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1 econstor Make Your Publications Visible. A Service of Wirtschaft Centre zbwleibniz-informationszentrum Economics Koshevoy, Gleb; Mosler, Karl Working Paper Multivariate Gini indices Discussion papers in statistics and econometrics, No. 7/95 Provided in Cooperation with: University of Cologne, Institute of Econometrics and Statistics Suggested Citation: Koshevoy, Gleb; Mosler, Karl (995) : Multivariate Gini indices, Discussion papers in statistics and econometrics, No. 7/95, Univ., Seminar für Wirtschafts- und Sozialstatistik, Köln This Version is available at: Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen wecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle wecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence.

2 DISCUSSION PAPERS IN STATISTICS AND ECONOMETRICS SEMINAR OF ECONOMIC AND SOCIAL STATISTICS UNIVERSITY OF COLOGNE No. 7/95 Multivariate Gini Indices by G.A. Koshevoy and K. Mosler June 995 Abstract The Gini index and the Gini mean dierence of a univariate distribution are extended to measure the disparity of a general d-variate distribution. We propose and investigate two approaches, one based on the distance of the distribution from itself, the other on the volume of a convex set in (d + )- space, named the lift zonoid of the distribution. When d =, this volume equals the area between the usual Lorenz curve and the line of zero disparity, up to a scale factor. We get two denitions of the multivariate Gini index, which are dierent (when d >) but connected through the notion of the lift zonoid. Both notions inherit properties of the univariate Gini index, in particular, they are vector scale invariant, continuous, bounded by 0 and, and the bounds are sharp. They vanish if and only if the distribution is concentrated at one point. The indices have aceteris paribus property and are consistent with multivariate extensions of the Lorenz order. Illustrations with data conclude the paper. Key words: Dilation; Disparity measurement; Gini mean dierence; Lift zonoid; Lorenz order. Seminar fur Wirtschafts{ und Sozialstatistik, Universitat zu Koln, Meister-Ekkehart-Str. 9/II, D-5093 Koln; Tel: +49// , e{mail: mosler@wiso.uni-koeln.de

3 Introduction To measure the disparity of a probability distribution, the Gini mean dierence and its scale invariant version, the Gini index, are most widely used. The Gini index is closely connected to the Lorenz curve; it amounts to twice the area between the Lorenz curve and the diagonal of the unit square. By this, the Gini index is consistent with the Lorenz order: It increases from one distribution to another if the rst Lorenz curve lies above the second. In this paper we investigate extensions of the Gini mean dierence and the Gini index to measure the disparity of a population with respect to several attributes s = ;...;d. The Gini mean dierence of a univariate distribution F is dened as the expected distance between two independent random variables which both follow the law F. Our rst notion will be an immediate extension of this (Section 4). For a d-variate empirical distribution F A ; with data matrix A =[a is ], it reads M D (F A )= n d n i= n d j= s= (a is, a js ) : () We call M D the distance{gini mean dierence. Our second notion, M V ; will be based on the volume of the lift zonoid and named the volume{gini mean dierence (Section 5). The lift zonoid of a d-variate distribution is a convex set in IR d+ which extends the generalized Lorenz curve; see Section 3 below. For F A we will get M V (F A )= d d s= n s+ i <...<i s+ n r <...<r sd j det(;a r ;...;r s i ;...;i s+ )j; () where is a column of ones, and A r ;...;r s i ;...;i s+ is the matrix obtained from the rows i ;...;i s+ and the columns r ;...;r s of the data matrix. For univariate data, the Gini index equals the Gini mean dierence of the relative data, which are the original data `scaled down' by their mean. Thus, for a d-variate distribution we will dene the distance-gini index and the volume-gini index by R D (F A )=M D (e FA ) and R V (F A )=M V (e FA ); (3) where F A is componentwise scaled down to e FA by its mean vector; see Section 3. Every d-variate Gini index should have at least the following properties: be equal to the usual Gini index in case d =,vary between 0 and, be scale invariant, and increase with a proper multivariate extension of the Lorenz order. This and

