International Journal of Mathematical Archive-6(1), 2015, Available online through ISSN

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1 Iteratoal Joural of Mathematcal Archve-6(), 5, Avalable ole through ISSN ORDER STATISTICS, LORENZ TRANSFORM AND THE CVAR RISK MEASURE Werer Hürlma* Swss Mathematcal Socety, Feldstrasse 45, CH-84 Zürch. (Receved O: 6--4; Revsed & Accepted O: 3--5) ABSTRACT The class of lear fuctos of order statstcs or L-estmates s cosdered. Uder fte varaces ad other sutable restrctos, t s kow that L-estmates coverge dstrbuto to a ormal dstrbuto as the sample sze creases to fty. Ths result s appled to obta appromate cofdece tervals for the Lorez trasform ad the codtoal value-at-rsk measure usg L-estmates case the data follows a appromate geeralsed Pareto dstrbuto wth fte varace. By fte varace, the goodess-of-ft of the L-estmate compared to the true Lorez trasform s measured usg the epected relatve error of appromato. Mathematcs Subject Classfcato: 6F5, 6E7, 6E, 6P5. Keywords: order statstcs, L-estmate, Lorez trasform, codtoal value-at-rsk, geeralsed Pareto, ormal dstrbuto, appromate cofdece terval. INTRODUCTION A mportat class of statstcs cossts of the lear fuctos of order statstcs, usually called L-estmates (e.g. Rychlk [7]). It appears to have bee frst etesvely studed by Percy Daell 9 (see Stgler []). Gve the order statstcs X...,..., of sze ad a sequece of real umbers ( ) X of a radom sample X X X ( ) c,..., c, the L-estmate s the statstcs defed by S c X S X X, the -trmmed mea.well-kow eamples clude the sample mea S ( ) X ( ) teger less tha or equal to, ad G s mea dfferece [ ] ( ) [4]). g ( ) X ( ), where [ ] deotes the greatest (G [6], Davd The preset work emphaszes some essetal statstcal propertes of the codtoal value-at-rsk or epected shortfall measure, whch has bee recogsed as a mportat rsk measure moder rsk maagemet (a etesve recet revew s Nadarajah et al. [5]). For a radom varable X wth dstrbuto fucto F( ), R, ad quatle fucto Q( u) f { F( ) u} level ( ) where, u ( ), s defed as follows (e.g. Hürlma [9], Proposto.): L X Q( u) du,, the codtoal value-at-rsk (CVaR) measure to the cofdece CVaR X E X L X { }, (.) deotes the Lorez trasform of X. The relatoshp (.) suggests two ways of statstcal estmato. By kow mea, oe uses the obvous L-estmate of the Lorez trasform, (.) L X X ( ) Correspodg Author: Werer Hürlma*, E-mal: whurlma@bluew.ch. Iteratoal Joural of Mathematcal Archve- 6(), Ja. 5 39

2 Werer Hürlma*/ Order Statstcs, Lorez Trasform ad The Cvar Rsk Measure / IJMA- 6(), Ja.-5. { } to estmate CVaR by ( ) E[ X ] L ( ) [ X ] mmedately the followg L-estmate: ( ) CVaR X X. Alteratvely, from (.) oe obtas through (.) ( ). (.3) I the rsk maagemet cotet, where these L-estmates are calculated usg market or smulated data, t s useful to kow the asymptotc dstrbuto of these quattes. Cosder L-estmates of the form where J u, u ( ) S J X ( ), (.4),, s a approprate weght fucto. Formulas for the asymptotc mea ad varace of such L-estmates have bee foud sce Jug []. Uder sutable restrctos, partcular fte varace of X,,...,, t has bee kow for a log tme that L-estmates coverge dstrbuto to a ormal dstrbuto as the sample sze creases to fty (e.g. Govdarajulu et al. [7], Cheroff et al. [3], Moore [4], Shorack [] ad Stgler [], [3]). A more detaled accout of the cotet follows. Secto summarses a ma result of Stgler [3] ad apples t to the L-estmates (.), (.3). Its use s llustrated wth the geeralsed Pareto dstrbuto Secto 3. Sce scaled ecesses over hgh thresholds are the lmt geeralsed Pareto dstrbuted by the theorem of Pckads [6], ad Balkema ad de Haa [], a dscusso of the dstrbuto propertes of these quattes should be based upo ths choce (e.g. McNel [], Secto 3). By fte varace, the L-estmates (.), (.3) have a asymptotc ormal dstrbuto. Ths allows the costructo of appromate cofdece tervals for the Lorez trasform ad the CVaR measure. To llustrate, we lst tabular form the crtcal sample sze requred to estmate these quattes wth a fed precso. I case the varace s fte, the precedg results do ot apply. To measure the goodess of appromato of the true Lorez trasform by the L-estmate (.) for the geeralsed Pareto wth fte varace, we calculate Secto 4 the epected relatve error of appromato. To obta gve relatve errors, a creasg sample sze s requred by creasg cofdece level.. ASYMPTOTIC DISTRIBUTION OF L-ESTIMATES BY FINITE VARIANCE The smplest ma result about the asymptotc ormalty of L-estmates of the form (.4) s due to Stgler [3], Theorem. The reader terested more detals ad up-to-date mathematcal treatmet s refereed to Serflg [9], Se [8], ad Jureckovc ad Se []. Theorem.: Let X X X,..., be a radom sample of sze such that E[ X ] <,,...,, ad let S J X be a L-estmate. If J u ( ), u (, ), s bouded ad cotuous almost everywhere at u F( ), R, the oe has [ ] z lm P S E S ( ) e dz S Φ, (.) π [ ] ad the asymptotc mea ad varace of the L-estmate are gve by Proof: See Stgler [3]. [ ] J, F lm E S Q( u) J( u) du, (.) J F S J F F J F y F y dy d < y (, ) lm [ ] [ ]. (.3) Let us apply ths mportat result to the L-estmates (.) ad (.3) of the Lorez trasform ad the CVaR measure. 5, IJMA. All Rghts Reserved 4

