Fitting bivariate losses with phase-type distributions

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1 Scandnavan Actuaral Journal ISSN: (Prnt) (Onln) Journal hompag: Fttng bvarat losss wth phas-typ dstrbutons Amn Hassan Zadh & Martn Blodau To ct ths artcl: Amn Hassan Zadh & Martn Blodau (23) Fttng bvarat losss wth phas-typ dstrbutons, Scandnavan Actuaral Journal, 23:4, , DOI:.8/ To lnk to ths artcl: Publshd onln: 22 Au. Submt your artcl to ths journal Artcl vws: 38 Vw rlatd artcls Ctng artcls: 3 Vw ctng artcls Full Trms & Condtons of accss and us can b found at Download by: [Bblothèqus d l'unvrsté d Montréal] Dat: 23 Octobr 25, At: 8:9

2 Scandnavan Actuaral Journal, 23 Vol. 23, No. 4, , Orgnal Artcl Fttng bvarat losss wth phas-typ dstrbutons AMIN HASSAN ZADEH a * AND MARTIN BILODEAU b Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 a Dpartmnt of Statstcal and Actuaral Scncs, Unvrsty of Wstrn Ontaro, London, Ontaro Canada N6A 5B7 b Départmnt d mathématqus t d statstqu, Unvrsté d Montréal, Montréal, Québc Canada H3C 3J7 (Accptd Jun 2) Maxmum lklhood stmaton and (paramtrc bootstrap) goodnss-of-ft tst ar consdrd for bvarat phas-typ dstrbutons ntroducd by Assaf and Collagus. In a spcal cas, th dpndnc structur of bvarat phas-typ dstrbutons s rvald. Th rsults ar usd to ft a ral b-dmnsonal data st rlatd to nsuranc losss (LOSS) and allocatd loss adjustmnt xpnss (ALAE). Th fttd bvarat phas-typ s usd to obtan condtonal quantls and man of ALAE for a gvn valu of LOSS. Th bvarat phas-typ dstrbuton mts all th rqurmnts lstd n th study by Klugman and Parsa. Kywords: bvarat phas-typ dstrbutons; dpndnt losss; contnuous-tm Markov chan; copulas; EM algorthm; paramtrc bootstrap goodnss-of-ft tst. Introducton Phas-typ (PH) random varabls ar dfnd as th tm untl absorpton n a st of absorbng stats n a contnuous tm Markov chan nvronmnt. Coxan, Erlang-n, hyprxponntal and mxtur of Erlang-n dstrbutons ar spcal cass of PH random varabls. Nuts (98) dfns th PH random varabl and stablshs ts thortcal proprts. PH dstrbutons ar dns among all dstrbutons wth postv support. In addton, thy hav dnsty, Laplac transform and all thr momnts n closd form and thus, varous probablty quantts can b obtand asly. Dspt th ntrstng proprts of PH varabls, som dffcults ars n statstcal stmaton. Nonunqunss of rprsntatons n som PH modls, as dscussd n O Cnnd (989), and ovrparamtrzaton s brfly mntond n Asmussn t al. (996). Asmussn t al. (996) study th paramtr stmaton by th EM algorthm, as wll as fttng othr dnsts on th postv ln wth PH dstrbutons. In Assaf and Lvkson (982) som proprts of PH varabls n rlablty ar nvstgatd. Phas-typ dstrbutons hav many appalng faturs whch mak t vry attractv to actuars. Many systms wth PH nput varabls yld an output varabl whch s also *Corrspondng author. E-mal: ahassan@stats.uwo.ca # 2 Taylor & Francs