4 more will be shown for our two notions. Also they will be investigated for general d-variate probability distributions. For univariate distributions, the Gini mean dierence increases with the dilation order, and the Gini index increases with the Lorenz order, which we call relative dilation because it amounts to dilation of the relative distributions. Of course, dilation implies relative dilation. We will consider several extensions of dilation to the multivariate case. The rst is classical d-variate dilation, which means that one distribution equals the other one plus `noise'. The second, directional dilation, has the following property. G is a directional dilation of F if and only if the lift zonoid of G includes that of F. Further, absolute and relative versions of these dilations are considered in Section 3. We will show in Section 6 that both notions of the Gini mean dierence are increasing with absolute dilation and directional absolute dilation. Similarly, both our Gini indices increase with relative dilation and directional relative dilation. Although M D and R D are obvious extensions of the univariate notions, most of their properties have not been explored so far. In particular we prove in Section 4 that R D varies between 0 and, and that the bounds are sharp. We also establish a connection between M D (F ) and the lift zonoid of F : M D (F ) is proportionate to the average area of certain two{dimensional projections of the lift zonoid (Remark 4.). There are several attempts in the literature to dene a multivariate Gini mean dierence. Wilks (960) proposes the volume of a convex body associated with F. Oja (983) shows that the Wilks index is the expected volume of a simplex generated by d+ random vertices which are independent and identically distributed according to F ; see also Giovagnoli and Wynn (995). In our framework, the Wilks index amounts to d + times the volume of the lift zonoid (Theorem 5.). Torgersen (99) uses, as a multivariate Gini mean dierence, the volume of the zonoid of the distribution, which is the projection of its lift zonoid on the last d coordinates. For a one-point distribution, both the Wilks{Oja and the Torgersen indices vanish. But also for many other distributions they are zero, which appears to be unsatisfactory. Our notion M V (F )avoids this drawback; it vanishes if and only if F is a one-point distribution. In addition, we provide the correct scaling factor which makes R V vary between 0 and. M V (F )isanaverage of projections of the lift zonoid on coordinate planes (Remark 5.). Another multivariate Gini index, associated with a concentration surface, has been introduced by Taguchi (98). For the relations between Taguchi's concentration surface and the lift zonoid, see Koshevoy and Mosler (995a). Overview: Some properties of the usual univariate Gini index will be surveyed in Section. Section 3 presents the denitions of six multivariate dilation orderings and

5 of the lift zonoid and the Lorenz zonoid of a d-variate distribution. Section 4 is about the multivariate distance{gini index and its properties. The multivariate volume{ Gini index is introduced and analyzed in Section 5. In Section 6 we demonstrate that our Gini indices are increasing with multivariate dilations. Section 7 concludes the paper with a numerical illustration. Notation: IR k (IR k +)isthek{dimensional Euclidean space of row vectors (nonnegative rowvectors). In IR k ;x T is the transpose of a vector x; the usual componentwise ordering, and S k the unit sphere. 0 stands for the origin, and x; y for the segment between x and y in IR k.[a ;...;a l ] denotes the l k matrix with rows a ;...;a l IR k.for D and E in IR k, D + E = fu : u = x + y; x D; y Eg is the Minkowski sum, and V k (D) isthek-dimensional volume of D. The univariate Gini index We will shortly survey the Gini mean dierence and the Gini index of a univariate distribution. Let F : IR![0; ] be a given R probability distribution function on IR which has a nite expectation (F )= xdf (x) > 0: Denition. (Gini mean dierence, Gini index) M(F )= IR IR jx,yjdf (x)df (y): (4) is the Gini mean dierence of F. R(F )=M(F)=j(F)jis the Gini index of F. M(F ) is the mean Euclidean distance between two independent random variables divided by two, where both random variables are distributed with F, and R(F )is the mean Euclidean distance divided by twice the expectation of F. The denition and, as we will see in Section 4, the following results hold also for distributions with (F ) < 0. Proposition. Let F (s) = inffx : F (x) sg;s ]0; ]; denote the inverse distribution function of F, and L F (t) =(F) R t 0 F (s)ds, t [0; ]: Then, if F (0) = 0, (i) M(F ) equals the area between the graphs of the two functions t 7! j(f )j L F (t) and t 7! j(f )j (, L F (, t)), t [0; ]. (ii) R(F ) equals the area between the graphs of the two functions t 7! L F (t) and t 7!, L F (, t), t [0; ]. 3

