The p-folded Cumulative Distribution Function and the Mean Absolute Deviation from the p-quantile

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1 The -folded Cumultive Distribution Function nd the Men Absolute Devition from the -quntile Jing-Ho Xue,, D. Michel Titterington b Dertment of Sttisticl Science, University College London, London WC1E 6BT, UK b School of Mthemtics nd Sttistics, University of Glsgow, Glsgow G12 8QQ, UK Abstrct The ims of this short note re two-fold. First, it shows tht, for rndom vrible X, the re under the curve of its folded cumultive distribution function equls the men bsolute devition from the medin (MAD). Such n equivlence imlies tht the MAD is the re between the cumultive distribution function (CDF) of X nd tht for degenerte distribution which tkes the medin s the only vlue. Secondly, it generlises the folded CDF to -folded CDF, nd derives the equivlence between the re under the curve of the -folded CDF nd the weighted men bsolute devition from the -quntile (MAD ). In ddition, such equivlences give the MAD nd MAD simle grhicl interrettions. Some other rcticl imlictions re lso briefly discussed. Keywords: Cumultive distribution function (CDF), Folded CDF, Men bsolute devition from the medin (MAD) Introduction The folded cumultive distribution function for rndom vrible cn be esily obtined by folding down the uer hlf of the cumultive distribution function (CDF). It is simle grhicl method for summrising distributions, nd hs been used for the evlution of lbortory ssys, clinicl trils nd qulity control (Monti, 1995; Krouwer nd Monti, 1995). Corresonding uthor. Tel.: ; Fx: Emil ddresses: jingho@stts.ucl.c.uk (Jing-Ho Xue), michel.titterington@gl.c.uk (D. Michel Titterington) Prerint submitted to Sttistics & Probbility Letters Februry 23, 2011

2 The men bsolute devition from the medin (MAD) is obtined by verging the bsolute devitions over oultion from its medin. It is summry sttistic for mesuring the vribility or disersion of distribution. This short note first shows tht the re under the curve of the folded CDF equls the MAD, nd then generlises the folded CDF to -folded CDF nd derives the equivlence between the re under the curve of the -folded CDF nd the weighted men bsolute devition from the -quntile, which hs been used s risk mesure for ortfolio otimistion (Ogryczk nd Ruszczyński, 2002; Ruszczyński nd Vnderbei, 2003). 2. Reltionshi between the folded CDF nd the MAD Consider univrite, continuous rndom vrible X, with robbility density function (PDF) f(x), with CDF F (x) nd with the suort of f(x) being the intervl [, b]. For discrete X, derivtion similr to the one below cn be obtined nd is thus omitted here The theoreticl cse The CDF F (x) is rel-vlued function in the rnge of [0, 1], defined s F (x) x f(y)dy. (1) The folded CDF, denoted by G(x) herefter, is obtined by folding down the uer hlf of the CDF. It is therefore rel-vlued function in the rnge of [0, 1 ], defined by 2 F (x), if F (x) 1 2 G(x), (2) 1 F (x), otherwise. A folded CDF is lso termed mountin lot, in view of its she. The MAD is defined by MAD x m f(x)dx, (3) where m is the medin of the distribution F (x) such tht m f(x)dx m 2 f(x)dx 1 2. (4)

