A distribution-free alternative to least-squares regression and its application to Rb/Sr isochron calculations |
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| Authors: | Raymond G Vugrinovich |
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| Institution: | (1) Geological Survey Division, P.O. Box 30028, 48909 Lansing, Michigan, USA |
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| Abstract: | A distribution-free estimator of the slope of a regression line is introduced. This estimator is designated Sm and is given by the median of the set of n(n – 1)/2 slope estimators, which may be calculated by inserting pairs of points (X
i, Yi)and (X
j, Yj)into the slope formula S
i = (Y
i – Yj)/(X
i – Xj),1  i < j n Once S
m is determined, outliers may be detected by calculating the  residuals given by Ri = Yi – SmXi where 1 i n, and chosing the median Rm. Outliers are defined as points for which |Ri – Rm| > k (median {|R
i – Rm|}). If no outliers are found, the Y-intercept is given by Rm. Confidence limits on Rm and Sm can be found from the sets of Ri and Si, respectively. The distribution-free estimators are compared with the least-squares estimators now in use by utilizing published data. Differences between the least-squares and distribution-free estimates are discussed, as are the drawbacks of the distribution-free techniques. |
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| Keywords: | Distribution-free estimator regression slope errors-in-variables Rb/Sr isochrons |
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