Decomposition of the mean difference of a linear combination of variates

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1 Decomposton of the mean dfference of a lnear combnaton of varates Paolo Radaell - Mchele Zenga Dpartmento d Metod Quanttatv per le Scenze Economche ed Azendal Unverstà degl Stud d Mlano-Bcocca 1 Introducton In ths paper we show, n an elementary way, that the mean dfference of the sum Y of the varates X 1, X 2,..., X,..., X c, can be obtaned as the dfference of the sum of the mean dfferences of each varate X wth a non-negatve quantty that measures the departure of the data matrx from the unform rankng (cograduaton) matrx. The mean dfference (Y ) of Y s equal to the sum (X ) of the mean dfferences of each varate X only f there s a unform rankng among the varates. By utlzng the decomposton of (Y ) we have decomposed, n an analogous way, the Gn s concentraton rato of a sum of non-negatve varates. A prevous verson of ths work can be found n Zenga and Radaell [2002] and has been presented at the Fourth Internatonal Conference on Statstcal Data Analyss based on the L 1 - Norm and Related Methods - Unversty of Neuchâtel, Swtzerland, see Radaell and Zenga [2002]. The present work reflects the common thnkng of the two authors, even f, more specfcally, M. Zenga wrote sectons 1, 2 and 3, whle P. Radaell wrote the remanng sectons. 1

2 P. Radaell and M. Zenga The result obtaned n ths paper shows that the Gn s mean dfference, lke the standard devaton, may be a useful measure varablty when a varate Y can be expressed as a sum of c varates. Fnally the decomposton s extended to a varate Y obtaned as a lnear combnaton of the varates X ; ( = 1,..., c). 2 Defntons and notatons Let X 1, X 2,..., X,..., X c be c varates observable on each of the N unts of a fnte populaton. In each of the N rows of matrx (2.1) the values of the c varates are reported: x x 1... x 1c x 1... x... x c. (2.1) x N1... x N... x Nc Wth Y = X we denote the sum of the c varates. The N values of Y arranged n ncreasng order of magntude are: y (1) y (2)... y ()... y (N). (2.2) The matrx (2.3) s obtaned from matrx (2.1) permutng the rows accordng to the N ncreasng values y () : x x 1... x 1c x 1... x... x c. (2.3) x N1... x N... x Nc In other words, n matrx (2.3) we have: x x x c = y () = 1, 2,..., N. (2.4) 2

3 Decomposton of the mean dfference of a lnear combnaton of varates In (2.3), the c values of a row belong to one of the N unts of the populaton. Furthermore, the ncreasng order of the sums y () does not mply that the same sortng s fulflled for each of the c varates, n other words t s not generally true that: x 1 x 2... x... x N = 1, 2,..., c. The matrx (2.5) s obtaned from matrx (2.3), arrangng n ncreasng order the values of each column: In other words: x (11)... x (1)... x (1c) x (1)... x ()... x (c). (2.5) x (N1)... x (N)... x (Nc) x (1) x (2)... x ()... x (N) = 1, 2,..., c. (2.6) Note that n the matrx (2.5) the values on a sngle row do not necessarly refer to the same unt of the populaton. Gven that the values of each column are arranged n ncreasng order, the matrx (2.5) can be defned unform rankng (cograduaton) matrx. Addng up each row of (2.5), we obtan the theoretcal values: y () = x (1) + x (2) x () x (c) = 1, 2,..., N. (2.7) Gn s mean dfference (wthout repetton) of the varate X that takes the values x 1,..., x,..., x N on the N unts of a fnte populaton s gven by: (X) = 1 N(N 1) N =1 It s well known (Gn [1914]) that the statstcs: S(X) = N x x l. (2.8) l=1 x x l (2.9) l 3