6 Proof. t 7! L F (t) is the Lorenz function, and its graph is the Lorenz curve of F.t7! j(f )j L F (t) isthegeneralized Lorenz function. Itiswellknown that R(F ) amounts to twice the area between the Lorenz curve and the main diagonal of the unit square. The area between the main diagonal and the graph of t 7! (,L F (,t)) is congruent to this rst area. Hence (ii). Part (i) follows immediately since M(F ) = j(f)jr(f): The special case of an empirical distribution is particularly important. Let F a denote the distribution function which gives equal weight tongiven points a i in IR, a... a n, a=(a ;...;a n ), and let a = n (a +...+a n ). Then the Lorenz curve of F a is the linear interpolation of the points (k=n; a =a +...+a k =a), k =;...;n; in two-space. M(a ;...;a n )=M(F a )= n n ja i, a j j (5) n is the Gini mean dierence of the sample a =(a ;...;a n ); and j= R(a ;...;a n )=R(F a )= a M(a ;...;a n ) (6) is the Gini index of a; provided the sample mean is not zero. The Gini index of a equals the Gini mean dierence of the `scaled down' sample ea =(a =a;...;a n =a); R(a ;...;a n )= n n j= n i= i= j a i a, a j a j : (7) The Gini index and the Gini mean dierence have interesting properties which we will extend to our multivariate notions. Here we state them for empirical distributions. They hold as well for general univariate distributions. Proposition. (i) Let (a ;...;a n )IR n + with P a i > 0. Then 0=R(a;...;a) R(a ;...;a n )R(0;...;0; n i= a i )=, n <; R(a ;...;a n ) = R(a ;...;a n ) for every >0; a R(a +;...;a n +) = a+ R(a ;...;a n ) for every >0: (8) (ii) R is strictly increasing with the Lorenz order, i.e., R(a ;...;a n ) >R(b ;...;b n ) if L Fa (t) L Fb (t) for all t and < for some t. (iii) R is a continuous function IR n! IR. 4

7 Proposition.3 (i) Let (a ;...;a n )IR n + with P a i > 0. Then 0=M(a;...;a) M(a ;...;a n )M(0;...;0; n i= M(a ;...;a n )=M(a ;...;a n )for every >0 a i )=a(, n ) < a M(a +;...;a n +)=M(a ;...;a n ) for every IR: (ii) M is strictly increasing with the Lorenz order. (iii) M is a continuous function IR n! IR. These and other properties have been investigated by many authors. For surveys and references, see Nygard and Sandstrom (98) and Giorgi (990, 99). 3 Multivariate dilations and the lift zonoid Let F d (F0) d be the class of probability distribution functions IR d! IR which have a nite (nite and non-zero) expectation vector, and let F d + F d 0 be the subclass of probability R distributions on the nonnegative orthant IR d :Given F + Fd, let (F )= IR d xdf (x) =( ;...; d ). For every F F d and =( ;...; d ) IR d, dene F (x ;...;x d )=F(x ;...;x d d ), and F + (x ;...;x d )=F(x + ;...;x d + d ). For F F d; e 0 F =F(F) is called the relative distribution function, namely, iff is the distribution function of a random vector =( ;...; d ), then F e is the distribution of e = j j ;...; d : j d j In the sequel, when using e F,we tacitly assume that F F d 0 : Given F and G in F d, let and Y be two random vectors from the same probability space which are distributed according to F and G, respectively. G is a dilation of F, F G; if there exists a random vector such that E( j ) = 0 and Y has the same distribution as +. The random variable may be interpreted as `noise', so that Y is distributed like plus some noise. We call G an absolute dilation of F; F a G; if, G,(G) is a dilation of F,(F ). Given F and G in F d 0, G is a relative dilation of F, F r G; if, e G is a dilation of e F: For F F d and p =(p ;...;p d )IR d,we denote F (t; p) = fxir d :xp T tg 5 df (x); t IR;

8 ef(t; p) = d F e (x); fxir d :xp T tg t IR: If F is the distribution function of the random vector in IR d, then F (;p)isthe distribution function of the random variable p +...+p d d in IR; similarly e F (;p) is the distribution function of p =j j +...+p d d =j d j. G is a directional dilation of F, F dir G; if, for every p S d, G(;p) is a dilation of F (;p). We will say that G is a directional relative dilation of F, F dirr G; if, for every p S d, e G(;p) is a dilation of e F(;p). Similarly, G is named a directional absolute dilation of F, F dira G; if, for every p S d, G(;p) is an absolute dilation of F (;p). All these dilations are partial orders (reexive, transitive and antisymmetric)onf d, and related by the following implications. F G =) F dir G + + F r G =) F dirr G F G =) F dir G + + F a G =) F dira G However, in general, no reverse implication holds. For proofs, see Section 6 below. Next we dene a multivariate generalization of the Lorenz curve and the generalized Lorenz curve. Denition 3. (Koshevoy and Mosler (995a,b)) Let F F d.for a measurable function h :IR d +![0; ], consider the vector (z 0 (F; h);z(f; h)) IR d+, where The set z 0 (F; h) = IR d h(x)df (x); z(f; h) = IR d h(x)xdf (x): b(f ) (z 0 (F; h);z(f; h)) : h :IR d +![0; ] measurable is called the lift-zonoid of F. L(F )=b ( e F)is called the Lorenz zonoid of F: The lift zonoid is a multivariate generalization of the generalized Lorenz curve, and the Lorenz zonoid is one of the Lorenz curve. The following theorem establishes the relation between the lift-zonoid and directional dilation. 6

