured, for example, temporally or spatially, as when the data are time series collected at different locations in space. We extend one-way functional principal component analysis (PCA) to two-way functional data by introducing regularization of both left and right singular vectors in the singular value decomposition (SVD) of the data matrix. We focus on a penalization approach and…

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用最小加性主成分分析加性依赖与曲线共线性AI 解读

Additive principal components are a nonlinear generalization of linear principal components. Their distinguishing feature is that linear forms $\sum_ia_i X_i$ are replaced with additive functions $\sum_i\phi_i(X_i)$. A considerable amount of flexibility for fitting data is gained when linear methods are replaced with additive ones. Our interest is in the smallest principal components, which is somewhat uncommon. Smallest additive principal components amount to data…

关于函数典型变量、交替最小二乘法与 ACE 的讨论AI 解读

We discuss properties of some data-analytic methods which are intimately related to each other: alternating least squares (ALS), correspondence analysis and more recently Breiman and Friedman's ACE algorithm. The application of these methods to regression produces nonparametric estimators of nonlinear transformations, both of the response and the predictors. These procedures are among the most powerful tools for data analysis, but missing awareness of some artifacts could…

线性平滑器与加性模型AI 解读

We study linear smoothers and their use in building nonparametric regression models. In the first part of this paper we examine certain aspects of linear smoothers for scatterplots; examples of these are the running-mean and running-line, kernel and cubic spline smoothers. The eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother, and several other topics such as the number of degrees of freedom of a…