GINI S MEAN DIFFERENCE IN THE THEORY AND APPLICATION TO INFLATED DISTRIBUTIONS

STATISTICA, anno LXIII, n. 3, 003 GINI S MEAN DIFFERENCE IN THE THEORY AND APPLICATION TO INFLATED DISTRIBUTIONS. INTRODUCTION In 9 Prof. Corrado Gn pubshed (n Itaan) a very vast statstca study ntatng consderaton on the mean caed ater n the terature Gn's mean dfference. Curousy enough, the subsequent (non-itaan) authors deang wth ths probem do not refer to that work. In the book by Kenda and Stuart (963), Gn's name was mentoned and hs work was nserted n the references, but wthout further partcuars. Therefore, t may be supported that, for many authors, the work n queston was dffcut of attanment. It s wrtten n an ancent stye: very engthy (56 pages) wth ong descrptons. It s hard to perceve any modern notaton n t. The perod of Word War I undoubtedy dsturbed the extenson of Gn's deas. In the twentes C. Gn referred to hs dea. In ths case he pubshed two papers n foregn journas (9, 96). We have faed to ascertan whether someone was deang wth the mean dfference n the thrtes and fortes. Maybe, the anguage barrer of the pubshed papers caused not wdespread popuarty among not-itaan theoretcans of statstcs. It was ony n fftes and further decades when the Itaan statstcans dscussed anew (n Itaan) the mean dfference. We see here Savemn (956 and 957), Mchett and Da'Ago (957), Casteano (965), Grone (968a, 968b), Zanard (973 and 974). However, t s worthy of notce snce, unke other quanttes desgned for measurng the dsperson of a random varabe, the mean dfference s ndependent of any centra measure of ocazaton, whch can be seen from ts defnton. (.) x y df( x ) df( y) When the random varabe X s dscrete (a case more often consdered) the formua has the form

470 j j, (.) j x x p p where p P( X x ), p P( X x ). j j The anaytc nvestgaton of the dscussed characterstc s made dffcut because of the absoute vaue occurrng n the formua. However, t factates the computatons on numerca data, whch aso concerns, as s we known, the mean devaton. Hence we sometmes encounter the nvestgatons concernng the mean dfference, connected wth the mean devaton. Ths s the case, for nstance, n Ramasubban (958). Methodcay, ths paper s based on operatons ndcated by Johnson (957) n the consderatons referrng to the mean devaton of the bnoma dstrbuton. The mean devaton, aso for the bnoma dstrbuton, was consdered earer by Frame (945). The dffcutes connected wth the absoute vaue can effcenty be overcome n the case of the mean devaton by usng ncompete moments. Ths was demonstrated n T. Gerstenkorn's paper (975). As far as the mean dfference s concerned, the nvestgaton of ths statstc and, n partcuar, of some of ts propertes referrng to a random sampe dd not gve any adequate resuts athough, n the case of a norma varabe, one can menton a few papers. For the norma dstrbuton, the exact standard error of the mean dfference was probaby gven for the frst tme by Nar (936), but wth the appcaton of a rather compcated method. Much ater, n 95, Lomnck obtaned the very resut by usng a smper method. A year ater, Kamat cacuated the thrd moment n the exact form and, n the consderatons on the skewness measure, nferred that, for a great n, the dstrbuton of the mean dfference may be the same as the -dstrbuton. Foowng Kamat, Ramasubban (956) obtaned an approxmaton of vaues for the fourth moment and showed that the concentraton measure (kurtoss) cacuated on ths bass, taken together wth the vaues for (obtaned by Kamat), seems to show the exactness of the - approxmatons, at east for sampe greater than 0 (n > 0). The same author tred to obtan an emprca dstrbuton of the mean dfference for sma sampes (n < 0), but we do not know whether the resuts were pubshed. The norma dstrbuton may be consdered as the mt case of the bnoma and the Posson one under certan assumptons. So, t s natura to examne the moments from the sampe for those dscrete dstrbutons and to try to adapt a sutabe dstrbuton for, even f an exact dstrbuton cannot be found easy. In order to make the task easer, at the frst step one derves the formuae for the absoute mean dfference gven as r r r x x j p p j. (.3) j

