Bias - Corrected Maximum Likelihood Estimation of the Parameters of the Generalized Pareto Distribution

Transcription

1 Ecoometcs Wokg Pape EWP090 ISSN Depatmet of Ecoomcs Bas - Coected Maxmum Lkelhood Estmato of the Paametes of the Geealzed Paeto Dstbuto Davd E. Gles Depatmet of Ecoomcs, Uvest of Vctoa Vctoa, B.C., Caada V8W Y Hu Feg Depatmet of Ecoomcs, Busess & Mathematcs Kg s Uvest College Uvest of Weste Otao Ra T. Godw Depatmet of Ecoomcs, Uvest of Vctoa Vctoa, B.C., Caada V8W Y Revsed, Apl 00 Abstact We deve aaltc expessos fo the bases, to O( - of the maxmum lkelhood estmatos of the paametes of the geealzed Paeto dstbuto. Usg these expessos to bas-coect the estmatos s foud to be extemel effectve tems of bas educto, ad geeall esults a small educto elatve mea squaed eo. I geeal, the aaltc bas-coected estmatos ae also foud to be supeo to the alteatve of bas-coecto va the bootstap. Kewods Bas educto; Exteme values; Geealzed Paeto dstbuto; Peaks ove theshold Mathematcs Subject Classfcato 6F0; 6F40; 6N0; 6N05 Autho Cotact: Davd E. Gles, Dept. of Ecoomcs, Uvest of Vctoa, P.O. Box 700, STN CSC, Vctoa, B.C., Caada V8W Y; e-mal: dgles@uvc.ca; Phoe: ( ; FAX: (

2 . Itoducto Ths pape dscusses the calculato of aaltc fst-ode bas expessos fo the maxmum lkelhood estmatos (MLE s of the paametes of the geealzed Paeto dstbuto (GPD. Ths dstbuto s wdel used exteme value aalss, ad the motvato fo ts use such studes ases fom asmptotc theo that s specfc to the tal behavou of the data. Accodgl, pactce, the paametes ma be estmated fom a elatvel small umbe of exteme ode statstcs (as s the case f the so-called peaks ove theshold pocedue s used, so the fte-sample popetes of the MLE s fo the paametes of ths dstbuto ae of patcula teest. Specfcall, we cosde the O( - bas fomula toduced b Cox ad Sell (968, as e-expessed b Codeo ad Kle (994. Ths methodolog s patculal appealg hee, as t eables us to obta aaltc bas expessos, ad hece bas-coected MLE s, eve though the lkelhood equatos fo the GPD do ot admt a closed-fom soluto. It should be oted that the Cox-Sell appoach that we adopt hee s coectve, the sese that a bas adjusted MLE ca be costucted b subtactg the bas (estmated at the MLE s of the paametes fom the ogal MLE. A alteatve pevetve appoach, toduced b Fth (99, volves modfg the scoe vecto of the log-lkelhood fucto po to solvg fo the MLE, but we do ot dscuss ths appoach hee. Iteestgl, Cba-Neto ad Vascocellos (00 fd that these two appoaches ae equall successful wth espect to (fte sample bas educto wthout loss of effcec the cotext of the MLE fo the paametes of the beta dstbuto. I that same cotext, the fd that the bootstap pefoms pool wth espect to bas educto ad effcec. We do ot pusue pevetve methods ths stud. We fd that bas-coectg the MLE s fo the paametes of the GPD, usg the estmated values of the aaltc O( - bas expessos, s extemel effectve educg absolute elatve bas. I addto, geeal ths s accompaed b a modest educto elatve mea squaed eo. We compae ths aaltc bas coecto wth the alteatve of usg the bootstap to estmate the O( - bas, ad the coectg accodgl. I commo wth othe elated studes, we fd that the bootstap bas-coecto s qute effectve, ad s ot to be ecommeded. Secto summazes the equed backgoud theo, ad ths s the used to deve aaltc expessos fo the fst-ode bases of the MLE s of the paametes of the geealzed Paeto dstbuto secto. Secto 4 epots the esults of a smulato expemet that evaluates the