3 29 30 By elementry lgebr nd interchnge of vribles for integrtion, it follows tht the re under the curve of G(x) is G(x)dx m m m y F (x)dx + 1 F (x)}dx m x } f(y)dy dx + m x m } y dx f(y)dy + m m } f(y)dy dx } dx f(y)dy y m f(y)dy. (5) Tht is, the re under the curve of G(x) equls the MAD The emiricl cse Suose tht we hve smle of N observtions from the distribution F (x) nd tht, mong the N observtions, there re n distinct vlues x i } n i1 with corresonding roortions (x i ). Without loss of generlity, let x 1 < x 2 <... < x n. By buse of nottion, we use the sme symbols for F (x), G(x), m, MAD nd their emiricl versions, when there is no mbiguity in the context. The emiricl CDF, F (x), cn be defined s F (x) x i x (x i ). (6) 40 Emiriclly, the medin m is ny oint such tht F (m) 1 2 nd x i m (x i ) 1 2. (7) If m x K nd m x K+1 both stisfy (7) then ny x-vlue such tht x K x x K+1 qulifies to be the smle medin. Otherwise, m is the unique x K for which (7) holds nd in this cse both inequlities re strict; this rgument includes the cse in which ll the N observtions re distinct. 3

4 45 Hence, the re under the curve of G(x) cn be exressed s K 1 i1 G(x i )(x i+1 x i )} + G(x K )(m x K ) + G(m)(x K+1 m) + K 1 i1 n 1 ik+1 G(x i )(x i+1 x i )} F (x i )(x i+1 x i )} + F (x K )(m x K ) + 1 F (m)} (x K+1 m) + n 1 ik+1 [1 F (x i )} (x i+1 x i )]. (8) If we substitute eqution (6) into eqution (8), the re becomes } K 1 i K (x i+1 x i ) (x j ) + (m x K ) (x j ) i1 + (x K+1 m) + jk+1 j1 (x j ) + n 1 ik+1 j1 (x i+1 x i ) ji+1 K (m x K + x K x K x j+1 x j )(x j )} j1 jk+1 (x j ) (x K+1 m + x K+2 x K x j x j 1 )(x j )} K (m x j )(x j )} + j1 jk+1 (x j m)(x j )} x j m (x j )}. (9) j1 As the MAD cn be defined s MAD x i m (x i )}, (10) i1 eqution (9) shows tht the re under the curve of G(x) equls the MAD. } 4

5 Furthermore, equtions (5) nd (9) suggest tht the MAD is the re, or mesure of bsolute difference, between F (x) nd the CDF for degenerte distribution which tkes the medin m s the only vlue Generlistions to the -folded CDF nd the MAD The folded CDF cn be generlised to -folded CDF, denoted by G (x) herefter nd given by F (x), if F (x), G (x) (11) 1 F (x), otherwise, where (0, 1). Similrly, the MAD cn lso be generlised to men bsolute devition from the -quntile, denoted by MAD herefter nd given by MAD x m f(x)dx, (12) where, for (0, 1), m F 1 () is the -quntile. Then, s imlied by eqution (5), the -folded CDF is relted to the MAD through G (x)dx MAD. In ddition, the MAD is mesure of bsolute difference between F (x) nd the CDF for degenerte distribution which tkes m s the only vlue. However, when is vlue other thn 1/2, G (x) is not continuous t m. Hence, here we define G (x) s weighted version of tht in eqution (11): 1 F (x), if F (x), G (x) (13) 1 F (x), otherwise, for (0, 1), such tht G (x) is continuous t m with G (m ) 1. Accordingly, the MAD is defined s weighted version of tht in eqution (12): } 1 MAD mx (m x), x m f(x)dx, (14) 5