Gn s mean dfference n the theory and appcaton to nfated dstrbutons 47 Ths probem s dscussed n Ramasubban's paper (959). An extenson of the probem can be found n the paper by Katt (960). The work of Gn s aso mentoned by Rao (98). Gn's mean dfference met wth no approbaton of the authors of handbooks. We dd not fnd t n any Posh handbook. It s concsey dscussed n the Engsh handbook by Kenda and Stuart (963). It s worth our whe to menton here the German textbook by Rnne (974) where a practca appcaton of ths statstc was dscussed.. PROPERTIES OF THE MEAN DIFFERENCE If a random varabe takes a fnte number of vaues then the expresson for the dfference s wrtten down n the form N j x x j nn j, (.) n n j where p P( X x ), p j P( X x j ) N N and n n... n N, and t s the so-caed dfference wth repettons. The dfference s sometmes defned n another way x x j nn j, j N( N ) j (.) as the mean dfference wthout repettons. Sometmes, the above formuae are wrtten down wthout takng the weghts nto account, and then N N N j x x j,, j,,..., N, (.a) N N x x j, j N( N ). (.a) j However, we most often use the foowng formuae N( N ) N N x x j (.b) j or

47 N( N ) N j x x j. (.c) j These formuae w be ustrated by an exampe from Rnne's book (p. 9) n n x k k x TABLE x x 5 6 6 8 0 3 3 6 n 4 5 5 7 9 5 90 5 3 5 8 8 7 54 6 0 4 7 7 0 6 46 6 4 7 7 0 6 46 8 5 5 8 4 34 0 3 3 6 4 3 0 3 9 3 3 9 6 6 6 x x 4 6 6 4 4 4 4 66 0 34 v x x From Tabe we get 0 9 34 7,. It mght seem that the dffcutes wth the occurrence of the absoute vaue w dsappear f, n pace of the mean dfference, we ntroduce the coeffcent. E ( x y) df( x ) df( y) However, after smpe cacuatons t turns out that E, s a doube varance. Nevertheess, ths nterestng reaton shows that the varance may be defned as a haf of the mean vaue of the squares of a possbe dfferences of vaues of the varabe, that s, n other words, one can defne t wth no need of turnng to a consderaton of devatons wth respect to the centra (mean) vaue. We found ths remark n Udny Yue and Kenda (953) (p. 46) as n Kenda and Stuart ony (p. 47; n Russ. ed. p. 74).

Gn s mean dfference n the theory and appcaton to nfated dstrbutons 473 The notaton of formuae (.) for the mean dfference may be so modfed that there w not be the absoute vaue sgn. Note that x x n n x x n n. (.3) j j j j j j j If the observatons are marked wth numbers n such a way that x x... x N, then formua (.3) may be wrtten down n the form j j, (.3a) j j ( x x ) n n and then formuae (.) and (.) w take the form N ( x x j ) nn j, (.4) j j N( N ) ( x x ) n n. j j (.5) j j Formuae (.4) and (.5) may be gven some other form. Note that, after carefu cacuatons, we have N N N ( x x ) k( N k)( x x ), j k k j j k therefore the mean dfference may be wrtten down as or N k( N k)( x xk ) (.6) k N k N k( N k)( x xk ). (.7) k N( N ) k These forms of the mean are partcuary handy when the dstances xk xk are the same. A further smpfcaton of the formuae can be obtaned by ntroducng a dstrbuton functon

