Tail Factor Convergence in Sherman s Inverse Power Curve Loss Development Factor Model

Transcription

1 T tor Covergee Sherm s Iverse Power Curve Loss Deveopmet tor Moe Jo Evs Astrt The fte prout of the ge-to-ge eveopmet ftors Sherm s verse power urve moe s prove to overge to fte umer whe the power prmeter s ess th - tertvey to verge to fty whe the power prmeter s - or greter or the overget prmeter vues smpe formu s erve terms of y fte prout of ge-to-ge ftors for the epots of terv otg the mt of the fte prout These epots overge to the mt s the fte tme utoff pot reses or y fte tme utoff the prout of ge-to-ge ftors es eow the terv thus the ower epot of the terv s wys etter estmte of the mt th the fte prout tsef Sever umer empes re ue for ustrto Keywors T tor Iverse Power Curve BACKGROUND AND INTRODUCTION t ft I 984 Sherm [5] fou tht verse power urve of the form empr ge-to-ge oss eveopmet ftors etter th sever other s futo forms he teste Lowe Mohrm 985 [3] epresse oer out the overgee of the prout of the ge-to-ge ftors Boor 2006 [ p 373] the CAS T tor Workg Prty 203 [2 p 52] ote tht there hs ee o kow ose form epresso tht ppromtes the t geerte y the verse power urve I prte the ge-to-ge eveopmet ftors proue y the urve re mutpe together out to some fte ge utoff suh s t = 80 to proue umutve eveopmet ftor The mpt of ftors eyo tht ge to utmte or the t ftor eyo the utoff ths se t = 8 s ssume to e egge Atertvey f the mpt of the t ftor s ot egge the some other moeg oserto must form the seeto of the utoff tme The potet ger the ssumpto of egge t ftor mpt s ustrte Te Two fferet sets of prmeters shre the sme t ge-to-ge ftor of 0 t t = the sme umutve ftor of 30 from t = to 00 However whe the umutve ftor for Empe usg power prmeter = 40 grows oy tte pst t = 00 Empe 2 usg = 05 ppers to zoom towr fty the t Csuty Atur Soety E-orum 204-Voume Copyrght 204 Nto Cou o Compesto Isure I A Rghts Reserve

2 Covergee Of Sherm s Iverse Power Curve T tor Te : Empes of Apprety Coverget Dverget T tors for the Iverse Power Curve Moe Prmeter Vues Prmeters Empe Empe Cumutve Deveopmet tors rom to Empe Empe E E E+4 Ths pper uses s re yss ([4] eg str tetook referee to prove tht the fte prout of the ge-to-ge ftors overges to fte umer whe the power prmeter s ess th - verges to + whe - Note ths pper we refer to sequee tht reses wthout y upper ou s vergg to + or hvg mt of + urthermore whe < - for y fte prout of the ge-to-ge ftors up to spef ge there s smpe formu for terv otg the mt of the fte prout As reses the terv eomes tghter the epots eh overge to the mt of the fte prout The ower epot of ths terv s wys etter estmte of the fte prout th the fte prout of the ge-to-ge ftors whh s wys ess th the ower epot It s worth otg g tht t vergee oes ot eessry me the moe s v ut smpy tht y spef fte utoff pot shou e otherwse justfe or overget t ether utoff pot must st e justfe y some other oserto or re must e tke tht the t ftor pst the utoff s resoy ose to The terv 2 Csuty Atur Soety E-orum 204-Voume Copyrght 204 Nto Cou o Compesto Isure I A Rghts Reserve

3 Covergee Of Sherm s Iverse Power Curve T tor estmte erve ths pper hep swer the tter questo The proof of overgee/vergee s out Seto 2 wth the proof of sever usefu emms Appe A The terv estmte s erve Seto 22 Numer empes of the progressve overgee/vergee of the fte prout the terv estmte of the fte prout for sever sets of prmeters re show Seto 23 2 CONVERGENCE THEOREM AND LIMIT ESTIMATION oowg the otto ovetos of the reet CAS workg prty [2] the remer of ths pper ste of t s use for ge or tme 2 Sttemet Proof of Prmry Theorem rst we w set up efto for the fte prout of the ge-to-ge ftors the verse power urve moe Defto: s postve teger where > 0 0 re re umers Note ths efto ues ses where egs t hgher vue th s the prmeter e rese to he suh ses It s so worth otg tht 0 tht w e use susequet ervtos Theorem key ft ( If - the m Csuty Atur Soety E-orum 204-Voume 3 Copyrght 204 Nto Cou o Compesto Isure I A Rghts Reserve

4 Covergee Of Sherm s Iverse Power Curve T tor ( If < - the m ests Proof: ( or y sequee of umers 0 where 2 the equty hos org to Lemm A3 Appyg ths we hve If - the m org to Lemm A osequety m ( By Lemm A2 Summg over gves og for y 0 so og og og If < - the L m ests s ess th + org to Lemm A Now ote tht og s resg sequee euse og mpes tht 0 s oue y L Cosequety m og ess th + So m ests s ess th + ests s 4 Csuty Atur Soety E-orum 204-Voume Copyrght 204 Nto Cou o Compesto Isure I A Rghts Reserve

5 Covergee Of Sherm s Iverse Power Curve T tor Csuty Atur Soety E-orum 204-Voume 5 Copyrght 204 Nto Cou o Compesto Isure I A Rghts Reserve 22 A Iterv Estmte for the Ifte Prout Lmt or the overget se of < - t s posse to ostrut usefu terv estmte for the fte prout Defto: The t upper ou ftor s ep U Defto: The t ower ou ftor s L Theorem 2 Let m If < - the: ( ( m U ( ( m L ( U L Proof: ( + < 0 mpes tht 0 m osequety m ep