9 Theorem 3. (Koshevoy and Mosler (995a,b)) For F; G F d +, (i) F dir G if and only if b (F ) b (G); (ii) F dirr G if and only if L(F ) L(G); (iii) F dira G if and only if b (F,(F ) ) b (G,(G) ): Proof. For part (i), see Koshevoy and Mosler (995b), for part (ii), Koshevoy and Mosler (995a), the part (iii) follows from the part (i). Both relative dilation and directional relative dilation are multivariate extensions of the usual univariate Lorenz ordering, i.e. the ordering of Lorenz curves. dirr has been named the multivariate Lorenz order in Mosler (994); see also Koshevoy and Mosler (995a). If we compare empirical distributions with the same number, say n, of support points in IR d, dilation and directional dilation correspond to majorization and directional majorization of n d matrices; see Marshall and Olkin (979, ch. 5). 4 The multivariate distance-gini index The denition of the univariate Gini mean dierence (4) has the following multivariate generalization. Denition 4. For F F d the distance-gini mean dierence is M D (F )= d IRd IR d jjx, yjj df (x) df (y) (9) where jjjj denotes the Euclidean distance inir d.r D (F)=M D (e F)is the distance- Gini index. In the case of an empirical distribution function F A,we get M D (F A )= dn R D (F A )= dn n j= n j= n d i= s= n d i= s= (a is, a js ) ; (0) (a is, a js ) a s : () Several properties of the distance{gini mean dierence and the distance{gini index follow easily from the denitions. Recall that, for =( ;...; d )IR d ; we denote F (x ;...;x d )=F(x ;...;x d d ) and F + (x ;...;x d )=F(x + ;...;x d + d ). 7

10 Proposition 4. For all F F d, (i) 0 M D (F ); (ii) M D (F )=0if and only if F is a one-point distribution. (iii) M D (F + )=M D (F)for all ;...; d. (iv) M D is continuous w.r.t weak convergence of distributions. Proposition 4. For all F F d 0, (i) 0 R D (F ): (ii) R D (F )=0if and only if F is a one-point distribution. (iii) R D (F )=R D (F)for all ;...; d >0. (iv) R D is continuous w.r.t weak convergence of distributions. Proposition 4.(iii) says that R D is vector scale invariant, while Proposition 4.(iii) states that M D is translation invariant. Regarding upper bounds we have the following result. Theorem 4. For F F d + ;the following inequalities hold. and the bounds are sharp. M D (F ) < d d j= j (F ); R D (F ) < ; We will prove the theorem at the end of this Section. Before we consider a property which is desirable for every index of multivariate disparity. It says that, if to a distribution in d attributes a (d + )-th attribute is added which does not vary in the population, then the disparity index remains essentially unchanged: It multiplies by a factor which depends only on d. Denition 4. (Ceteris paribus property) Let J d be a real valued function which is dened on a subset D d of F d ;din:we say that J d ;d IN; has the ceteris paribus property if J d+ (F E 0 )=(d)j d (F) for all F D d ; 0 IR;dIN: () Here E 0 denotes the univariate one-point distribution at 0, and (d) isaconstant for every d. Theorem 4. M D and R D have the ceteris paribus property with (d) = 8 d d+ :

11 The proof is obvious from the denition of M D. Theorem 4.3 Let dp denote the rotation invariant area element on the sphere S d ; d. There holds M D (F )= R D (F )=,( d+ ) 4d d,( d+ ) 4d d psd + + psd + + ju, vj df (u; p) df (v; p)dp; (3) ju, vj d e F(u; p) d e F (v; p)dp: (4) Proof. We use the following formula by Helgason (980, Lemma 7.). For every z IR d and k>0 holds ps d jzp T j k dp = From this formula with k =,we conclude that M D (F ) = d = d = d = IRd,( d+ ) d,( d+ ) d,( d+ ) 4d d IR d jjx, yjjdf (x) df (y) IRd ( IR d IRd ( ps d psd + + d,( k+ ),( d+k ) jjzjj k : (5) ps d jxp T, yp T j dp) df (x) df (y) IR d jxp T, yp T j df (x) df (y))dp ju, vj df (u; p) df (v; p)dp: (6) This proves (3). The result for R D follows immediately with e F in place of F. Recall that the area of S d equals d =,( d ). Equation (3) in Theorem 4.3 says that the distance-gini mean dierence M D is a constant times the average, over all directions p in the sphere, of the Gini indices of all univariate distribution functions F (;p); " d+,( M D (F)= ),( d # ) d,( d) M(F (;p))dp ; (7) d ps d and R similarly for R D (F ). Recall, that the Euler Gamma-function,(s) = t s e,t dt has the following properties: p =,( ) and,(s +) = s,(s); and 0 9