474 F P( X x ) F( x ). k k k In the case when we get F k k and the dstances are dentca and equa to unty, N N N NF ( N NF ) F ( F ) N. (.8) k k k k k k If we denote by Gk NFk the cumuated frequency, then we sha obtan N N Gk ( N Gk ). (.9) k Ths form s convenent for practca computatons, whch s demonstrated by the Tabe (Kenda and Stuart, pp. 50-5; Russ. ed. p. 78) Heght, nches Frequency Gh TABLE N G h Gh N G 57 8583 7,66 58 4 6 8579 5,474 59 4 0 8565 7,300 60 4 6 854 59,964 6 83 44 844,5,504 6 69 33 87,589,36 63 394 707 7878 5,569,746 64 669 376 709 9,99,584 65 990 366 69 4,74,54 66 3 3589 4996 7,930,644 67 39 498 3667 8,034,306 68 30 648 437 4,98,676 69 063 7 374 9,907,94 70 646 7857 78 5,79,896 7 39 849 336,77,664 7 0 845 34,3,434 73 79 8530 55 469,50 74 3 856 3 96,96 75 6 8578 7 60,046 76 5 8583 7,66 77 8585 Totas 8585 05,990,850 h From Tabe we obtan 05990850,88. 8585 To make the presentaton fu, t s worth our whe to menton the so-caed concentraton coeffcent of Gn. Gn was engaged n the queston of concentraton as eary as 90, but he gave t a proper form n the paper of 94 presented on the 9 th of May (.e. shorty before the outbreak of Word War I) at the meetng of the Roya Veneta Insttute for Scence, Letters and the Arts:

Gn s mean dfference n the theory and appcaton to nfated dstrbutons 475 G, m E( X ), f t exsts, m or G x whch s, of course, an abstract number. In statstca practce we aso use of the so-caed concentraton curve of Lorenz (905). It s a curve whose ponts have the co-ordnates ( F( x ), ( x )) where ( x ) xdf( x ) x m s the so-caed ncompete moment (0 ( x ) ). The curve s convex. It can be shown that the area S, contaned between the concentraton curve and the straght ne F, s equa numercay to G. The proof can be found n Kenda and Stuart (p. 49; Russ. ed. pp. 76-77). 3. THE MEAN DIFFERENCE FOR INFLATED DISTRIBUTIONS In many feds of scence, one appes the we-known probabty dstrbutons of a dscrete random varabe. However, there happen stuatons n whch we are ready to admt that a gven phenomenon s subject to a typca probabty dstrbuton, under the condton that we sha expose ths dstrbuton to some deformaton. In such a case, we appy most frequenty the so-caed mxture of dstrbutons. As the smpest mxture we may cassfy the so-caed nfated dstrbuton whch conssts n composng any dscrete dstrbuton wth the degenerate (.e. one-pont) dstrbuton. We sha ntroduce the foowng notatons for the dscrete dstrbuton: P( X ) h( ), 0,,,... We then have Defnton 3.. We say that a dscrete random varabe Y s subject to the nfated dstrbuton (deformed at the pont = 0) f ts probabty functon s expressed by the formua h(0) f 0, P( Y ) h( ) f, (3.)

476 where (0,], whe. The deformaton of a dstrbuton may aso take pace at any pont of the dstrbuton. Defnton 3.. We say that a dscrete random varabe Y s subject to the generazed nfated dstrbuton (.e. the one wth a deformaton at any pont = ) f h( ) f, P( Y ) h( ) f 0,,,...,,,..., (3.) where (0,] and. In partcuar, for the bnoma dstrbuton, we have Defnton 3.3. We say that a dscrete random varabe Y s subject to the nfated bnoma dstrbuton P( X ) (deformed at the pont = 0) f ts probabty functon s expressed by the formua n q f 0, P( Y ) n p q n f,,..., n, (3.3) where (0,], whe, 0 p, p q. If, then the above dstrbuton reduces to the bnoma dstrbuton n n P( X ) p q for 0,,,..., n. Defnton 3.4. We say that a dscrete random varabe Y s subject to the generazed nfated bnoma dstrbuton f ts probabty functon s expressed by the formua n p q n f, P( Y ) n p q n f 0,,,...,,,..., n, (3.4) where 0,, 0 p, p q. Of course, formuae (3.) (3.4) present probabty dstrbutons, whch foows from smpe cacuatons. Infated dstrbutons were ntroduced nto the terature by S. N. Sngh (963) for the case of the Posson dstrbuton, and next made a thorough study of for

