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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. STAT PROBABIL LETT , 81 (8) 1179 - 1182. 10.1016/j.spl.2011.03.014.

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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 (MAD(p)). In addition, such equivalences give the MAD and MAD(p) simple graphical interpretations. Some other practical implications are also briefly discussed. (C) 2011 Elsevier B.V. All rights reserved.

Type:Article
Title:The p-folded cumulative distribution function and the mean absolute deviation from the p-quantile
DOI:10.1016/j.spl.2011.03.014
Keywords:Cumulative distribution function (CDF), Folded CDF, Mean absolute deviation (MAD) from the median, COHERENT MEASURES, RISK, AGREEMENT
UCL classification:UCL > School of BEAMS > Faculty of Maths and Physical Sciences > Statistical Science

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