12 R the Euler Beta-function B(a; b) = 0 ta (, t) b dt is equal to,(a),(b)=,(a + b): Therefore,,( d+ ) d+,( d,( d) = ),( ),( d+) = B( d+ ; ) : By the mean value theorem we conclude: Corollary 4. For every F there exist some p and ~p S d such that M D (F ) = R D (F ) = d+ B( ; ) d+ B( ; ) M(F (;p)) R( e F (; ~p)): The corollary says that, for every distribution F, there are directions p and ~p which reect the dependence structure of F, i.e. the interplay between the attributes, for the Gini mean dierence and the Gini index, respectively. Remark 4. M D (F ) is related to the lift zonoid (F b ) as follows. For p = (p ;...;p d ) S d, let pr p denote the projection of IR d+ on the two dimensioned plane which is spanned by the vectors (; 0;...;0) P and (0;p ;...;p d ). Then, for z =(z 0 ;z ;...;z d ) IR d+,we get pr p (z) =(z 0 ; z i p i ) with respect to this base. The projection of the lift zonoid by pr p equals the lift zonoid of F (;p) (Koshevoy and Mosler 995b). So, we can state that M D (F )isb( d+ ; )= times the average area of these two dimensioned projections of the lift zonoid. The following proof of Theorem 4. uses this fact. Proof of Theorem 4.. For F F d, holds b + (F ) [0; ] [0;(F)]. Therefore, in view of Remark 4., holds M(F (;p)) = + + and ju, vj df (u; p) df (v; p) = V (pr p ( b (F )) V (pr p ([0; ] [0;(F)])): (8) Recall that V denotes the two-dimensional volume. Thus, by (7), M D (F ),( d+ ) d d ps d V (pr p ([0; ] [0;(F)]))dp: Given p S d P, the projection pr p ([0; ] [0;(F)]) is P a rectangle whose edges have length and j j (F )p j j and whose area amounts to j j (F )p j j: Therefore, d M D (F ),( d+ ) d d ps d j= 0 j j (F )p j jdp

13 =,( d+ ) d d d j= ps d j j (F )p j jdp: (9) In view of (5), we get,( d+) j d j (F )p j jdp = jj(0;...;0; j (F);0;...;0)jj = j (F ): (0) ps d P Thus, (9) and (0) yield M D (F ) d j j(f ): The strict inequality is due to the fact that every lift zonoid is contained in the (d + ){dimensional rectangle [0; ] [0;(F)], but the latter is no lift zonoid. P It is easily seen that the upper bound d j j(f ) cannot be improved. For example, consider the n d matrix A whose j-th row is(0;...;0;n j (F);0;...;0), j = ;...;d; P while other rows are (0;...;0). P Then lim n! M D (F A ) = n,d lim n! n d j j(f ), which shows that d j j(f ) is the least upper bound for the mean distance{gini mean dierence. The least upper bound for the distance{gini index is established by passing from F to e F: Recall that j ( e F)=forj=;...;d. 5 The multivariate volume{gini index Here we start with the denition of the univariate Gini index as twice the area between the Lorenz curve and the diagonal and extend it to the multivariate case. Given F F d ;let ; ;...; d be independent random vectors each of which is distributed according to F. Q denotes the (d +)(d+ ) matrix having rows (;);(; );...;(; d ), and Ej det Qj is the expectation of the modulus of its determinant. The term (d!) Ej det Qj was called a multivariate Gini index by Wilks (960); see Oja (983) and Giovagnoli and Wynn (995). Oja (983) has interpreted it via the average volume of random simplexes with vertices ; ;...; d : The following theorem shows that ((d + )!) Ej det Qj equals the volume of the liftzonoid of F: Theorem 5. Let F be a given distribution function in IR d. Let ; ;...; d be independent random vectors each of which is distributed according to F, and let Q denote the (d +)(d+) matrix having rows (;);(; );...;(; d ). Then V d+ ( (F b )) = Ej det Qj: (d + )!