3 2 A. Hassan Zadh and M. Blodau 242 Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 PH. For xampl, n Drkc t al. (24), n th Sparr Andrsn rnwal modls wth PH dstrbutd clams, th dstrbuton of dfct at run s PH wth an ntrprtabl rprsntaton. Asmussn (2) appls PH dstrbutons to rsk thory. L and Garrdo (24) consdr run probablty n rsk thory for Erlang-n dstrbutons, a spcal cas of PH dstrbutons. Tractablty and as of calculatons of PH dstrbutons nctd som rsarchrs to gnralz th noton to th multvarat cas. In th study by Assaf t al. (984), a multvarat PH dstrbuton s dfnd. In Kulkarn (989), a nw class of multvarat PH dstrbuton s ntroducd. In th multvarat cas, th structur of dpndnc undr som condtons s studd by L (23). Th condtonal tal xpctaton for multvarat PH dstrbutons s obtand n Ca and L (25a). Multvarat PH rsk modl s th subjct of Ca and L (25b). Ths papr s organzd as follows. Unvarat and multvarat PH varabls, wth thr proprts, ar brfly dfnd n Scton 2. Scton 3 covrs paramtr stmaton of bvarat PH (BPH) dstrbutons va th EM algorthm. In Scton 4, a mthod to smulat a BPH dstrbuton s usd n a small smulaton study on th bas and standard dvaton of th EM stmator. A (paramtrc bootstrap) goodnss-of-ft tst for BPH dstrbutons s proposd n Scton 5. Scton 6 ncluds a data analyss of th ALAE data by fttng a BPH dstrbuton. It also gvs xprssons for th condtonal quantls and condtonal man. Ths artcl xtnds th works of Asmussn t al. (996), Assaf t al. (984) and Åhlström t al. (999) to problms of statstcal natur n BPH dstrbutons, namly, th statstcal stmaton by th EM algorthm and a paramtrc bootstrap goodnss-of-ft tst. To our knowldg, ths s th frst papr n whch BPH dstrbutons ar appld n th contxt of a ral data analyss. 2. Prlmnars Consdr {J t, t ] } a rght contnuous Markov procss on th fnt stat spac G{, 2,..., m, m} wth ntal probablty vctor a and nfntsmal gnrator matrx A. Suppos that G and G 2 ar two nonmpty stochastcally closd substs (EƒG s sad to b stochastcally closd f, onc J t has ntrd E, t nvr lavs) of G such that G SG 2 {m}. Stats,...,mar transnt and stat m s absorbng. Hnc, absorpton nto stat m s crtan to happn. As a convnton, all vctors ar column vctors and suprscrpt T dnots th transpos of a matrx. Th matrx A can b wrttn as A ¼ T t T ; () whr th matrx T(t j )smm and t (t j )sanm-dmnsonal vctor. Ths lmnts satsfy t B,,...,m,t j ], "j, and Tt, whr s a vctor of ons. Stats,

4 Fttng bvarat losss , m ar transnt f and only f T s nonsngular (s Nuts 994). In ths artcl, w always suppos that a m, and hnc, a can b wrttn as a ¼ p : Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 Lt X and X 2 b th tms untl absorpton n G and G 2, rspctvly. W call th jont dstrbuton of (X,X 2 ) a bvarat PH (BPH) dstrbuton wth rprsntaton (p, T, G, G 2 ). Th margnal dstrbutons of X and X 2 hav unvarat PH dstrbutons. If (X, X 2 ) has a BPH dstrbuton, by usng Markov chan thory, t s shown n Assaf t al. (984) that th jont survval functon s PðX > x ; X 2 > x 2 Þ¼ pt Tx g Tðx 2 x Þ ; x 2 x p T Tx 2 g2 Tðx x 2 Þ g ; x x 2 ; (2) whr g k, k, 2, s an mm dagonal matrx whos th dagonal lmnt s f 2 C c k, and othrws. Assaf t al. (984) also provd th Laplac transform /ðu ; u 2 Þ¼E u X u 2 X 2 ¼ p T ½ðu þ u 2 ÞI TŠ fg 2 ½u I TŠ Tg þ G ½u 2 I TŠ T ½Tg G G 2 Šg whr G k Tg k g k T, k, 2, s th commutator. In gnral, th jont dstrbuton F has a sngular componnt on th st x x 2, whch can b avodd by supposng that t, for 2 C c \ Cc 2, s Assaf t al. (984). Hraftr, w also suppos that p, for 2, and as a rsult PðX > ; X 2 > Þ ¼. By mposng ths structur on th ntal probablty vctor p w hav that g k pp, k, 2, and hnc, th margnal survval functons can b obtand asly from Equaton (2). Usng th fact that d Tx /dxt Tx Tx T, th dnsty of th absolutly contnuous componnt can b drvd from Equaton (2): f ðx ; x 2 Þ¼ pt Tx G Tðx 2 x Þ T ; x 2 x p T Tx 2 G2 Tðx x Þ 2 Tg ; x x 2 : (3) Th sngular componnt on x x 2 may b usful n som applcatons rlatd to lf nsuranc. It s gvn n Assaf t al. (984) wth a furthr smplfcaton as PðX ¼ X 2 > xþ ¼p T Tx T Tg G G 2 (4) ¼ p T Tx T g T: Hnc, PðX ¼ X 2 Þ¼p T T g T; s obtand wth th valuaton of Equaton (4) at x. Thus, wth a corrcton to th statmnt mad n Assaf t al. (984), th sngular part s zro f and only f