4 P. Radaell and M. Zenga s gven by: S(X) = 2 x () (2 N 1). (2.10) 3 Decomposton of the mean dfference of a sum The mean dfference of the varate Y = X s gven by: (Y ) = S(Y ) N(N 1) (3.1) where, accordng to (2.10): S(Y ) = y y l = 2 Substtutng (2.4) n (3.2) we have: ( N ) S(Y ) =2 x (2 N 1) l =1 y () (2 N 1). (3.2) = 2 x (2 N 1). (3.3) We can rewrte (3.3) as follows: S(Y ) = 2 ( x() x () + x ) (2 N 1) ( ) x() x (2 N 1). (3.4) = 2 x () (2 N 1) 2 From (2.10): thus: N S(X ) = 2 x () (2 N 1), =1 ( ) x() x (2 N 1). (3.5) S(Y ) = S(X ) 2 The sum ( ) x() x (2 N 1) s equal to: ( ) ( ) x() x (N + 1) x() x. = 2 4

5 Decomposton of the mean dfference of a lnear combnaton of varates Now, snce the sum ( ) x() x = 0, for each = 1, 2,..., c, t derves that: ( ) ( ) x() x (2 N 1) = 2 x() x and, substtutng n (3.5), we get: S(Y ) = S(X ) 4 ( ) x() x. (3.6) By theorem 368 of Hardy et al. [1952, p. 261], the sum: N x =1 s greatest when the values x, = 1,..., N are arranged n ncreasng order, that s: N x =1 Therefore, for each = 1,..., c, we have: N x (). =1 ( ) x() x 0 (3.7) wth equalty only f x = x (), = 1,... N. It derves that: ( ) x() x 0 (3.8) wth equalty only f x = x (), ; that s when the matrx (2.3) s equal to the unform rankng matrx (2.5). From (3.6) and (3.8) we get: S(Y ) = S(X 1 + X X c ) S(X ). (3.9) The non-negatve term 4 ( x() x ) can be nterpreted as a measure of departure of the data matrx (2.3) from the unform rankng (cograduaton) matrx (2.5). 5

6 P. Radaell and M. Zenga Dvdng (3.6) by N(N 1) we obtan the subtractve decomposton for the Gn s mean dfference of Y : (Y ) = (X X c ) = (X ) 4 N(N 1) ( ) x() x. (3.10) Obvously: (Y ) = (X 1 + X X c ) (X ) (3.11) wth equalty only n the case of unform rankng of the c varates. Zenga [2003] derved from decomposton (3.10) the followng normalzed dstrbutve compensaton ndex: C = 1 (Y ) c (X ) = 1 S(Y ) c S(X ) whose range s [0; 1] n partcular: C = 0 f there s no compensaton among the c varates.e. there s unform rankng (co-graduaton); C = 1 f there s maxmum compensaton among the c varates.e. Y s constantly equal to M 1 (Y ). The compensaton ndex C was also nvestgated by Maffenn [2003] who decomposes C n order to evaluate the contrbuton of each varate to the overall compensaton; furthermore the author studes the ndex behavour n the case of ndependence among the varates and apples the methodology to talan famles ncomes. Moreover Borron and Zenga [2003] propose a test of concordance based on the dstrbutve compensaton rato C and compare t wth other classcal rank correlaton methods such as the Spearman s rho and the Kendall s tau. Some developments on the power of ths test are the obect of the nvestgaton by Borron and Cazzaro [2005]. 6