14 Proof. onoids are limits of zonotopes. Recall, that a zonotope in IR k is the Minkowski sum of line segments, say 0;y ;y n IR k with some given y i IR k : () It has volume (see, e.g., Shephard 974) i <...i k n j det[y i ;...;y ik ]j: () For a given F; there exists a sequence F ;IN;of distribution functions with nite supports in IR d + which converges weakly to F, i.e., lim R gdf = R gdf for every continuous and bounded function g :IR d!ir:due to the continuity of zonoids with respect to weak convergence (Bolker 969), we have lim (b (F ); b (F )) = 0, where is the Hausdor distance. The volume is a continuous function with respect to the Hausdor distance. Therefore, V d+ ( b (F )) = lim V d+ ( b (F )). Each volume V d+ (b (F )) can be calculated by the formula (). Let F have atoms at x ;...;x m with probabilities q ;...;q m. Then b (F )=0;(q ;q x )+...+0;(q m ;q m x m ):Hence V d+ (b (F )) = = = i <...i d+ m (d + )! j det[(q i ;q i x i );...;(q id+ ;q id+ x id+ )]j m i ;...;i d+ = (d + )! Ej det Q F j: q i...q id+ j det[(;x i );...;(;x id+ )]j This completes the proof. However, the volume of a lift-zonoid equals zero rather often, also if F is no onepoint distribution. Observe, that if the vectors x ;...;x n are linearly dependent, then the volume of the zonotope in () equals zero. Thus, whenever the support of F is contained in a linear subspace of IR d+ with dimension less than d +;then the volume of the lift zonoid is zero. In the case of an empirical distribution F, if, e.g., one of the attributes is equally distributed in the population, or if two attributes have the same distribution then V d+ (b (F ))=0. The volume of the Lorenz zonoid is given by the following formula. V d+ (L(F )) = Q d j= j jj V d+( b (F )): (3) In Mosler (994) the (d + )-dimensional volume of L(F ) has been introduced as a multivariate Gini index, called the Gini zonoid index. Although this index shows

15 a number of useful properties (boundedness between 0 and, 0 at one-point distributions, vector scale invariance, weak monotonicity with multivariate dilations), it may be zero also at distributions which are not concentrated at one point. To avoid this drawback of the Gini zonoid index, we propose the following denition. Let C d = f(z 0 ;z ;...;z d ) IR d+ : z 0 =0;0z s ;s=;...;dg, which isa d-dimensional cube in IR d+. Instead of the volume of the lift zonoid, we use the volume of the lift zonoid `expanded' by this cube. Denition 5. The volume-gini mean dierence is dened by M V (F)= d R V (F )=M V (e F)is the volume-gini index. V d+ ( b (F )+C d ) : (4) Let d =:Since jx, yj = j det( )j we conclude that the the distance{gini index x y and the volume-gini index are the same. This observation allows us to extend Proposition.(ii) to an arbitrary distribution F F0, dropping the assumption that F (0) = 0: The choice of the constant =( d, ) in (4) will be explained in the following theorem. We need some notations: For a nonempty subset K f;...;dg;f (K) denotes the marginal distribution with respect to the coordinates indexed by K: Theorem 5. M V (F )= d R V (F )= d ;6=Kf;...;dg V jkj+ (b (F (K) )); (5) V jkj+ (b ( F e(k) )): (6) ;6=Kf;...;dg Note that Formula (), M V for empirical distributions, follows from () and (5). Remark 5. By Equation (5), the volume-gini mean dierence is the average of the volumes of projections of the lift zonoid on coordinate subspaces. They are spanned by (;0;...;0) and (0; e r );rk; K f;...;dg: Here e r is the r-th coordinate unit vector in IR d : Proof of Theorem 5.. We will prove (5) for an empirical distribution F. Then an approximation argument yields (5) for a general distribution. (6) obviously 3