6 69 such tht G (x)dx m m 1 b F (x)dx + m 1 F (x)}dx 1 (m y)f(y)dy + (y m )f(y)dy m } 1 mx (m y), y m f(y)dy;. (15) tht is, the weighted MAD equls 70 G (x)dx, the re under the curve of 71 G (x). 72 From eqution (14), we cn mke the following observtions. First, when 73 1/2, the MAD reverts to the MAD. Secondly, the reltive weight re ceived by the vlues of X lrger thn m is. When > 1/2, hence, the vlues of X lrger thn m receive hevier weight thn tht 76 received by the vlues smller thn m, nd the lrger the, the lrger the reltive weight. Such ttern reverses if < 1/2. In both cses, it 1 78 seking, devition from m to more extreme sit- 79 ution receives hevier weight thn devition from m to less extreme 80 sitution, when the overll vribility is summrised by the MAD. 81 Therefore, such n MAD cn be used s mesure of risk, s doted 82 in men-risk models for ortfolio otimistion by Ogryczk nd Ruszczyński 83 (2002), Ruszczyński nd Vnderbei (2003), Miller nd Ruszczyński (2008) 84 nd Choi nd Ruszczyński (2008), for exmle. These studies hve discussed 85 the reltionshi between the MAP nd exected shortfll, sometimes termed 86 conditionl vlue t risk, verge vlue t risk or exected til loss Imlictions for rctice Our results hve number of rcticl imlictions. First, nlogously to the Blnd-Altmn difference lot (Altmn nd Blnd, 1983; Blnd nd Altmn, 1986, 1999), which is oulr in medicl sttistics nd nlytic chemistry, the folded CDF is lso grhicl tool for ssessing greement between two ssys or methods, often by reresenting the difference between the two ssys by rndom vrible X. Both lots cn be redily understood by the users who my not be sttisticins or oertions reserch nlysts. 6

7 Comred with the Blnd-Altmn difference lot, the folded CDF stresses more the medin nd tils of the difference. If the two ssys re unbised with ech other (Krouwer nd Monti, 1995), the medin would be close to zero. If the vribility between the two ssys is lrge, the width ner the bottom of the folded CDF would be lrge, nlogously to confidence intervl. Comlementry to such width, the re under the curve of the folded CDF is nother mesure of the vribility between the two ssys, roughly through visul insection or recisely through quntittive comuttion. Therefore, the equivlence between the under-curve re nd the MAD suggests, nd rovides theoreticl justifiction of, this mesure. Secondly, the weighted men bsolute devition from the -quntile, shown s the MAD in eqution (14), includes the MAD s secil cse nd, more imortntly, hs been doted s risk mesure in men-risk models for ortfolio otimistion. It is well defined nd investigted (Ruszczyński nd Vnderbei, 2003). Moreover, it is very generic mesure of disersion or risk, nd cn be used in other risk-mngement rctice. Lstly but imortntly, the equivlences give the MAD nd MAD simle grhicl interrettions for rctitioners from outside the sttistics nd oertions reserch communities. Acknowledgments This work ws rtly suorted by funding to J.-H.X. from the Internl Visiting Progrmme, under the EU-funded PASCAL2 Network of Excellence. Thnks to the Associte Editor for the comments. References Altmn, D. G., Blnd, J. M., Mesurement in medicine: the nlysis of method comrison studies. Journl of the Royl Sttisticl Society, Series D (The Sttisticin) 32 (3), Blnd, J. M., Altmn, D. G., Sttisticl methods for ssessing greement between two methods of clinicl mesurement. The Lncet 327 (8476), Blnd, J. M., Altmn, D. G., Mesuring greement in method comrison studies. Sttisticl Methods in Medicl Reserch 8 (2),

8 Choi, S., Ruszczyński, A., A risk-verse newsvendor with lw invrint coherent mesures of risk. Oertions Reserch Letters 36 (1), Krouwer, J. S., Monti, K. L., A simle, grhicl method to evlute lbortory ssys. Euroen Journl of Clinicl Chemistry nd Clinicl Biochemistry 33 (8), Miller, N., Ruszczyński, A., Risk-djusted robbility mesures in ortfolio otimiztion with coherent mesures of risk. Euroen Journl of Oertionl Reserch 191 (1), Monti, K. L., Folded emiricl distribution function curves mountin lots. The Americn Sttisticin 49 (4), Ogryczk, W., Ruszczyński, A., Dul stochstic dominnce nd relted men-risk models. SIAM Journl on Otimiztion 13 (1), Ruszczyński, A., Vnderbei, R. J., Frontiers of stochsticlly nondominted ortfolios. Econometric 71 (4),