7 Decomposton of the mean dfference of a lnear combnaton of varates 4 Decomposton of Gn s concentraton rato of a sum In ths secton we assume that the c varates X are non-negatve and that ther mean values are postve: M 1 (X ) = 1 x > 0 = 1, 2,..., c. (4.1) N Obvously: M 1 (Y ) = M 1 (X ) > 0. (4.2) Gn s concentraton rato of a non-negatve varate X wth M 1 (X) > 0, s defned by: R(X) = Dvdng both terms n (3.10) by 2M 1 (Y ), we obtan: R(Y ) = (Y ) 2M 1 (Y ) = (X 1) (X c ) 2M 1 (Y ) (X) 2M 1 (X). (4.3) 4 2M 1 (Y )N(N 1) (X 1 ) 2M = 1 (X 1 ) 2M 1(X 1 ) (X c) 2M 1 (X c ) 2M 1(X c ) 2M 1 (Y ) = R(X 1)M 1 (X 1 ) R(X c )M 1 (X c ) M 1 (X 1 ) M 1 (X c ) ( ) x() x 2 ( x () x ) M 1 (Y )N(N 1) 2 ( ) x () x. M 1 (Y )N(N 1) The share of the varate X on the sum Y s gven by: ω = x x = NM 1(X ) NM 1 (Y ) = M 1(X ) M 1 (Y ) ; (4.4) obvously ω = 1. Thus the decomposton of R(Y ) can be rewrtten as: R(Y ) = R (X ) ω 2 N(N 1)M 1 (Y ) ( ) x() x. (4.5) 7

8 P. Radaell and M. Zenga Equaton (4.5) s an easy decomposton that allows Gn s concentraton rato of a sum to be obtaned as the dfference of the weghted arthmetc mean of concentraton ratos of each varate X wth a non-negatve quantty that measures the departure of the data matrx (2.3) from the unform rankng (cograduaton) matrx (2.5). 5 Comparson wth others Gn s concentraton rato decompostons 5.1 The decomposton proposed by Rao Rao [1969] proposed two decompostons of Gn s concentraton rato: the frst one s by sub-populatons, the second one s by components of ncome 1. In ths secton we compare the latter wth the decomposton (4.5). Let x be the ncome of the th famly ( = 1,..., N) due to the th component X ( = 1,..., c). The total ncome of the th famly s gven by: y = x. The whole ncome composton of the N famles can be reported n a matrx smlar to (2.1). Suppose to permute the rows of the matrx (2.1) so that the famles are arranged n ncreasng order accordng to ther total ncomes y () (matrx (2.3)). The obectve of ths approach s to explan the nequalty of total ncomes by the nequaltes observable for each of the c ncome sources. In order to reach ths result, Rao consders, for each ncome component, two dfferent sortngs of the N famles: ) n the frst one the values of each ncome component are sorted n ncreasng order of magntude; n other words for the th component we have: x (1)... x ()... x (N) 1 The subect s also dscussed n Kakwan [1980]. 8

9 Decomposton of the mean dfference of a lnear combnaton of varates The result s a unform rankng (cograduaton) matrx (2.5); ) n the second one the ncome components are sorted n ncreasng order accordng to ther total ncomes (matrx (2.3)). The famles are then parttoned, for both sortngs, n k subsets so that each set ncludes N/k famles. In order to make the comparson wth decomposton (4.5) easer we set k = N so that each subset ncludes only one famly. Let q = t=1 x t = 1,..., N ndcate the cumulatve sums of the th component ncomes n matrx (2.3); q = t=1 x (t) be the same cumulatve sums n matrx (2.5). = 1,..., N The value of Gn s concentraton rato for the th ncome component ( th column of matrx (2.5)) R(X ) = (X )/2M 1 (X ) can be obtaned by the followng formula related to the Lorenz curve (Gn [1914]): ( M1 (X ) q ) R(X ) = N 1 =1 N 1 =1 M 1(X ) = N 1 =1 (p q ) N 1 =1 p (5.1) where p = N and q q = N M 1 (X ). Rao apples formula (5.1) also on the cumulatve sums q obtanng the statstcs: where q = R(X ) = q N M 1 (X ). N 1 =1 ( M1 (X ) q ) N 1 =1 M 1(X ) = N 1 =1 (p q ) N 1 =1 p (5.2) 9