3 Werer Hürlma*/ Order Statstcs, Lorez Trasform ad The Cvar Rsk Measure / IJMA- 6(), Ja ASYMPTOTIC DISTRIBUTION OF THE SAMPLE CVaR Cosder the L-estmate (.4) wth J( u) f u ad J( u) ( ) f u >, whch yelds (.3). From (.) oe obtas for the asymptotc mea ( J, F) Q( u) du CVaR [ X ], (.4) where the last equalty follows by Defto (.) ad Proposto. Hürlma [9]. Therefore, the mea of the CVaR L-estmate s asymptotcally ubased. The asymptotc varace s determed by the varace of the stop-loss radom varable ( X d ) wth value-at-rsk d Q ( ), amely J, F Var X Q π Q π Q where π ( k ) k ( ) E ( X ), k, Q( ) detfes wth the usual value-at-rsk fuctoal VaR [ X ], (.5), deote the stop-loss trasforms of degree oe ad two, ad the quatle. The formula (.5) s derved as follows. Usg (.3) ad the facts (e.g. Hürlma [8], Theorem.) π ( ) ( ( ) F ( y) dy, F ( y) F( y), π ) ( ) π ( y) dy, oe gets J, F F ( y) dy F( ) d π ( ) F( ) d Q( ) Q( ) π ( ) d π ( ) F ( ) d π Q π Q, Q( ) Q( ) where the value of the last tegral follows from the relato (use partal tegrato) π Q( ) () ( [ Q( ) ] (.6) () ) F ( ) d π () π ( ) F ( ) d. (.7) Q( ).. ASYMPTOTIC DISTRIBUTION OF THE SAMPLE LORENZ TRANSFORM The L-estmate (.) of the Lorez trasform ca be wrtte the form (.4) wth J( u) f u ad J( u) f u >. The asymptotc mea of ths estmate equals ( J, F) Q( u) du L [ X ], (.8) whch shows that the mea of the Lorez trasform L-estmate s asymptotcally ubased. To determe the asymptotc varace usg (.3), we assume that X, as wll be the case our applcato to the geeralsed Pareto dstrbuto Secto 3. We show the formula Usg (.3) we have () () () () ( J, F ) π ( ) π [ Q( ) ] π ( ) π [ Q( ) ] () () [ Q( ) π ( )] π [ Q( ) ]. Q( ) Q( ) Q( ) ( J, F) F ( y) dy F( ) d { π ( ) π [ Q ]} F( ) d Q( ) Q( ) Q( ) Q( ) π ( ) d π ( ) F ( ) d π [ Q ] d F ( ) d. (.9) 5, IJMA. All Rghts Reserved 4