3 popetes of bas-coected estmatos that ae based o ou aaltc esults, as well as the coespodg bootstap bas-coected MLE s. Some cocludg emaks appea secto 5.. Fst-ode bases of maxmum lkelhood estmatos Fo some abta dstbuto, let l ( be the (total log-lkelhood based o a sample of obsevatos, wth p-dmesoal paamete vecto, θ. l( s assumed to be egula wth espect to all devatves up to ad cludg the thd ode. The jot cumulats of the devatves of l( ae deoted: k j E l / ;, j =,,., p ( ( j k jl E l / ;, j, l =,,., p ( ( j l k E( l / ( l / ;, j, l =,,., p. ( j, l j l The devatves of the cumulats ae deoted: k ( l j k / j l ;, j, l =,,., p. (4 ad all of the k expessos ae assumed to be O(. Extedg eale wok b Tuke (949, Batlett (95a, 95b, Haldae (95, Haldae ad Smth (956, Sheto ad Wallgto (96 ad Sheto ad Bowma (96, Cox ad Sell (968 showed that whe the sample data ae depedet (but ot ecessal detcall dstbuted the bas of the s th elemet of the MLE of θ ( ˆ s: p p p s jl Bas( ˆ k k 0.5k k O( ; s =,,., p. (5 s j l jl j, l whee k j s the (,j th elemet of the vese of the (expected fomato matx, K k }. Codeo ad Kle (994 oted that ths bas expesso also holds f the data ae o-depedet, povded that all of the k tems ae O(, ad that t ca be e-wtte as: p p p s ( l jl Bas( ˆ k k 0.5k k O( ; s =,,., p. (6 s jl j jl Notce that (6 has a computatoal advatage ove (5, as t does ot volve tems of the fom defed (. { j

4 ( l j ( l j Now, let a k ( k /, fo, j, l =,,., p; ad defe the followg matces: ( l l j jl A { a } ;, j, l =,,., p (7 ( ( ( p A A A... A. (8 Codeo ad Kle (994 show that the expesso fo the O( - bas of the MLE of θ ( ˆ ca be ewtte the coveet matx fom: Bas( ˆ K Avec( K O(. (9 A bas-coected MLE fo θ ca the be obtaed as: whee ~ ˆ ˆ ˆ ( ˆ K A vec K, (0 Kˆ ( K ad Aˆ ( A. ˆ ˆ It ca be show that the bas of ~ wll be O( -.. Applcato to the geealzed Paeto dstbuto We ow tu to the poblem of educg the bas of the MLE s fo the paametes of a dstbuto that s wdel used the cotext of the peaks ove theshold method exteme value aalss. The geealzed Paeto dstbuto (GPD was poposed b Pckads (975, ad t follows dectl fom the geealzed exteme value (GEV dstbuto (Coles, 00, pp.47-48, that s used the cotext of block maxma data. The dstbuto ad dest fuctos fo the GPD, wth shape paamete, o tal dex, ξ ad (modfed scale paamete σ, ae: F( / exp( / ; / ; 0, 0 0 ( f ( (/ / (/ exp( / ; / ; 0, 0 0 ( espectvel. Note that 0 f 0, ad 0 / f 0. Ou patcula teest ths dstbuto ases the cotext of modelg exteme values the etus o facal assets. 4