10 P. Radaell and M. Zenga Note that R(X ) s not Gn s concentraton rato snce the N values x n a column of matrx (2.3) are not necessarly n ncreasng order. 2 Furthermore Rao shows that: R(X ) R(X ) R(X ) (5.3) where the lower and the upper bounds are reached respectvely f n the th column of matrx (2.3) the N famles th ncomes are n descendng or ascendng order. Rao shows that the concentraton rato R(Y ), computed on the total ncomes, can be obtaned as the dfference of the weghted arthmetc mean of components concentraton ratos, wth weghts gven by the shares (4.4) of each component on the total ncome, wth a non-negatve quantty that Rao defned an overall measure of the extent to whch component-nequaltes offset each other. Usng the notaton above, Rao s decomposton s: R(Y ) = R(X ) ω R(X ) ω [ 1 R(X ] ). (5.4) R(X ) Comparng decompostons (4.5) and (5.4) we note that, for both, we have to subtract a non-negatve term from the weghted arthmetc mean of concentraton ratos computed on each ncome component. These terms are respectvely: and: 2 N(N 1)M 1 (Y ) ( ) x() x (5.5) R(X ) ω [1 R(X ] ). (5.6) R(X ) Obvously (5.5) s equal to (5.6) but they are dfferent n the way they have been obtaned and n ther nterpretaton. 2 It s not necessary true that: The term depends, n both cases, on the p q = 1,..., N 1. 10

11 Decomposton of the mean dfference of a lnear combnaton of varates dfferent famles sortng for ndvdual components wth respect to the one obtaned for the total ncome. The nterpretaton s clear n (5.5) gven that we consder the ndvdual weghted dfferences ( x () x ), whle on the contrary, n (5.6) the nterpretaton s not clear snce the term to be subtracted depends on the ratos R(X )/R(X ). Furthermore, we do not know whether t s advsable to compute concentraton ratos on values n non ascendng order and obtan values whch may be negatve for a measure that by defnton (and tradtonal use n lterature) should le n the nterval [0; 1]. In other words the measure R(X ) can be hardly explaned. 5.2 The decomposton proposed by Lerman and Ytzhak Lerman and Ytzhak [1984; 1985] propose a decomposton of the overall Gn coeffcent by ncome sources. In partcular the authors show that each source s contrbuton to the Gn coeffcent may be vewed as the product of three factors: the source s Gn coeffcent; the source s share of total ncome; the Gn correlaton between the source and the rank of total ncome. The pont of departure s the relatonshp between the Gn s mean dfference and the covarance; ths relaton was ponted out by De Vergottn [1950] 3 n a paper concernng a general expresson for concentraton ndexes 4 despte t s frequently ascrbed to Stuart [1954]. 5 Ths relaton states that the Gn s mean dfference wth repetton of a varable X s equal to four tmes the covarance between the varable and ts rank: (X) = 4 Cov [X; F X (X)] (5.7) 3 See also Zenga [1987, p. 47]. 4 Ths approach has been followed also n Dancell [1987]. 5 See for example Davd [1968], Lerman and Ytzhak [1984] and Balakrshnan and Rao [1998, p. 497]. 11

12 P. Radaell and M. Zenga where F X denotes the cumulatve dstrbuton of X. For a varate X that takes the values x 1,..., x,..., x N on the N unts of a fnte populaton, (5.7) can be rewrtten as: (X) = 1 N 2 N =1 =4 Cov N x x l l=1 [ X, r(x) ] N = 2 N 2 x [2r(x ) N 1] = 2 N 2 x() [2 N 1] (5.8) where r(x ) s the rank of th value. In ths framework, Ytzhak and Olkn [1991] (see also Olkn and Ytzhak [1992] and Schechtman and Ytzhak [1999]) defne, for two random varables X and Y wth contnuous dstrbuton functons F X and F Y, respectvely, and a contnuous bvarate dstrbuton F X,Y, the Gn covarance between X and Y as: Gcov (X, Y ) Cov [X; F Y (Y )]. (5.9) A measure of assocaton between X and Y, see Schechtman and Ytzhak [1987] for detals, can be defned as: Γ (X, Y ) = Cov [X; F Y (Y )] Cov [X; F X (X)] (5.10) and: Γ (Y, X) = Cov [Y ; F X (X)] Cov [Y ; F Y (Y )]. (5.11) In our framework, the mean dfference (wth repetton) of the varate Y = X 12

