On the Link Between the Concepts of Kurtosis and Bipolarization. Abstract

Transcription

1 O the Lk etwee the Cocepts of Kurtoss ad polarzato Jacques SILE ar-ila Uversty Joseph Deutsch ar-ila Uversty Metal Haoka ar-ila Uversty h.d. studet) Abstract I a paper o the measuremet of the flatess of a come dstrbuto erreb ad Slber 989) showed how t was possble to derve from the dex a measure of the degree of Kurtoss of a dstrbuto whose defto made t qute smlar to the more famous earso measure of Kurtoss. Ths ote shows that t s possble to derve from the dex of flatess proposed by erreb ad Slber 989) a measure of bpolarzato that has all the mportat propertes oe would lke a bpolarzato dex to have. Ths paper was wrtte whle Jacques Slber was vstg the Laboratoro ccardo evell at the Collego Carlo Alberto, Mocaler, Italy. He s very thakful to the Laboratoro ad partcular to ts drector, ruo Cot, for ther very warm hosptalty. Ctato: SILE, Jacques, Joseph Deutsch, ad Metal Haoka, 007) "O the Lk etwee the Cocepts of Kurtoss ad polarzato." Ecoomcs ullet, Vol., No. 36 pp. -5 Submtted: September 5, 007. Accepted: October, 007. UL:

2 . Itroducto I a crtcal revew of the cocept of Kurtoss, alada ad Macllvray 988) wrote that t s best to defe kurtoss vaguely as the locato- ad scale-free movemet of probablty mass from the shoulders of a dstrbuto to ts cetre ad tals. As stressed by uesch 007) f oe starts wth a ormal dstrbuto ad moves scores from the shoulders to the ceter ad the tals, keepg varace costat, kurtoss s creased. The dstrbuto wll lkely appear more peaked the ceter ad fatter the tals. I a paper o the measuremet of the flatess of a come dstrbuto erreb ad Slber 989) showed how t was possble to derve from the dex a measure of the degree of Kurtoss of a dstrbuto whose defto made t qute smlar to the more famous earso measure of Kurtoss. The purpose of ths ote s to show that t s also possble to derve from the dex of flatess proposed by erreb ad Slber 989) a measure of bpolarzato that has all the mportat propertes oe would lke a bpolarzato dex to have. Secto II recalls the ma results obtaed by erreb ad Slber 989) whle Secto III proves the lk betwee ther measure of the flatess of a dstrbuto ad the cocept of bpolarzato.. O the Measuremet of the Flatess of a Icome Dstrbuto: earso s 895) famous Kurtoss dex s defed as x x) K ) x x) where, our case, x would be the come of dvdual, the total umber of dvduals ad x the average come the populato. Expresso ) may be also expressed as / x x) + / ) + x x ) K ) / x x) + x x) / ) + Assumg that x x Kx Kx, erreb ad Slber 989) have proposed a alteratve measure of Kurtoss defed as / [ + ) x m] + [ m 3 ) x ] K 3) / [ + ) x m] + [ m ) x ] / ) + / ) + where m s the meda of the come dstrbuto. The smlarty betwee ) ad 3) s clear. I earso s 895) formulato the cetral value of referece s the arthmetc mea of the dstrbuto whle the formulato suggested by erreb ad Slber 989) the cetral value s the meda. ut ote that both formulatos the gaps wth respect to the cetral value are gve a hgher weght the umerator tha the deomator.

3 erreb ad Slber 989) have however show that 3) could also be expressed as + ) K where, ad are respectvely the mea dfferece of the whole dstrbuto, of the dstrbuto of the rch dvduals, the latter beg defed as those wth a come hgher tha the meda come, ad of the poor dvduals, the latter beg defed as those wth a come lower tha the meda come. More precsely, ad are defed as x x 5) / / / ) x x 6) 7) x x / ) / ) + / ) + 3. The Lk etwee The Idex of Flatess ad the Measuremet of polarzato: Let us ow defe a dex as 8) K Sce the come dstrbutos of the rch ad of the poor do ot overlap, t ca be show see, Nygärd ad Sadström, 98) that such a case / ) + ) + 9) where s the betwee groups mea dfferece, the groups represetg the poor ad the rch as they were defed before. It s fact easy to prove see, erreb ad Slber, 989) that, sce these two groups are of equal sze, the betwee groups mea dfferece, whch assumes that all the rch ear the average come y of the rch ad all the poor ear the average come y of the poor, may be expressed as / ) y y ) 0) Combg ), 8) ad 9) we ed up wth [/ ) + ) / )] [ / ) + )) / ] [ / ) + ) + ) / ) + ))]/ / ) + ))]/ ) [ / ) + ))]/[ + / ) + ))] ) [ Expresso ) shows clearly that wll decrease whe the wth groups dsperso creases, that s, whe or creases. I addto, sce the weght of ) s greater ts

