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Erschienen in:
2018 | OriginalPaper | Buchkapitel
Elemental Estimates, Influence, and Algorithmic Leveraging
verfasst von : K. Knight
Erschienen in: Nonparametric Statistics
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Abstract
It is well-known (Subrahmanyam, Sankhya Ser B 34:355–356, 1972; Mayo and Gray, Am Stat 51:122–129, 1997) that the ordinary least squares estimate can be expressed as a weighted sum of so-called elemental estimates based on subsets of p observations where p is the dimension of parameter vector. The weights can be viewed as a probability distribution on subsets of size p of the predictors {x
i : i = 1, ⋯ , n}. In this contribution, we derive the lower dimensional distributions of this p dimensional distribution and define a measure of potential influence for subsets of observations analogous to the diagonal elements of the “hat” matrix for single observations. This theory is then applied to algorithmic leveraging, which is a method for approximating the ordinary least squares estimates using a particular form of biased subsampling.