4 Werer Hürlma*/ Order Statstcs, Lorez Trasform ad The Cvar Rsk Measure / IJMA- 6(), Ja.-5. The formula (.9) follows by otg that Q( ) π ( ) d { π ( ) π [ Q ]}, π ( ) F ( ) d π ( ) π [ Q ] Q( ) [ Q ] F ( ) d π ( ) π 3. GENERALISED PARETO WITH FINITE VARIANCE. { } Q( ), As lmtg dstrbuto of scaled ecesses over hgh thresholds, the geeralsed Pareto dstrbuto (GPD) s a approprate parametrc dstrbuto for use facal rsk maagemet (e.g. Embrechts et al. [5], McNel et al. [3]). Its survval fucto s descrbed by F ( ), >, >, >. (3.) The r-th momet ests oly f < r. Uder the assumpto < oe has E[ X ] <, Var[ X ] <. (3.) Through calculato oe gets the stop-loss trasforms of degree oe ad two ( π ) ( ) F ( y) dy F ( ) π Iserted (.6) usg that F[ Q ], (3.3) π ( y) dy F ( ). (3.4) ( )( ) oe obtas ( J F), ( ) ( ) ( ). (3.5) Sce the L-estmate (.3) has a asymptotc ormal dstrbuto, a appromate ε -cofdece terval for the codtoal value-at-rsk from a GPD wth fte varace reads CVaR where Z Φ ( / ) s the ( ε / ) precso ε ε J F ( J, F) Z (, ) [ X ] Z, CVaR [ X ] ε ε ( J, F) ε Table 3. lsts the crtcal sample sze, (3.6) -quatle of the stadard ormal dstrbuto. Ths terval has the Z. (3.7) J F Z (, ) ε 4 fed precso 5%, ε 5%,, but by varyg < ad. requred to estmate CVaR [ X ] usg (3.6) wth 5, IJMA. All Rghts Reserved 4

5 Werer Hürlma*/ Order Statstcs, Lorez Trasform ad The Cvar Rsk Measure / IJMA- 6(), Ja.-5. Table-3.: Crtcal sample sze by fed precso for CVaR estmato 95% 99% 99.9% ' '98 97 ' The asymptotc varace of the L-estmate (.) s obtaed from (.9). We eed the quatle fucto of the GPD, that s Q( u) [( u) ], u (, ), (3.8) whch s obtaed from (3.). Isertg ths ad the stop-loss trasform formulas (3.3) ad (3.4) to (.9) oe obtas ( J, F) ( ) ( ) ( ) ( ) ( ) ( ). (3.9) Aga, sce the L-estmate (.) has a asymptotc ormal dstrbuto, a appromate ε -cofdece terval for the Lorez trasform from a GPD wth fte varace reads L ( J, F) ( J, F) Z L X [ X ] Z, L [ X ] ε ε Table 3. lsts the crtcal sample sze requred to estmate 5%, ε 5%,, but by varyg < ad.. (3.) usg (3.6) wth fed precso Table 3.: Crtcal sample sze by fed precso for Lorez trasform estmato 95% 99% 99.9% Some geeral commets cocerg Tables 3. ad 3. ad ther comparso are order. As < the value, a creasg crtcal sample sze s requred. For small < 4 (3.6) for codtoal value-at-rsk. By kow mea ad 4 trasform t usg the formula (.). 4. GENERALISED PARETO WITH INFINITE VARIANCE comes closer to, oe should use the cofdece terval, t s preferable to use the cofdece terval (3.) ad I case the varace s fte, the results of Sectos ad 3 do ot apply. Ths occurs for the GPD wth parameter [, ), for whch the mea s however fte. A theoretcal justfcato of ths surace rsk model s foud Aeb et al. []. A straghtforward calculato usg (3.8) yelds the Lorez trasform ( ) L [ X ] Q u du 5, IJMA. All Rghts Reserved 43. (4.)

6 Werer Hürlma*/ Order Statstcs, Lorez Trasform ad The Cvar Rsk Measure / IJMA- 6(), Ja.-5. To measure the goodess of appromato of the Lorez trasform by the L-estmate (.) for the GPD wth fte varace, let us calculate the epected relatve error of appromato defed by E [ ] L [ X ] E L X L X 95% 99% 99.9% , IJMA. All Rghts Reserved 44. (4.) Usg the eplct formulas for the dstrbuto fuctos of order statstcs ad makg a trasformato of varables oe obtas that [ ] E L X I, I Q( u) N ( u) du, N u u ( u),,...,. Note that the N s s the Berste bass of the polyomals of degree ot eceedg. Sce > by assumpto, oe obtas [( ) ] ( ) N u u u u du B(, ) B(, ),,...,, q ( p) ( q) B p q d p where (, ) ( ) ( p q) s a beta coeffcet. Observe that ( ) ( ) B( )!!,! ad, usg the recurrece relato ( ) ( ), oe gets B(, ) k ( k ) ( ) ( k ) k Isertg above, the mea of the L-estmate (.) equals [ [ X ]] E L ( ) ( ) ( ) ( ) ( k ) k k ( k ) (4.3) (4.4), (4.5) (4.6) For the typcal parameter values (,, ) (,, ) relatve error of appromato by varyg ad the sample sze. k k. k k C, C k k. (4.7) 7 (e.g. McNel [], p. 9), the Table 4. dsplays the epected Table-4.: Epected relatve error of appromato for the sample Lorez trasform %