16 follows from (5). Let F have atoms at x ;...;x m in IR d with probabilities q ;...;q m. Then Hence, by () V d+ (b (F )+C d ) = b(f )+C d =0;(q ;q x )+...+0;(q m ;q m x m )+ i <...i d+ m + d l= d s= 0; (0; e s ): j det[(q i ;q i x i );...;(q id+ ;q id+ x id+ )]j i <...i d+,l m s <...s l d j det[(q i ;q i x i );...;(q id+,l ;q id+,l x id+,m ); (0; e s );...;(0; e sl )]j + m i= j det[(q i ;q i x i );(0; e );...;(0; e d )]j: Let l d, and s <...s l d be xed, K = f;...;dgnfs ;...;s l g. Then we have V jkj+ (b (F (K) )) = (7) i <...i d+,l m In view of q +...+q m =; det[(q i ;q i x i );...;(q id+,l ;q id+,l x id+,l ); (0; e s );...;(0; e sl )]j: m i= j det[(q i ;q i x i )(0; e );...;(0; e d )]j =: (8) (7) and (8) yield (5). The following three theorems establish properties of R V and M V : Proposition 5. For all F F d, (i) 0 R V (F ), (ii) R V (F )=0if and only if F is a one-point distribution, (iii) R V (F )=R V (F)for all ;...; d >0. (iv) R V is continuous w.r.t. weak convergence of distributions. (v) If F F d, then R + V (F ) < and the bound is sharp. Proof. (i) The volume is a nonnegative function. (ii): If F is a one-point distribution, then, for every K, b ( e F (K) ) is the main diagonal 4

17 of the unit hypercube in IR fjkj+g and has volume zero. Therefore R V (F ) = 0. If F is no one-point distribution, at least one of its univariate marginals, say F (j ),isthe same. Then the univariate Gini index R(F (j ) ) is positive. Since V ( b ( e F (j ) )) = R(F (j ) ), at least one summand in (6) does not vanish, and therefore R V (F ) > 0. (iii): The vector scale invariance is obvious from the denition of R V (F ), since it is based on the relative distribution e F only. (iv) follows from Theorem 7. in Koshevoy and Mosler (995b). (v): For every K, b ( e F (K) ) is contained in the unit hypercube of IR jkj+, hence 0 V jkj+ (b ( e F (K) )) <, and, by (6), 0 R V (F ) <. It is easily seen that the upper bound cannot be improved. For example, consider the distribution F (x) = Q d i= F i(x i ) where F i (x i )=0ifx i <0;F i (x i )=(n)=n if 0 x i < ; F i (x i )=ifx i :Then R V (F )!, for n!. Proposition 5. For all F F d, (i) 0 M V (F ), (ii) M V (F )=0if and only if F is a one-point distribution, (iii) M V (F + )=M V (F)for all ;...; d. (iv) M V is continuous w.r.t. weak convergence of distributions. (v) If F F d +, then M V (F ) < d, ;6=Kf;...;dg Y ik and the rst inequality cannot be improved. The proof is similar to that of Proposition 5.. i d, (max i Theorem 5.3 M V and R V have the ceteris paribus property with (d) = d d+, : i +) d Proof. It is easily seen, that V jkj+ ( b ((F E ) (K) )=0ifd+K.Ifd+6 K then F (K) =(FE ) (K) :This and (5) yield the proposition. 6 Consistency with multivariate dilations The univariate Gini index respects dilation and Lorenz order. We will show that our distance-gini and volume-gini indices do the same for properly dened extensions of these orderings. 5

18 Proposition 6. The following implications holds (i) F G ) F r G ) F dirr G. (ii) F G ) F a G ) F dira G. (iii) F G ) F dir G ) F dirr G and F dira G. (iv) F dirr G ) R(F (;p)) R(G(;p)) for all p S d. RProof. A standard R characterization of dilation says that F G if and only if (x)df (x) (x)dg(x) holds for all convex functions IR d! IR; see, e.g., the references in Mosler (994). Further, F G implies (F ) = (G). (i): Assume F G, and let :IR d!irbeconvex. Then, with ( ;...; d )= (F)=(G), the function x 7! ( x ;...; x d d )isconvex, too. We conclude (x)de F(x) = x ;...; x d d df (x) x ;...; x d d dg(x) = (x)de G(x): Therefore F r G. Now assume that F r G. Let p S d, :IR!IR convex. Then the function x 7! (xp T )isconvex, and from F r G follows that (u)de F(u; p) = (xp T )d e F (x) (xp T )d e G(x) = (u)de G(u; p); hence F dirr G. (ii): The proof is similar to that of (i). (iii): Dilation implies directional dilation. The rest follows from parts (i) and (ii) with d =. (iv): If F dirr G and p S d, then F (;p) is smaller than G(;p) in relative dilation (= usual Lorenz order). As the usual Gini index is consistent with Lorenz order, we conclude (iv). Note that, besides the implications given in Proposition 6., in general no other implications hold between the various multivariate dilations. Proposition 6. (i) ; dir are partial orders (reexive, transitive, antisymmetric) in F d. (ii) r and dirr are preorders (reexive, transitive) in F d 0. (iii) a and dira are preorders (reexive, transitive) in F d. 6