13 Decomposton of the mean dfference of a lnear combnaton of varates s, accordng to (5.7): (Y ) =4 Cov [Y ; F Y (Y )] =4 Cov [X ; F Y (Y )] Cov [X ; F Y (Y )] =4 Cov [ X ; F X (X ) ] Cov [ X ; F X (X ) ] = Γ (X ) (5.12) where: Γ = Γ (X, Y ) = Cov [X ; F Y (Y )] Cov [ X ; F X (X ) ] (5.13) s the Gn correlaton (5.10) between the th component X and the sum Y. It must be observed that (5.13) s equvalent to the rato between Rao s (5.2) and (5.1): Γ = R(X ) R(X ). (5.14) In order to compare (5.12) wth the decomposton here proposed, we rewrte (3.10) for the mean dfference wth repetton: (Y ) = (X ) 4 N 2 N ( ) x() x (5.15) =1 and (5.12) as: (Y ) = (X ) (X ) (1 Γ ). (5.16) 13

14 P. Radaell and M. Zenga Clearly: 6 4 N 2 For a fxed : 4 N 2 N ( ) x() x = (X ) (1 Γ ). =1 N ( ) x() x = (X ) (1 Γ ) (5.17) =1 vanshes f and only f Γ = +1 that s when there s a perfect postve Gn correlaton between X and Y or, equvalently, when X and Y are cograduated. Ths comparson hghlghts that the term to be subtracted to the sum of the mean dfferences of the varates X, ( = 1,..., c) n order to obtan the mean dfference of Y n (5.15) can be nterpreted as a measure of departure from the stuaton of perfect postve Gn correlaton between each varate X and the sum Y. Dvdng both terms n (5.16) by 2M 1 (Y ), we obtan: R(Y ) = (Y ) 2M 1 (Y ) = (X ) Γ 2M 1 (Y ) (X ) M 1 (X = Γ 2M 1 (X ) M 1 (Y ) = Γ R(X ) ω (5.18) 6 (X ) (1 Γ ) = =4 = 2 N 2 = 4 N 2 = 4 N 2 4 Cov [ X ; F X (X ) ] ( ) 1 Cov [X ; F Y (Y )] Cov [ X ; F X (X ) ] { [ Cov X ; F X (X ) ] Cov [X ; F Y (Y )] } { N } N x [2r(x ) N 1] x [2r(y ) N 1] =1 =1 =1 N x [r(x ) r(y ))] N ( ) x() x. =1 14

15 Decomposton of the mean dfference of a lnear combnaton of varates where ω s the s share of total ncome (see (4.4)). In order to pont out the relaton between Rao s and Lerman and Ytzhak s decompostons, we observe that (5.18) can be rewrtten as: R(Y ) = R(X ) ω R(X ) ω [1 Γ ] (5.19) that, gven (5.14), becomes: R(Y ) = R(X ) ω [ R(X ) ω 1 R(X ] ) R(X ) (5.20) whch s Rao s decomposton (5.4). 6 Decomposton of the mean dfference of a lnear combnaton In ths secton we provde an extenson of the decomposton shown n secton 3 to the more general case of a lnear combnaton of varates. Let Y = α 1 X α X α c X c = α X (6.1) denotes the lnear combnaton of the c varates X wth coeffcents α 0 ( = 1,..., c). If we denote wth: Z = α X = 1,..., c Y n (6.1) s smply the sum of the new varates Z, ( = 1,..., c) so we can decompose Gn s mean dfference of Y accordng to (3.10) as: (Y ) = (Z Z c ) = (Z ) 4 N(N 1) ( ) z() z (6.2) where, for each, the values z are sorted accordng to the ther ncreasng total y () and the values z () are sorted themselves. 15