Gn s mean dfference n the theory and appcaton to nfated dstrbutons 477 the bnoma dstrbuton by M. P. Sngh for deformatons at the nta pont (965/66) and at an arbtrary one (966). Infated dstrbutons were beng deat wth by many authors. Many papers on varous subjects were wrtten. The probems concernng these dstrbutons were dscussed, for nstance, by T. Gerstenkorn (977). The mean dfferences for dstrbuton (3.) and for nfated bnoma dstrbuton (3.3) were dscussed by T. Gerstenkorn n the paper of 997. Here we sha dea wth the mean dfference for generazed nfated dscrete dstrbuton (3.). Theorem 3.. Gn's mean dfference for the generazed nfated dscrete dstrbuton s expressed by the formua {[ F( ) ] m m ( )} (3.5) j j [ ( j ) h( ) h( j ) ( j ) h( ) h( j )], j 0 j 0 where: m the expected vaue of an unnfated dstrbuton, m( ) the rght-hand ncompete moment (.e. the one wth the truncaton of the vaue of the varabe to x ncusve) of the unnfated dstrbuton, F( ) the dstrbuton functon of the unnfated dstrbuton at a pont x. Proof. From the defnton we have j ( j ) P( Y ) P( Y j ) ( j ) P( Y ) P( Y j ) 0 j j 0 j ( ) ( ) ( ) ( 0) ( ) ( ) ( ) ( )... j 0 j P Y P Y j P Y P Y P Y P Y + P( Y ) P( Y ) ( j ) P( Y ) P( Y j ) j ( ) ( ) ( ) ( 0) ( ) ( ) ( ) ( )... j 0 j j P Y j P Y P Y P Y P Y P Y + P( Y ) P( Y ) ( ) P( Y ) P( Y )

478 j ( ) ( ) ( ) (0)( ( )) j 0 j h h j h h ( ) h()( h( ))... h( )( h( )) ( j )( h( ) h( j )) ( j ) h( ) h( j ) j j 0 j h(0)( h( )) ( ) h()( h( ))... h( )( h( )) ( )( h( ) h( ) j j h h j h h h j 0 ( ) ( ) ( ) (0) (0) ( )) ( ) h() ( ) h() h( )... h( ) h( ) h( ) ( j ) h( j ) ( j ) h( ) h( j ) j j [ h(0) ( ) h(()... h( )] ( j ) h( j ) j ( j ) h( ) h( j ) j 0 j ( j ) h( j ) ( j ) h( j ) j 0 j j j ( j ) h( ) h( j ) ( j ) h( ) h( j ) j 0 j 0 h( j ) jh( j ) jh( j ) j 0 j 0 j j j h( j ) ( j ) h( ) h( j ) ( j ) h( ) h( j ) j j 0 j 0

Gn s mean dfference n the theory and appcaton to nfated dstrbutons 479 F( ) ( h( j ) jh( j ) jh( j ) j 0 j 0 j j j jh( j ) ( j ) h( ) h( j ) ( j ) h( ) h( j ) j j 0 j 0 [ F( ) ( F( )] [ m m ( )] j j m ( ) ( j ) h( ) h( j ) ( j ) h( ) h( j ), j 0 j 0 whch aready mpes formua (3.5). Basng ourseves on (3.5), we sha demonstrate what forms the mean dfference takes for generazed nfated bnoma dstrbuton (3.4). For the purpose, we sha make use of reaton (.4), p. 550 from Ramasubban's paper (958) as we as formua (.4) for the ncompete moment of the bnoma dstrbuton, cted n T. Gerstenkorn's paper (97) as the resut gven by Rsser and Traynard (933, pp. 30-3 or 957, pp. 9-93). Namey, n n m( ) ( ) p q npm0( ) n where m0( ) P( X ) F( ). Takng the above nto account, we sha get n 4 F( ) np 4 ( ) p q npm ( ) n 0 npq p q p q 0 n n n n n n n j ( j ) h( ) h( j ) j 0 n 4 F( ) np 4 ( ) p q n

