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The p-folded cumulative distribution function and the mean absolute deviation from the p-quantile

Xue, JH; Titterington, DM; (2011) The p-folded cumulative distribution function and the mean absolute deviation from the p-quantile. Statistics & Probability Letters , 81 (8) 1179 - 1182. 10.1016/j.spl.2011.03.014. Green open access

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Abstract

The aims of this short note are two-fold. First, it shows that, for a random variable X, the area under the curve of its folded cumulative distribution function equals the mean absolute deviation (MAD) from the median. Such an equivalence implies that the MAD is the area between the cumulative distribution function (CDF) of X and that for a degenerate distribution which takes the median as the only value. Secondly, it generalises the folded CDF to a p-folded CDF, and derives the equivalence between the area under the curve of the p-folded CDF and the weighted mean absolute deviation from the p-quantile (MADp). In addition, such equivalences give the MAD and MADp simple graphical interpretations. Some other practical implications are also briefly discussed.

Type: Article
Title: The p-folded cumulative distribution function and the mean absolute deviation from the p-quantile
Open access status: An open access version is available from UCL Discovery
DOI: 10.1016/j.spl.2011.03.014
Publisher version: https://doi.org/10.1016/j.spl.2011.03.014
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Cumulative distribution function (CDF), Folded CDF, Mean absolute deviation (MAD) from the median
UCL classification: UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Statistical Science
URI: https://discovery.ucl.ac.uk/id/eprint/1316866
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