6 Covergee Of Sherm s Iverse Power Curve T tor 6 Csuty Atur Soety E-orum 204-Voume Copyrght 204 Nto Cou o Compesto Isure I A Rghts Reserve ( + < 0 mpes m m ( Tkg the ogrthm of the t ftor ppyg oug tehques esre Lemms A A2 og Epoettg proues ep Cosequety ( ( ( U Smry usg tehques from Lemms A A3 proues Cosequety ( ( ( L Ths ompetes the proof of Theorem 2 The ower epot of the estmto terv s wys etter estmte of the fte prout se L osequety L The t ou ftors re omputtoy smpe eve for rge vues of gve mesure of the retve wth of the estmto terv pror to og the omputtoy tese uto of the fte prout or empe to heve ert trget U for the

7 Covergee Of Sherm s Iverse Power Curve T tor upper ou requres ogu A more reevt mesure of retve error ut wthout y smpe formu for tht the uthor s wre of s the rto of the t upper ou ftor to the t ower ou ftor U / L ep Empe : A upper ou ftor trget set t U = 0 for the prmeter vues = = -40 = requres 78 However y 29 the rto of the t upper ou ftor to the t ower ou ftor s out 0 23 More Numer Empes Te 2 shows s fferet sets of prmeters eh of whh proues ge-to-ge ftor t = of 0 umutve ftor from = to 00 of 30 The prmeter sets re ee y set of vues { } for the power prmeter or = - the vergee hppes very sowy ut for = - the overgee hppes remrky sowy To heve 0 U for the = - prmeter set wou requre though y U / L 0 st stroomy sow rte of overgee Csuty Atur Soety E-orum 204-Voume 7 Copyrght 204 Nto Cou o Compesto Isure I A Rghts Reserve

8 Covergee Of Sherm s Iverse Power Curve T tor Te 2: Empes of te Deveopmet tor Prouts Iterv Estmtes or Ifte Deveopmet tor Prouts Prmeter Vues Prmeters Empe 3 Empe 4 Empe Cumutve Deveopmet tor Prout (Ifte Prout Lower Bou Ifte Prout Upper Bou Empe 3 Empe 4 Empe 5 00 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Prmeter Vues Prmeters Empe 6 Empe 7 Empe Cumutve Deveopmet tor Prout Empe 6 Empe 7 Empe E E E E+94 8 Csuty Atur Soety E-orum 204-Voume Copyrght 204 Nto Cou o Compesto Isure I A Rghts Reserve

9 Covergee Of Sherm s Iverse Power Curve T tor Akowegmet The uthor s grety thkfu to Joh Roertso D Corro Le Herk for revew ommets o ths pper Appe A - Lemms Lemm A Let e postve teger 0 k ( If the m k k ( If the m k ests Proof: It suffes to show overgee or vergee for m k se k s fte umer k or k 0 k therefore m k or k 0 k s strty eresg futo of k therefore there s swh equty k k t t k k t k t osequety t t k k t t Sovg the tegrs whe Csuty Atur Soety E-orum 204-Voume 9 Copyrght 204 Nto Cou o Compesto Isure I A Rghts Reserve

10 Covergee Of Sherm s Iverse Power Curve T tor k k or tkg mts proues m k k I ths se the upper ou of the equty s fte umer mpes overgee to fte umer se the sequee of prt sums the me s o-eresg k or 0 tkg mts resuts m k se ths se the ower ou of the erer equty verges m or the se tegrto of the erer equty es to og k k og Oe g tkg mts es to m k k from the ower ou of the equty vergg m og Lemm A2 If > 0 the og 0 Csuty Atur Soety E-orum 204-Voume Copyrght 204 Nto Cou o Compesto Isure I A Rghts Reserve

11 Covergee Of Sherm s Iverse Power Curve T tor Csuty Atur Soety E-orum 204-Voume Copyrght 204 Nto Cou o Compesto Isure I A Rghts Reserve Proof: If t the 0 / t osequety 0 / t t So 0 / t t t sovg the tegrs proues 0 og Lemm A3 for y sequee of umers 0 teger 2 Proof: We proee y uto or 2 se 0 2 t foows tht Assume the ouso of the emm s true for where 2 We w show tht the emm s the true for + I geer But so

12 whh estshes the emm Covergee Of Sherm s Iverse Power Curve T tor REERENCES [] Boor J Estmtg T Deveopmet tors: Wht to o Whe the Trge Rus Out Csuty Atur Soety orum Wter 2006 [2] CAS T tor Workg Prty The Estmto of Loss Deveopmet T tors: A Summry Report Csuty Atur Soety orum 203 [3] Lowe S P Mohrm D Dsusso of Etrpotg Smoothg Iterpotg Deveopmet tors PCAS LXXII 985 [4] Ru W Prpes of Mthemt Ayss 3r e MGrw-H I 976 [5] Sherm R E Etrpotg Smoothg Iterpotg Deveopmet tors PCAS LXXI 984 Bogrphy of the Author Joth Evs CAS SA CA CERA MAAA WCP s tury t the Nto Cou o Compesto Isure Bo Rto L Hs work prmry voves reserh eveopmet of NCCI s rtemkg reservg tstrophe moeg proeures 2 Csuty Atur Soety E-orum 204-Voume Copyrght 204 Nto Cou o Compesto Isure I A Rghts Reserve