16 P. Radaell and M. Zenga In order to express (Y ) as a functon of the orgnal varates X we observe that: (Z ) = α (X ) = 1,..., c; (6.3) z = α x = 1,..., c; = 1,..., N; (6.4) α x () f α > 0 z () = α x (N +1 ) f α < 0 = 1,..., c; = 1,..., N. (6.5) If we defne the functon: = N [ 2 N 1 2 ] f α > 0 sgn (α ) = N + 1 f α < 0 = 1,..., c (6.6) where: +1 f k > 0 sgn (k) = 1 f k < 0 t s possble to get a unque expresson for (6.5): z () = α x ( ) = 1,..., c; = 1,..., N. (6.7) Fnally (6.2) can be rewrtten wth respect to the orgnal varates X as follows: (Y ) = α (X ) 4 N(N 1) ( ) α x( ) x (6.8) 7 Decomposton of Gn s concentraton rato of a lnear combnaton As n secton 4 we assume that the c varates Z are non-negatve and that ther mean values are postve: M 1 (Z ) = 1 z > 0 N = 1, 2,..., c (7.1) 16

17 Decomposton of the mean dfference of a lnear combnaton of varates so that: M 1 (Y ) = M 1 (Z ) > 0. (7.2) Ths means that, wth respect to the orgnal varates X, we should have, for each = 1,..., c: α > 0 f X 0 α < 0 f X 0. From now on we suppose, wthout loss of generalty, that α > 0 and X 0 for = 1,..., c. Gn s concentraton rato of Y can be decomposed as follows: R(Y ) = 2 ( ) R(Z ) ω z() z. (7.3) N(N 1)M 1 (Y ) where: ω = NM 1(Z ) NM 1 (Y ) = M 1(Z ) M 1 (Y ) denotes the share of the varate Z on the sum Y. Wth respect to the orgnal varates X we have: and: R(Z ) = (Z ) 2 M 1 (Z ) = α (X ) 2 α M 1 (X ) = R(X ) = 1,..., c M 1 (Y ) = M 1 (Z ) = α M 1 (X ). Fnally we can rewrte decomposton (7.3) as follows R(Y ) = 2 R(X ) ω N(N 1) ( ) α α x() x. (7.4) M 1 (X ) Concludng remarks In ths paper we show an easy subtractve decomposton for the Gn s mean dfference (Y ) of a varate Y obtaned as the sum of c varates. In ths decomposton 17

18 P. Radaell and M. Zenga the unform rankng (cograduaton) matrx (2.5) plays a central role gven that Gn s mean dfference of the sum s no greater than the sum of Gn s mean dfferences of the varates added up wth equalty only f the data matrx (2.3) s a unform rankng (cograduaton) matrx. By utlzng the decomposton of (Y ) we get a smple analogous decomposton for the Gn s concentraton rato R(Y ). Furthermore we compare the decomposton obtaned for R(Y ) wth the decompostons proposed by Rao [1969] and Lerman and Ytzhak [1984; 1985]. Fnally n sectons 6 and 7 we extend the decompostons of the Gn s mean dfference and concentraton rato to the more general case of a lnear combnaton of varates. Key words Gn s Mean dfference; Subtractve decomposton; Unform rankng (cograduaton) matrx; Gn s concentraton rato 18