5 The (tege-ode cetal momets of the GPD ca be show (see pat A of the Appedx to be: E( Y! / ( ; =,,. ad the th momet exsts f /. We wll be coceed wth the MLE fo ' (,. The fte-sample popetes of ths estmato have ot bee cosdeed aaltcall befoe, although Jodeau ad Rockge (00 povde some lmted Mote Calo esults fo a modfed MLE of the shape paamete the elated GEV dstbuto. Othe estmatos ae avalable fo example, Hoskg ad Walls (987 dscuss the method of momets (MOM ad pobablt-weghted momets estmatos of θ; Castllo ad Had (997 popose the elemetal pecetle method ; ad Luceño (006 cosdes vaous maxmum goodess of ft estmatos, based o the empcal dstbuto fucto. The above codto fo the exstece of momets ca, of couse, lmt the applcablt of MOM estmato fo ths dstbuto. I what follows, t s mpotat to ote that the MLE s also defed ol ceta paamete ages. Moe specfcall, the MLE s of ξ ad σ do ot exst f because that case the dest (4. teds to ft whe teds to /. I addto, the usual egulat codtos (Dugué, 97; Camé, 946, p.500 do ot hold f / (Luceño, 006, p.905. Essetall fo these easos, maxmum lkelhood estmato of the paametes of the GPD ca be challegg pactce, as s dscussed moe full b Castllo ad Had (997. Assumg depedet obsevatos, the full log-lkelhood based o ( s: So, l(, l( ( / l( /. ( l / l( / ( (4 l / { ( } (5 l / { l( / } ( (6 l / { ( ( } (7 l / {( } (8 5

6 6 l 4 ( / ( } / l( { / (9 l ( } ( ( { / (0 l / ( } { / ( l } ( { / ( I ths case the fst-ode codtos that ae obtaed b settg (4 ad (5 to zeo do ot admt a closed-fom soluto. Howeve, we ca stll deteme the bas of the MLE s of the paametes ad the obta the bas-adjusted coutepats b modfg the umecal solutos (estmates to the lkelhood equatos b the extet of the (estmated bas. As s show pat B of the Appedx, the followg esults ae obtaed eadl b dect tegato afte a smple chage of vaable: ( E ( ( ( E (4 ( ( E (5 ( ( E (6 ( ( ( E (7 ( ( 6( E (8 ( ( ( E (9 ( ( E (0 ( ( ( E (

7 Usg the same chage of vaable, ad tegatg b pats, we also have: E l( /. ( We ca ow evaluate the vaous tems eeded to deteme the Cox-Sell bases of the MLE s of ξ ad σ, as dscussed secto. Note that the k j,l tems defed ( ae ot eeded the Codeo- Kle vaat of the Cox-Sell bas fomula see (9 ad the assocated deftos. The followg esults ae easl obtaed: k ( ( k / ( (4 k / ( ( (5 k 4 ( ( (6 k 4 / ( (7 k 8 / ( ( ( (8 k 4 / ( ( (9 ( k ( 4 ( (40 ( k 0 (4 ( k / ( (4 ( k / ( (4 ( k ( 4 / ( ( (44 ( k / ( ( (45 Note that all of ( to (45 ae O(, as s equed fo the Cox-Sell esult. The fomato matx s The elemets of ( / ( ( K (46 / ( ( / ( ( A ae: ( a ( 4 ( ( ( (47 ( a / ( / ( ( (48 7

8 ( ( a a ( 4 / ( ( 4 / ( ( (, (49 ad the coespodg elemets of ( A ae: ( a 4 / ( ( ( (50 ( a / ( / ( (5 ( ( a a / ( ( / ( ( (5 ( ( Defg A A A, the Cox-Sell/Codeo-Kle expesso fo the bases of the MLE s of ξ ad σ to ode O( - s ˆ B Bas ˆ K Avec( K whch ca be evaluated b usg (46 (5., (5 Notg that all of the (l aj tems ae of ode, ad that (fom (46 K s of ode -, we see that the bas expesso (5 s deed O( -, as equed. Fall, a bas-coected MLE fo the paamete ~ vecto ca be obtaed as (, ~ ' ( ˆ, ˆ' Bˆ ', whee Bˆ ca be obtaed b eplacg ad σ (5 wth the MLE s. Ths modfed estmato s ubased to ode O( Smulato esults The bas expesso (5 s vald ol to O (. The actual bas ad mea squaed eo (MSE of the maxmum lkelhood ad bas-coected maxmum lkelhood estmatos have bee evaluated a Mote Calo expemet. The smulatos wee udetake usg the R statstcal softwae evomet (R, 008. Geealzed Paeto vaates wee geeated usg the evd package (Stepheso, 008, ad the log-lkelhood fucto was maxmzed usg the maxlk package (Toomet ad Hegse, 008, pmal usg the Newto-Raphso method. The use of ths algothm smla poblems s suppoted b Pescott ad Walde (980, fo example. The Nelde- Mead algothm was used fo 50 Table to avod the ve occasoal falue of the Newto- Raphso algothm whe the sample sze s ths small. Chaouche ad Baco (006 dscuss some elated ssues assocated wth solvg umecall fo the MLE of the GPD paamete vecto. Each pat of ou expemet uses 50,000 Mote Calo eplcatos. 8

