Connecting model-based and model-free approaches to linear least squares regression
Abstract
In a regression setting with response vector $\mathbf{y} \in \mathbb{R}^n$ and given regressors $\mathbf{x}_1,\ldots,\mathbf{x}_p \in \mathbb{R}^n$, a typical question is to what extent $\mathbf{y}$ is related to these regressors, specifically, how well can $\mathbf{y}$ be approximated by a linear combination of them. Classical methods for this question are based on statistical models for the conditional distribution of $\mathbf{y}$, given the regressors $\mathbf{x}_j$. In the present paper it is shown that various p-values resulting from this model-based approach have also a purely data-analytic, model-free interpretation. This finding is derived in a rather general context. In addition, we introduce equivalence regions, a reinterpretation of confidence regions in the model-free context.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2018
- DOI:
- 10.48550/arXiv.1807.09633
- arXiv:
- arXiv:1807.09633
- Bibcode:
- 2018arXiv180709633D
- Keywords:
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- Mathematics - Statistics Theory;
- 62J05