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On a non-parametric confidence interval for the regression slope. (English) Zbl 1392.62211

Summary: We investigate an application of the Tukey’s methodology in H. Theil’s regression [Proc. Akad. Wet. Amsterdam 53, 386–392 (1950; Zbl 0036.21601); ibid. 53, 521–525 (1950; Zbl 0036.21602); ibid. 53, 1397–1412 (1950; Zbl 0038.29504)] to obtain a confidence interval for the true slope in the straight line regression model with not necessarily normal errors. This specific approach is implemented since 2005 in an R package; however, without any theoretical background. We illustrate by Monte Carlo, that this methodology, unlike the classical Theil’s approach, seriously deflates the true confidence level of the resulting interval. We provide also rigorous proofs in case of four (in general) and five data points (under some additional conditions); together with a real life usage example in the latter case. Summing up, we demonstrate that one should never combine statistical methods without checking the assumptions of their usage and we also give a warning to the already wide community of R users of Theil’s regression from various fields of science.

MSC:

62J05 Linear regression; mixed models
62G15 Nonparametric tolerance and confidence regions
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