480 4 npm0( ) npq p q p q 0 j. ( j ) h( ) h( j ) j 0 Fnay we obtan n n n n n n n Coroary 3.. Gn's mean dfference n the case of the generazed nfated bnoma dstrbuton s expressed by the formua n F ( ) np( m ( ) ) ( ) p q 0 n npq p q p q 0 n n n n n n n j ( j ) h( ) h( j ). (3.6) j 0 The vaue of the dstrbuton functon F( ) of the bnoma dstrbuton, occurrng n (3.6), can be read n the avaabe statstca tabe of, for exampe, Zesk (97, p. 50). One can aso obtan another form of ths reaton by usng formua (.8), p. 550, from the paper by Ramasubban: n 4 F( ) np 4 ( ) p q n n 4 npm0 ( ) pq( ) p q 0 whence the mpcaton of n j j 0 ( j ) h( ) h( j ) Coroary 3.. Gn's mean dfference for the generazed nfated bnoma dstrbuton s expressed by the formua n F( ) np( m ( ) ) ( ) p q 0 n j n pq ( ) p q ( j ) h( ) h( j ) 0 j 0. (3.7) n

Gn s mean dfference n the theory and appcaton to nfated dstrbutons 48 One can show (after rather tosome cacuatons) that the mean dfference for the unnfated bnoma dstrbuton may be wrtten down n the form ' ' n n pq( ) p q 0 n 4 pq ( n )( n ) 3 (4 pq) npq!!! 3! n n ( n )... (n ) n (4 pq)... ( ). ( n )! n n n ( n )! If we adopt the notatons a ( n ), b, c, x 4 pq, then ' ab x a( a ) b( b ) x npq... c! c( c )! n a( a )... ( a n ) b( b )... ( b n ) x c( c )... ( c n ) ( n )!, whch can be wrtten down n a smper way as ' npqf( a, b, c, x ) npqf ( n ),,, 4 pq, (3.8) that s, n the form of the hypergeometrc seres F( a, b, c, x ) [ n, ] [ n, ] n a b x, [ n, ] n c n! [ n] where a a( a )( a )... ( a n ) s the so-caed factora poynoma (the generazed power). Takng account of (3.8), we fnay obtan Coroary 3.3. Gn's mean dfference of the generazed nfated bnoma dstrbuton s expressed by the asymptotc formua

48 n F ( ) np( m ( ) ) ( ) p q 0 n j npqf ( n ),,,4 pq ( j ) h( ) h( j ). (3.9) j 0 In the case 0, the formuae gven here are consderaby smpfed. As has been mentoned, ther fu forms can be found n T. Gerstenkorn's paper (997). Defnton 3.5. We say that a dscrete random varabe Y s subject to the generazed nfated negatve bnoma dstrbuton f ts probabty functon s expressed by the formua n n p q f q P(Y=)= n n p q f 0,,,...,,... q (3.0) where 0<a, a+=, 0<p<, q-p=. T. Gerstenkorn (997) has shown that the mean dfference of an nfated dstrbuton s gven by =m + ( j ) h( ) h( j ) (3.) j 0 where m s the expected vaue of the dstrbuton consdered wthout nfaton. Ramasubban (958) has gven formuae ((.) and (.)) for Gn's mean dfference of the negatve bnoma dstrbuton. By usng these reatons we obtan Coroary 3.4. Gn's mean dfference for the nfated (=0) negatve bnoma dstrbuton s expressed by the formua or =np + npq n ( )! ( ) p q 0!( )! (3.) np + npq F(n+, ½,, -4pq), (3.3)