9 The esults Table ae pecetage bases ad MSE s, the latte beg defed as 00(MSE / ξ ad 00(MSE / σ. These quattes ae vaat to the value of the scale paamete, so we have set. We epot esults fo seveal postve values of that ae the age cosdeed b Castllo ad Had (997 the smulato stud of othe estmatos fo the GPD, ad ae also of pactcal elevace. We estct ou atteto to postve values fo the shape paamete as (at least postve etus o facal assets ae potetall ubouded. Howeve, othe computatos that we have udetake show that the qualtatve atue of ou coclusos ae ualteed f 0. The age of sample szes that we cosde s also motvated b pactcal applcatos. Fo example, Books et al. (005 deal wth sample szes of 40, whle Bal ad Neftc (00 have a sample wth = 00, each case the cotext of fttg the GPD to facal data. I Table we see that the MLE s of the shape ad scale paametes ae egatvel ad postvel based, espectvel. I addto, the pecetage bases of the MLE s fo (ad decease (cease absolute value as the tue value of the shape paamete ceases. Of couse, these absolute bases decle mootocall as the sample sze ceases. The aaltc bas coecto pefoms extemel well all cases, geeall educg the pecetage bases b at least a ode of magtude. I some cases thee s a ove-coecto, wth the pecetage bas chagg sg. It s ecouagg that, wth ol oe excepto, the educto elatve bas fo the (coected estmatos of ad s accompaed b a small mpovemet elatve mea squaed eo whe 50. Smla esults ae epoted b Cba-Neto ad Vacocellos (00 the case of the beta dstbuto ad Gles (009 fo the half-logstc dstbuto. I addto to ˆ, ~, ˆ ad ~, we have also cosdeed the bootstap-bas-coected estmato (Efo, 979. The latte s obtaed as N B ˆ (/ ˆ(, whee ˆ ( j s the MLE of N B j j obtaed fom the j th of the N B bootstap samples, ad s ethe o. See Efo (98, p.. Ths estmato s also ubased to O (, but pactce t s kow that ths ofte comes at the expese of ceased vaace. I ths pat of the Mote Calo expemet we have used 50,000 eplcatos ad,000 bootstap samples wth each eplcato a total of 50 mllo evaluatos fo each assocated et Table. 9