7 Werer Hürlma*/ Order Statstcs, Lorez Trasform ad The Cvar Rsk Measure / IJMA- 6(), Ja.-5. We ote that the sample Lorez trasform overestmates average the theoretcal Lorez trasform. Ths meas that the correspodg sample CVaR the tet after formula (.) uderestmates geeral the theoretcal value. To obta relatve errors of a order less tha a gve percetage, a creasg sample sze s requred for creasg values of. REFERENCES. Aeb, M., Embrechts, P. ad Th. Mkosch. A large clam de. Bullet of the Swss Assocato of Actuares (99), Balkema, A. ad L. de Haa (974). Resdual lfe tme at great age. Aals of Probablty (974), Cheroff, H., Gastwrth, J.L. ad M.V. Johs. Asymptotc dstrbuto of lear combatos of fuctos of order statstcs wth applcatos to estmato. Aals of Mathematcal Statstcs 38 (967), Davd, H.A. (98). Order Statstcs ( d ed.). Joh Wley, New York, Embrechts, P., Klüppelberg, C. ad Th. Mkosch. Modellg Etremal Evets for Isurace ad Face. Applcatos of Mathematcs Stochastc Modellg ad Appled Probablty, vol. 33. Sprger, New York, G, C. (9). Varabltà é mutablta, cotrbuto allo studo delle dstrbuzo e delle relazo statstche. Stud Ecoomco-Gurdc della R. Uverstà d Caglar 3 (9), part, -, Govdarajulu, Z., LeCam, L. ad M. Raghavachar. Geeralsatos of theorems of Cheroff ad Savage o the asymptotc ormalty of test statstcs. Ffth Berkeley Symp. Math. Statst. Prob. (965), Hürlma, W. Hgher-degree stop-loss trasforms ad stochastc orders (II) applcatos. Blätter der Deutsche Gesellschaft für Vers.math. 4(3) (), Hürlma, W. Codtoal value-at-rsk bouds for compoud Posso rsks ad a ormal appromato. Joural of Appled Mathematcs 3(3) (3), Jug, J. O lear estmates defed by a cotuous weght fucto. Arkv fur Matematk 3 (955), Jureckova, J. ad P.K. Se. Robust Statstcal Procedures: Asymptotcs ad Iterrelatos. Joh Wley, New York, McNel, A.J. Estmatg the tals of loss severty dstrbutos usg etreme value theory. ASTIN Bullet 7 (997), McNel, A.J., Frey, R. ad P. Embrechts. Quattatve Rsk Maagemet: Cocepts, Techques ad Tools. Prceto Seres Face, Prceto, New Jersey, Moore, D.S. A elemetary proof of asymptotc ormalty of lear fuctos of order statstcs. Aals of Mathematcal Statstcs 39 (968), Nadarajah, S., Zhag, B. ad S. Cha (4). Estmato methods for epected shortfall. Quattatve Face 4() (4), Pckads, J. Statstcal ferece usg etreme order statstcs. The Aals of Statstcs 3 (975), Rychlk, T. Bouds for epectatos of L-estmates. I: Balakrsha, N. ad C.R. Rao (Eds.). Order Statstcs : Theory ad Methods. Hadbook of Statstcs, vol. 6. Elsever Scece B.V., Amsterdam, Se, P.K. Sequetal Noparametrc: Ivarace Prcples ad Statstcal Iferece. Joh Wley, New York, Serflg, R.J. Appromato Theorems of Mathematcal Statstcs. Joh Wley, New York, 98.. Shorack, G.R. Asymptotc ormalty of lear combatos of fuctos of order statstcs. Aals of Mathematcal Statstcs 4 (969), Stgler, S.M. Lear fuctos of order statstcs. Aals of Mathematcal Statstcs 4 (969), Stgler, S.M. Smo Newcomb, Percy Daell, ad the hstory of robust estmato Joural of the Amerca Statstcal Assocato 68 (973), Stgler, S.M. Lear fuctos of order statstcs wth smooth weght fuctos. The Aals of Statstcs (974), Source of support: Nl, Coflct of terest: Noe Declared [Copy rght 4. Ths s a Ope Access artcle dstrbuted uder the terms of the Iteratoal Joural of Mathematcal Archve (IJMA), whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted.] 5, IJMA. All Rghts Reserved 45