Gn s mean dfference n the theory and appcaton to nfated dstrbutons 483 where F, as prevousy, s a notaton of a hypergeometrc seres F(,,, x) wth parameters = n+, = ½, =, x = -4pq. In the case when a deformaton of the negatve bnoma dstrbuton takes pace n pont =, we make use of (3.5) and formua (.) for an ncompete moment of that dstrbuton gven by T. Gerstenkorn (97): m (+) = (+) n (-) + p + q n- + np. After sutabe cacuatons we get then Coroary 3.5. Gn's mean dfference for the generazed nfated negatve bnoma dstrbuton s gven by the formua or n n F( ) np( m0( ) ) ( )( ) p q j n ( )! npq ( ) p q ( j ) h( ) h( j ). 0!( )! j 0 (3.4) n n { F( ) np( m ( ) ) ( )( ) p q } 0 npqf( n,, 4 pq). (3.5) Defnton 3.6. We say that a random varabe Y s a subject to the generazed nfated Posson dstrbuton f ts probabty functon s gven by the formua e da! P(Y=)= e da 0,,...,,,...! (3.6) where 0<, + =, > 0. Makng use of (3.) and of formuae (.4), (.5), (.8)-(.0) gven by Ramasubban (958), we get

484 Coroary 3.6. Gn's mean dfference of nfated (=0) Posson dstrbuton s gven by = + e - 0!! 0!( )! (3.7) or repacng these sums by modfed Besse functon of the frst knd, we get =+ e - [I 0 () + I ()] (3.8) or aso =+ e I0( ) d (3.9) or n the form 0 + F(½,, -4). (3.0) In the case when = we make use of (3.5) and of formua (.6) by T. Gerstenkorn (97) e m (+) = [ F( ) ].! After sutabe cacuatons, we get then Coroary 3.7. Gn's mean dfference of the generazed nfated Posson dstrbuton s gven by e =[F(+)(-)-++ or n the form! ]+ e - [I 0 ()+I ()] ( j ) h( ) h( j ) j j 0 (3.) e [F(+)(-)-++! j ]+ F(½,, -4)- ( j ) h( ) h( j ). j 0 (3.) Defnton 3.7. We say that a random varabe Y s a subject to the generazed nfated ogarthmc dstrbuton f ts probabty functon s gven by the formua P(Y=) = p c f p c f,,...,,... (3.3)

Gn s mean dfference n the theory and appcaton to nfated dstrbutons 485 where c = -, 0<p<. n( p) Foowng as above, we get Coroary 3.7. Gn's mean dfference for the nfated (=0) ogarthmc dstrbuton s gven by p n ( )( ) =m - p p, ( p)n ( p) (3.4) p where m=- ( p )n( p ) (see: T. Gerstenkorn (97), formua (.3)). In the case when = we make use of (3.5) and of formua (.6) by Ramasubban (958) and aso of formua m (+) = - p ( p)n( p) (see: T. Gerstenkorn (97), formua (.)). Then, Gn's mean dfference for the generazed nfated ogarthmc dstrbuton s gven by p =[F(+)-+ - ] ( p )n( p ) ( p)n( p) p n ( )( ) p p + ( p)n ( p) j j 0 p ( j ) h( ) h( j ). (3.5) Defnton 3.8. We say that a random varabe s a subject to the generazed nfated geometrc dstrbuton f ts probabty functon s gven by qp f P(Y=) = p q f 0,,...,,,... (3.6) where +=, p+q =, p>0, q>0, 0, >0. Makng use of (3.) and (.8) by Ramasubban (958), we get Coroary 3.9. Gn's mean dfference for the nfated (=0) geometrc dstrbuton s gven by p p = q p, (3.7)