19 Note that the preorders r, dirr, a and dira are also antisymmetric when applied to the proper factor space. Proof. (i): The antisymmetry of dir is proven in Koshevoy and Mosler (995 b). The antisymmetry of follows from the antisymmetry of dir and Proposition 6.. (ii) and (iii) follow from (i) and Proposition 6.. Theorem 6. The distance-gini index R D and the volume-gini index R V are strictly increasing with (i) dilation, (ii) directional dilation, (iii) relative dilation, (iv) directional relative dilation. Proof. In view of Proposition 6., only (iv) has to be shown. Suppose F dirr G, hence R(F (;p)) R(G(;p)) for all p S d : Then + + ju, vj de F(u; p) d e F (v; p) + + ju, vj d e G(u; p) d e G(v; p) for all p: Therefore, + + ps d psd + + ju, vj d e F(u; p) d e F (v; p)dp ju, vj d e G(u; p) d e G(v; p)dp for all p. This yields, according to Proposition 4.3, R D (F ) R D (G): The result for R V follows immediately from Theorems 3., 5. and the following Proposition 6.3. That the indices are strictly increasing is seen from Theorems 3., 5. and the following Theorem 6.. Proposition 6.3 (Koshevoy and Mosler (995a)) Let F dirr G. Then F (K) dirr G (K) for all K; _6=Kf;...;dg: Theorem 6. (Koshevoy and Mosler (995b)) b (F )= b (G)i F = G: For the distance-gini and the volume-gini mean dierences, we have an analogous theorem. 7

20 Theorem 6.3 The distance-gini mean dierence M D and volume-gini mean difference M V are strictly increasing with (i) dilation, (ii) directional dilation, (iii) absolute dilation, (iv) directional absolute dilation. Proof. Proofs of (i) and (ii) are similar to those of (i) and (ii) in Theorem 6.. (iii) and (iv) follow from Propositions 4. and 5. respectively. 7 Conclusions We have presented two dierent approaches to extend the usual Gini index and Gini mean dierence to the multivariate case. Both approaches preserve important properties of the univariate notions, are increasing with proper multivariate dilations and have the ceteris paribus property. The distance{gini index and the volume{ Gini index of a given empirical distribution are easily calculated, but the latter needs more computation time. A computer program, written in GAUSS, can be obtained from the authors. Many other multivariate denitions are possible. A popular approach is to use the arithmetic mean, M S resp. R S, of the univariate indices, M S (F A )= n d R S (F A )= n d n i= n i= n j= n j= d s= d s= ja is, a js j; (9) j a is a i, a js a j j: (30) This is tantamount to employing the L distance instead of the Euclidean distance in our distance{gini notions. It can be shown that always R D (F ) R S (F ) and R V (F ) R S (F ) hold. But this approach, as the index depends on the marginals only, does not reect the dependency structure of the underlying distribution. To illustrate and contrast our notions, we calculate them for R. A. Fisher's Iris data (Fisher 936). The data include the measurements of four attributes, sepal length and width and petal length and width, of fty plants for each of three types of Iris, Iris setosa, Iris versicolor and Iris virginica. The data have been used to test the 8

21 hypothesis that Iris versicolor is a polyploid hybrid of the two other species which is related to the fact that Iris setosa is a diploid species with 38 chromosomes, Iris virginica is a tetraploid, and Iris versicolor is a hexaploid with 08 chromosomes. Iris setosa Iris versicolor Iris virginica R D R V R S R[] R[] R[3] R[4] Table. The multivariate Gini indices R D ;R V and R S for three types of Iris; data from Fisher (939). For further contrast, the univariate Gini index R[k] isgiven for each attribute k, k =;;3;4. As we can see from the Table, the four attributes are most variable at dierent types of Iris, as measured by their univariate Gini indices. E.g., the rst attribute, petal length, varies most with Iris virginica, while the second attribute, petal width, has its maximum Gini index with Iris setosa. But our three multivariate Gini indices, R D, R V ; and R S, order the variability of the three samples in the same way, Iris setosa < Iris versicolor < Iris virginica. Note, however, that no two of these multivariate indices are order equivalent in general. Under the assumptions that () a hybrid has an intermediate number of chromosomes compared to its origins and () that a higher number of chromosomes implies more variability, we may conclude that all three multivariate Gini indices back the hypothesis that Iris versicolor is a hybrid of the two others species. Acknowledgements We thank Stephan Erkel for his comments on a previous version and Ulrich Casser for writing the computer program and calculating the numerical example. 9

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