4 umerator tha ts deomator, t s also easy to see that wll crease whe the betwee groups dsperso creases. These are however the two prcpal features of a bpolarzato dex whch are ofte called, the lterature, the axoms of No-Decreasg Spread ad No-Decreasg polarty see, Esteba ad ay, 99, olfso, 99 ad 997, ag ad Tsu, 000, Chakravarty ad Maumder, 00 ad Chakravarty et al., 007) e should also remember that the dex for the whole come dstrbuto, the betwee groups dex, the dex amog the poor ad the dex amog the rch may be respectvely be expressed see, Kedall ad Stuart, 969, for a geeral defto of the dex) as / ) 3) / ) ) / ) / y ) 5) / ) / y ) 6) Fally, the case of o-overlappg groups, the overall dex may be expressed see, Slber, 989b) as + 7) where refers to the wth groups dex ad s wrtte, our case, as f s + f s 8) where f,f, s ad s refer respectvely to the populato shares of the groups of poor ad rch ad to the come shares of these two groups. Sce we assumed that f f / 9) ad sce our case s / ) y 0) ad s / ) y ) we ed up, combg expressos 8) to ) wth / ) y + / ) y ) If we combe ow expressos ), ), 5), 6), 7) ad ) t s easy to show that we wll ed up wth 3

5 [/ ) y + / ) y ]) / ) 3) / Note frst the relatve smlarty betwee the defto of the bpolarzato dex gve 3) ad the polarzato dex proposed by olfso 99) whch was defed as ) y / m) ) Secod ote also the smlarty betwee the polarzato dex suggested by Kabur ad Zhag 00) who defed ther dex as I / Σ g wg I g ) 5) where I refers to ay betwee groups equalty dex, I g to the correspodg equalty dex wth group g ad w g to the weght of group g geerally a populato weght but the case of the dex t has to be the product of the populato ad come weght of group g, as dcated Slber, 989). I other words the case of the dex, would be defed as / 6) whch s a ubouded dex at the dfferece of the dex proposed ths paper. Note also the lk betwee the dces ad whe the latter s defed o the bass of the Idex. Combg 7), 3) ad 6) we derve that ) / + ) [ / ) ]/[ / ) + ] ) / + ) 7) Clearly both dces move the same drecto sce / > 0.. Cocluso: e have attempted ths ote to show that, at least the case of two o-overlappg groups of equal sze, there was a clear lk betwee the cocept of bpolarzato ad that of the kurtoss of a come dstrbuto. The aalyss was lmted to the case of two o-overlappg groups of equal sze. It seems that the defto of the polarzato dex could be easly exteded to that of several o overlappg groups but the exstece of a lk such a case wth the cocept of kurtoss remas to be prove. The exteso of the use of the dex to the case of overlappg groups s probably more problematc ad addtoal work s certaly requred before some coclusos may be draw. efereces alada, K.. ad H. L. Macllvray 988) Kurtoss: Statstca, : -9. A Crtcal evew, Amerca erreb, Z. M. ad J. Slber 989) Deprvato, the Idex of Iequalty ad the Flatess of a Icome Dstrbuto, Mathematcal Socal Sceces 8: 9-37.

6 Chakravarty, S.. ad A. Maumder 00) Iequalty, olarzato ad elfare: Theory ad Applcatos, Australa Ecoomc apers, 0: -3. Chakravarty, S.., A. Maumder ad S. oy 007) A Treatmet of Absolute Idces of olarzato, Japaese Ecoomc evew, 58: Esteba, J.-M. ad D. ay 99) O the Measuremet of olarzato, Ecoometrca, 6: Kedall, M.. ad A. Stuart 969) The Advaced Theory of Statstcs, Charles rffe: Lodo. Nygärd, F. ad A. Sadström 98) Measurg Icome Iequalty, Almqvst ad ksell Iteratoal: Stockholm. earso, K. 895) Cotrbutos to the mathematcal theory of evoluto, II: Skew varato homogeeous materal, hlosophcal Trasactos of the oyal Socety of Lodo, 86: 33-. Slber, J. 989) Factor Compoets, opulato Subgroups ad the Computato of the Idex of Iequalty, evew of Ecoomcs ad Statstcs 7: ag, Y. Q. ad K. Y. Tsu 000) olarzato Ordergs ad New Classes of olarzato Idces, Joural of ublc Ecoomc Theory, : olfso, M. C. 99) he Iequaltes Dverge, Amerca Ecoomc evew, apers ad roceedgs, 8: olfso, M. C. 997) Dverget Iequaltes: Theory ad Emprcal esults, evew of Icome ad ealth, 3: 0-. uesch, K., Skewess, Kurtoss ad the Normal Curve, Karl uesch s Statstcal Lessos, avalable o 5