486 where m= p q. In the case when = we make use of (3.5) and of formua for the ncompete moment of ths dstrbuton p m (+) = (+)p + + ( F( )) (see: T. Gerstenkorn (97), formua (.7). q We then have Coroary 3.0. Gn's mean dfference for the generazed nfated geometrc dstrbuton s gven by p p =[F(+)-- ( F( )) +(+)p + ] q q p [ p j + ( j ) h( ) h( j ) ]. (3.8) j 0 Unversty of Trade Facuty of Mathematcs, Unversty of ód Char of Statstca Methods, Insttute of Econometrcs and Statstcs, Facuty of Economy and Socoogy, Unversty of ód TADEUSZ GERSTENKORN JOANNA GERSTENKORN REFERENCES V. CASTELLANO (965), Corrado Gn: a memor, Metron 4 (-4), pp. 3-35. E.L. CROW (958), The mean devaton of the Posson dstrbuton, Bometrka, 45 (3-4), pp. 556-559. G. DALL'AGLIO (965), Comportamento anntotco dee stme dea dfferenza meda e de rapporto d concentrazone, Metron 4 (-4), pp. 379-44. J.S. FRAME (945), Mean devaton of the bnomna dstrbuton, Amercan Mathematca Monthy, 5, pp. 377-379. T. GERSTENKORN (97), The recurrence reatons for the moments of the dscrete probabty dstrbutons, Dssertatones Mathematcae, 83, pp. -46. T. GERSTENKORN (975), Bemerkungen über de zentraen unvoständgen und absouten Momente der Póya-Verteung, Zastosowana Matematyk Appc. Math., 4 (4), pp. 579-597. T. GERSTENKORN (977), Jednowymarowe rozkady dyskretne ze zneksztacenem, n: Metody Statystyczne w Sterowanu Jakoc. Praca zborowa pod red. prof. Szymona Frkowcza, Ossoneum, Wrocaw, Sprawozdane z Konferencj PAN w Jabonnej 4-8 utego 975 r., pp. 95-08. (One and mutvarate dscrete probabty dstrbutons, n: Statstca Methods n Quaty Contro, compete edton by prof. Szymon Frkowcz - Reports of the conference of the Posh Academy of Scences at Jabonna, November 3-8, 975), Ossoneum, Wrocaw, Posh Academy of Scences, pp. 63-93, n Posh).

Gn s mean dfference n the theory and appcaton to nfated dstrbutons 487 T. GERSTENKORN (997), The Gn s mean dfference of an nfated dscrete dstrbuton, Proceedngs of 6 th Intern. Conf. on Mutvarate Statstca Anayss MSA 97, November 7-9 997, ód Unversty, Char of Statstca Methods, ed. Czesaw Domask, Darusz Parys, pp. 47-5. C. GINI (90), Indc d concentrazone e d dpendenza, Att dea III Runone dea Soceta Itaana per progresso dee scenze, Padua 90, pp. 453-469. C. GINI (9), Varabtà e mutabtà contrbuto ao studo dee dstrbuzon e dee reazon statstche, Stud Economco Gurdc dea R. Unverstà d Cagar, vo. III, Parte II, pp. 3-59. C. GINI (94), Sua msura dea concentrazone e dea varabta de caratter, Att de R. Isttuto Veneto d SS.LL.AA., a.a. 93-94, 73, parte II, pp. 03-48. C. GINI (9), Measurement of nequaty of ncomes, Economc Journa 3 (), pp. 4-6. C. GINI (96), The contrbuton of Itay to modern statstca methods, Journa of the Roya Statstca Socety, 89 (4), pp. 703-74. G.M. GIORGI (990), Bbographc portrat of the Gn concentraton rato, Metron, 48, pp. 83-. G. GIRONE (968a), Su moment e sua dstrbuzone dea dfferenza meda d un campone casuae d varab esponenza, Anna dea Facotà d Economa e Commerco de'unverstà deg Stud d Bar,, pp. 97-3. G. GIRONE (968b), La dstrbuzone dea dfferenza meda d un campone bernouano d varab esponenza, op. ct. pp. 5-3. N.L. JOHNSON (957), A note on the mean devaton of the bnoma dstrbuton, Bometrka, 44 (3-4), pp. 53-533. A.R. KAMAT (953), The thrd moment of Gn s mean dfference, Bometrka, 40 (3-4), pp. 45-45. S.K. KATTI (960), Moments of the absoute dfference and absoute devaton of dscrete dstrbutons, Annas of Mathematca Statstcs, 3 (), pp. 78-85. M.G. KENDALL, A. STUART (963), The advanced theory of statstcs, Vo., Dstrbuton Theory, II ed., Chares Grffn & Comp., London, Sec..0.3; Russan edton:,., 966. Z.A. LOMNICKI (95), The standard error of Gn s mean dfference, Annas of Mathematca Statstcs, 3, pp. 635-637. M.O. LORENZ (905), Methods of measurng the concentraton of weath, Journa of the Amercan Statstca Assocaton, 9, p. 09. B. MICHETTI, G. DALL'AGLIO (957), La dfferenza sempce meda, Statstca, 7 (), pp. 59-55. U.S. NAIR (936), Standard error of Gn s mean dfference, Bometrka, 8, p. 48. T.A. RAMASUBBAN (956), A -approxmaton to Gn s mean dfference, Journa of the Indan Socety of Agrcuture Statstcs, 8, p. 6. T.A RAMASUBBAN (958), The mean dfference and the mean devaton of some dscontnuous dstrbutons, Bometrka, 45 (3-4), pp. 549-556. T.A. RAMASUBBAN (959), The generazed mean dfferences of the bnoma and Posson dstrbutons, Bometrka, 46 (-), pp. 3-9. C.R. RAO (98), Dversty: ts measurement, decomposton, apportonment and anayss, Sankhy: The Indan Journa of Statstcs, Seres A, 44, Part, pp. -. H. RINNE (974), Statstk I: Voresungsunteragen für das Grundstudum der Wrtschaftswssenschaften m Fach Mathematk, Band I, Justus-Lebg-Unverstät, Gessen, R. RISSER, C.E. TRAYNARD (933, 957) Les Prncpes de a Statstque Mathématque, Lvre I, Séres Statstques, Gauther-Vars, Pars 933; ed. 957.