10 The bootstap-coected estmato also acheves a easoable degee of bas educto, especall wth egad to the scale paamete. I the latte case % Bas( s less tha % Bas ( ~, absolute tems, ~ fo two-thds of the etes Table. I cotast, % Bas ( s less tha % Bas (, absolute tems, fo two-thds of the cases cosdeed. Whe estmatg the shape paamete, the aaltcal coecto esults a ga elatve bas that s at least a ode of magtude bette tha that of the bootstap coecto a umbe of cases e.g., whe = 50 ad ξ =.0, o = 00 ad ξ = 0.5. The pecetage mea squaed eos of the bootstap-coected estmatos ae geeall ve close to those of the aaltcall coected estmatos. Wth the excepto of two cases (whe = 5, the aaltcal coecto leads to slghtl lowe mea squaed eo tha does the bootstap coecto. I addto, a few cases (e.g., fo whe = 50 ad ξ =.0 the vaace flato that s duced b the bootstap coecto actuall ceases the pecetage mea squaed eo above that of the ogal MLE. Oveall, ou esults povde a stog case fo usg the Cox-Sell aaltc bas coecto fo the MLE s of the paametes of the GPD 5. Coclusos We have deved aaltc expessos fo the bas to O( - of the maxmum lkelhood estmatos of the paametes of the geealzed Paeto dstbuto. These have the bee used to bas-coect the ogal estmatos, esultg modfed estmatos that ae ubased to ode O( -. We fd that the egatve elatve bas of the shape paamete estmato, ad the postve elatve bas of the scale paamete estmato ae each educed b usg ths coecto. Ths educto s especall otewoth the case of the shape paamete. Impotatl, these gas ae usuall obtaed wth a small mpovemet elatve mea squaed eo, at least fo sample szes of the magtude lkel to be ecouteed pactce. Usg the bootstap to bas-coect the maxmum lkelhood estmato s also qute effectve fo ths dstbuto. Howeve, o balace t s feo to the aaltc coecto, especall oce the effect o mea squaed eo s cosdeed. Whle educg the fte-sample bas of the MLE s of the paametes of the GPD s mpotat ts ow ght, thee s also cosdeable teest maagg the bas of the MLE s of ceta fuctos of these paametes. Specfcall, sk aalss we ae coceed wth value at sk (VaR ad the expected shotfall (ES, both of whch ae o-lea fuctos of the shape ad scale paametes whe the GPD s used the cotext of the peaks ove theshold method. Wok pogess b the authos addesses ths ssue b devg the Cox-Sell O( - bases fo the estmatos of VaR ad ES, ad evaluatg the bas-coected estmatos a mae smla to that adopted the peset pape. 0

11 Ackowledgmet We ae gateful to patcpats at the 009 Hawa Iteatoal Cofeece o Statstcs, Mathematcs ad Related Felds, ad a sema at the Uvest of Vctoa, fo the helpful commets. We also thak Lef Bluck fo povdg access to the addtoal computg facltes eeded to complete the smulato expemets. The secod autho ackowledges facal suppot fom Kg s College at the Uvest of Weste Otao.

12 Table : Pecetage bases ad MSE s ˆ ~ % Bas ( % Bas ( % Bas ( % Bas ( ˆ % Bas ( ~ % Bas ( ~ % MSE ( ˆ % MSE ( % MSE ( % MSE ( ˆ % MSE ( ~ % MSE ( ξ = ξ = ξ =

13 Refeeces Bal, T. G. ad S. N. Neftc (00, Dstubg extemal behavo of spot ate damcs, Joual of Empcal Face, 0, Batlett, M. S. (95a, Appoxmate cofdece tevals, Bometka, 40, -9. Batlett, M. S. (95b, Appoxmate cofdece tevals II. Moe tha oe ukow paamete, Bometka, 40, Books, C., A. D. Clae, J. W. Dalle Molle ad G. Pesad (005, A compaso of exteme value theo appoaches fo detemg value at sk, Joual of Empcal Face,, 9-5. Castllo, E. ad A. S. Had (997, Fttg the geealzed Paeto dstbuto to data, Joual of the Ameca Statstcal Assocato, 9, Chaouche, A. ad J-N. Baco (006, Statstcal feece fo the geealzed Paeto dstbuto: Maxmum lkelhood evsted, Commucatos Statstcs - Theo ad Methods, 5, Coles, S. (00, A Itoducto to Statstcal Modelg of Exteme Values (Spge-Velag, Lodo. Codeo, G. M. ad R. Kle (994, Bas coecto ARMA models, Statstcs ad Pobablt Lettes, 9, Cox, D. R. ad E. J. Sell (968, A geeal defto of esduals, Joual of the Roal Statstcal Socet, B, 0, Camé, H. (946, Mathematcal Methods of Statstcs, Pceto Uvest Pess, Pceto N.J. Cba-Neto, F. ad K. L. P. Vascocellos (00, Neal ubased maxmum lkelhood estmato fo the beta dstbuto, Joual of Statstcal Computato ad Smulato, 7, Dugué, D. (97, Applcato des popétés de la lmte au ses du calcul des pobabltés à l étude de dveses questos d estmato, Joual de l École Poltechque,, Fth, D. (99, Bas educto of maxmum lkelhood estmates, Bometka,, 80, 7-8. Gles, D. E. A. (009, Bas educto fo the maxmum lkelhood estmato of the scale paamete the half-logstc dstbuto, Ecoometcs Wokg Pape EWP090, Depatmet of Ecoomcs, Uvest of Vctoa. Gadshte, I. S. ad I. W. Rzhk (965, Table of Itegals, Sees ad Poducts, 4 th ed., Academc Pess, New Yok. Haldae, J. B. S. (95, The estmato of two paametes fom a sample, Sakhā,, -0. Haldae, J. B. S. ad S. M. Smth (956, The samplg dstbuto of a maxmum lkelhood estmate, Bometka, 4, 96-0.