19 References Balakrshnan, N. and Rao, C. (1998). Order Statstcs: Thoery & Methods, volume 16 of Handbook of Statstcs. North-Holland. Borron, C. and Cazzaro, M. (2005). Some Developments about a New Nonparametrc Test Based on Gn s Mean Dfference. (to be presented n Internatonal Conference n Memory of Two Emnent Socal Scentsts: C. Gn and M. O. Lorenz. Ther mpact n the XX-th century development of probablty, statstcs and economcs - Sena). Borron, C. and Zenga, M. (2003). A Test of Concordance Based on the Dstrbutve Compensaton Rato. Rapporto d Rcerca 51, Dpartmento d Metod Quanttatv per le Scenze Economche ed Azendal - Unverstà degl Stud d Mlano-Bcocca. Dancell, L. (1987). In Tema d Relazon e d Dscrodanze fra Indc d Varabltà e d Concentrazone. In Zenga, M., edtor, La Dstrbuzone Personale del Reddto: Problem d formazone, d rpartzone e d msurazone. Vta e Pensero. Davd, H. (1968). Gn s Mean Dfference Redscovered. Bometrka, 55(3): De Vergottn, M. (1950). Sugl Indc d Concentrazone. Statstca, X(4): Gn, C. (1914). Sulla Msura della Concentrazone e della Varabltà de Caratter. In Att del Reale Isttuto Veneto d Scenze, Lettere ed Art. Anno Accademco , volume Tomo LXXIII - Parte seconda, pages Veneza - Premate Offcne Grafche C. Ferrar. Hardy, G., Lttlewood, J., and Pólya, G. (1952). Inequaltes. Cambrdge Unversty Press, 2nd edton. Kakwan, N. (1980). Income Inequalty and Poverty. Methods of Estmaton and Polcy Applcatons. Oxford Unversty Press. 19

20 REFERENCES Lerman, R. and Ytzhak, S. (1984). A Note on The Calculaton and Interpretaton of the Gn Index. Economcs Letters, 15: Lerman, R. and Ytzhak, S. (1985). Income Inequalty Effects by Income Source: A New Approach and Applcatons to the Unted States. The Revew of Economcs and Statstcs, 67(1): Maffenn, W. (2003). Osservazon sull Indce d Compensazone Dstrbutva. Rapporto d Rcerca 57, Dpartmento d Metod Quanttatv per le Scenze Economche ed Azendal - Unverstà degl Stud d Mlano-Bcocca. (Forthcomng n Statstca). Olkn, I. and Ytzhak, S. (1992). Gn Regresson Analyss. Internatonal Statstcal Revew, 60(2): Radaell, P. and Zenga, M. (2002). Decomposton of the Mean Dfference of the Sum of K Varables. Abstract of contrbuted papers. Fourth Internatonal Conference on the L 1 Norm and Related Methods - Unversty of Nuechâtel, Swtzerland. Rao, V. (1969). Two Decompostons of Concentraton Rato. Journal of the Royal Statstcal Socety. Seres A (General), 132(3): Schechtman, E. and Ytzhak, S. (1987). A Measure of Assocaton Based on Gn s Mean Dfference. Communcatons n Statstcs, 16(1): Schechtman, E. and Ytzhak, S. (1999). On the Proper Bounds of the Gn Correlaton. Economcs Letters, 63: Stuart, A. (1954). The Correlaton Between Varate-Values and Ranks n Samples From a Contnuos Dstrbuton. The Brtsh Journal of Statstcal Psychology, VII(1): Ytzhak, S. and Olkn, I. (1991). Concentraton Indeces and Concentraton Curves. In Mosler, K. and Scarsn, M., edtors, Stochastc Orders and Decson Under 20

21 Rsk, volume 19 of Lecture Notes - Monograph Seres. Insttute of Mathematcal Statstcs - Hayward Calforna. Zenga, M. (1987). Concentraton Measures. In Naddeo, A., edtor, Italan Contrbutons to the Methodology of Statstcs, pages Socetà Italana d Statstca - Cleup Padova. Zenga, M. (2003). Dstrbutve Compensaton Rato Derved from the Decomposton of the Mean Dfference of a Sum. Statstca & Applcazon, I(1): Zenga, M. and Radaell, P. (2002). Decomposton of the Mean Dfference of the Sum of Varates. Rapporto d Rcerca 42, Dpartmento d Metod Quanttatv per le Scenze Economche ed Azendal - Unverstà degl Stud d Mlano-Bcocca. 21