488 G. SALVEMINI (956), Varanza dea dfferenza meda de campon ottenut secondo o schema d estrazone n bocco, Metron, 8 (-), pp. 33-6. G. SALVEMINI (957), Dstrbuzone dea dfferenza meda de campon rcavat da una massa dscreta equdstrbuta, Att dea XVII Runone Scentfca dea Socetà Itaana d Statstca, Roma, pp. 69-88. S.N. SINGH (963), A note on nfated Posson dstrbuton, Journa of the Indan Statstca Assocaton, (3), pp. 40-44. M.P. SINGH (965/66), Infated bnoma dstrbuton, Journa of Scentfc Researches Banares Hndu Unversty 6 (), pp. 87-90. M.P. SINGH (966), A note on generazed nfated bnoma dstrbuton, Sankhyã: The Indan Journa of Statstcs, 8 () p. 99. G. UDNY YULE, M.G. KENDALL (958), An Introducton to the Theory of Statstcs, Chares Grffn & Co, London, 3rd ed. G. ZANARDI (973), La dfferenza sempce meda ne campone: schema con rpetzone, Laboratoro d Statstca - Facotà d Economa e Commerco, Unverstà d Veneza, pp. -. G. ZANARDI (974), La stma dea varanza dea dfferenza meda camponara: schema con rpetzone, Rvsta Itaana d Economa - Demografa e Statstca, 8 (4), pp. 67-87. R. ZIELISKI (97), Statstca Tabes, PWN, Warsaw. RIASSUNTO La dfferenza meda d Gn: teora e appcazone ae nfated dstrbutons In questo avoro vengono dscusse acune nteressant propretà dee dfferenze mede d Gn. I avoro, ne quae vengono consderat sa artco su rvste che monografe, costtusce un mportante ntegrazone de ampa rassegna, effettuata da G.M. Gorg ne 990, de avor basat sue dee d Gn. Vene anche presentata un appcazone dee dfferenze mede ae cosddette nfated dstrbutons, ampamente utzzate nea statstca matematca. SUMMARY Gn s mean dfference n the theory and appcaton to nfated dstrbutons In the paper we gve nterestng propertes of Gn s mean dfference. We thoroughy consder the approprate terature takng account of book pubcatons and artces. It consttutes an mportant compement to the extensve bbography of papers based on Gn s deas, presented by G.M. Gorg n 990. We show an appcaton of the mean dfference to nfated dstrbutons whch are of weght and nterest n statstca probems.