14 Hoskg, J. R. M. ad J. R. Walls (987, Paamete ad quatle estmato fo the geealzed Paeto dstbuto, Techometcs, 9, Jodeau, E. ad M. Rockge (00, Testg fo dffeeces the tals of stock-maket etus, Joual of Empcal Face, 0, Luceño, A. (006, Fttg the geealzed Paeto dstbuto to data usg maxmum goodesssof-ft estmatos, Computatoal Statstcs ad Data Aalss, 5, Pckads, J. (975, Statstcal feece usg exteme ode statstcs, Aals of Statstcs,, 9-. Pescott, P. ad A. T. Walde (980, Maxmum-lkelhood estmato of the paametes of the theepaamete geealzed exteme-value dstbuto, Bometka, 67, R (008, The R Poject fo Statstcal Computg, Sheto, L. R. ad K. Bowma (96, Hghe momets of a maxmum-lkelhood estmate, Joual of the Roal Statstcal Socet, B, 5, Sheto, L. R. ad P. A. Wallgto (96, The bas of momet estmatos wth a applcato to the egatve bomal dstbuto, Bometka, 49, Stepheso, A. (008, evd: Fuctos fo exteme value dstbutos, Toomet, O. ad A. Hegse (008, maxlk: Maxmum lkelhood estmato, poject.og ; Tuke, J. W. (949, Stadad cofdece pots, Memoadum Repot No. 6, Statstcal Reseach Goup, Stafod Uvest. 4

15 Appedx Mathematcal Devatos A. Momets of the Geealzed Paeto Dstbuto A geeal expesso fo the momets of the GPD does ot appea to be well documeted, so t s deved hee. Whe 0 the GPD collapses to the expoetal dstbuto, whose momets ae well kow, so we cosde ol the case whee 0. We wll use the followg esult (Gadshte, ad Rzhk, 965, p. 9; tegal.4, o.4: x 0 ( m m / ( p qx dx (/ p ( p / q ( / ( m / / ( m (fo 0 / m. (A. Usg the GPD dest fo Y (4., E( Y ( / d. (A. 0 / Applg (A. wth ; m / ; q / ; ad p : E( Y ( / ( ( ( / ( /. (A. (! ( ( / ( / Repeatedl applg the ecuso fomula, ( a a( a, the ato of gamma fuctos (A. educes to (/ (/ (/...(/, so that E( Y ( /! (! / ( ( ( ( / ( / (...( (A.4 The paamete costats fo (A. to be vald amout to 0 ( ( /, o 0 (/. Ths codto pecsel matches the codto gve b Hoskg ad Walls (987, p. 4 fo the exstece of the th cetal momet of the GPD. Fall, ote that (A.4 collapses to the usual expesso fo the th momet of the expoetal dstbuto whe 0, ad to that of the ufom dstbuto o (0, σ whe. 5

16 B. Devato of Equato ( Settg /, ad usg ( wth 0, Let E E ( x, so that d dx /, ad 0 / d E ( ( ( ( x x x / / dx x / Results (4 to ( follow a smla mae. 6