5 4 A. Hassan Zadh and M. Blodau 244 [Tg G G 2 ], whch s quvalnt to t ¼ ; 2 C c \ Cc 2. Whn p, for 2 on can assum wthout loss of gnralty that p p ð;2 Þ A A ð;2þ B ðþ B ðþ 2 T A ðþ A ðþ 2 A Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 whr th partton corrsponds to th thr substs C c \ Cc 2, G \{m}, and G 2 \{m}. Thn, th jont dnsty Equaton (3) can b rwrttn, wth a corrcton to Assaf t al. (984), as f ðx ; x 2 Þ¼ pð;2þt Að;2Þ x B ðþ AðÞ ðx 2 x Þ A ðþ ; x 2 x ; p ð;2þt Að;2Þ x 2B ð2þ Að2Þ ðx x Þ 2 A ð2þ ; x x 2 : Th multvarat vrson of phas-typ random varabls (MPH) s also dfnd n Assaf t al. (984). Suppos that {J t, t ] } s a rght contnuous Markov chan on a fnt stat spac G. Lt G,..., G n b nonmpty stochastcally closd substs of G, such that \ n ¼ C has just on mmbr, namly m, and absorpton nto m s crtan. Th matrx A s stll th nfntsmal gnrator as n Equaton (). Dfn X k nf{t]nj t G k }, k,2,...,n. W assum that p for 2[ n ¼ C. Th jont MPH dstrbuton of (X...X n ) has th rprsntaton (p, T, G,..., G n ). For Bx 5 x x n, Sðx ;...; x n Þ¼PðX > x ;...; X n > x n Þ ¼ p T Tx g Tðx 2 x Þ... Tðx n x Þ n g n : Th MPH dstrbuton s absolutly contnuous f and only f t j, whnvr 2 C c k \ Cc l and j G k SG l, whr k " l. Laplac transform s gvn n Assaf t al. (984) and t can b usd to calculat all th momnts. As n th unvarat cas, MPH has th closur proprty. Lt T(T,..., T n ) and W(W,..., W m ) b ndpndnt MPH random vctors. Thn, th conjuncton (T, W)(T,..., T n, W,..., W m ) s an MPH random vctor. S Marshall and Shakd (986) for a proof. Morovr, MPH dstrbutons ar closd undr fnt mxtur and convoluton, s Assaf t al. (984) and Kulkarn (989). Ca and L (25b) gv an xplct rprsntaton for th convoluton of MPH dstrbutons. As for unvarat PH, th class of n-dmnsonal MPH dstrbutons s dns n th st of all dstrbutons on [, ] n, s Assaf t al. (984) for a proof. (5)

6 Fttng bvarat losss EM algorthm 3.. Gnral EM algorthm Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 Th EM (Expctaton-Maxmzaton) algorthm of Dmpstr t al. (977) s a gnral tratv mthod for fndng th maxmum lklhood stmat of th paramtrs, whn th data ar ncomplt or hav mssng valus. It fnds ts usfulnss whn th lklhood functon of th ncomplt (obsrvd) data s ntractabl but that of th complt (unobsrvd or mssng) data s of a smplr form whch can b analytcally optmzd. Th EM algorthm s not guarantd to fnd th global maxmum, t may convrg to a local maxmum or vn a saddl pont of th lklhood surfac, s Wu (983). Assum that th data x s obsrvd and gnratd by som dstrbutons, say f(xjf) wth log-lklhood functon L(f)log f (xjf). W call x th ncomplt data and rfr to L(f) as th ncomplt log-lklhood functon. Suppos that an unobsrvd (complt) data y, whr xx (y), has pdf g(yjf). Assum Qð/j/ ðpþ Þ¼E / ðpþ½log gðyj/þjxš; xsts for all pars (f (p), f). Ths notaton s for th condtonal xpctaton of log g (yjf) gvn x and th currnt valu f (p) of th paramtr. Anothr notaton oftn usd s E½log gðyj/þjx; / ðpþ Š. Th EM traton f (p) f (p) s dfnd as follows: E-stp: Comput Q(fjf (p) ). M-stp: Fnd f (p) that maxmzs Q(fjf (p) )ovrf. Smplfcatons occur whn th complt data dnsty functon s a mmbr of th xponntal famly gðyj/þ ¼bðyÞ xp½/ T tðyþš að/þ; (6) whr f s th vctor paramtr, t(y) s th vctor of complt data suffcnt statstc. If Equaton (6) holds, Dmpstr t al. (977) prsnt smplfd xprssons for th E and M stps: E-stp: Estmat th complt data suffcnt statstcs t(x) by fndng t ðpþ ¼ E / ðpþ½tðyþjxš: M-stp: Dtrmn f (p) as th soluton for f of th quaton E / ½tðyÞŠ ¼ t ðpþ : 3.2. EM algorthm for BPH dstrbuton EM algorthms n a Markovan chan nvronmnt ar not nw. Thr ar som works don n ths contxt, ncludng thos of Brur (22), Rydn (996), Robrts t al. (26), Robrts and Ephram (28), Asmussn t al. (996) and Åhlström t al. (999). Th

7 6 A. Hassan Zadh and M. Blodau 246 Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 lattr on s th EM algorthm for a spcal cas of BPH dstrbutons satsfyng x Bx 2 and wth cnsorng condtons on th data. Our work compard to Åhlström t al. (999) mght b consdrd as an ncrmntal work to gnral BPH dstrbutons. By dfnton a BPH random vctor has two componnts rprsntng th tms untl absorpton n two stochastcally closd substs G and G 2. Ths can b consdrd as an ncomplt data n th sns that thy only provd nformaton about th tm of httng G and G 2, not about th whol path of J t. Th ntal stat, th stats that hav bn vstd, and th tm spnt n ach vstd stat ar not obsrvd. Hnc, th hddn nformaton can hlp to maxmz th ncomplt lklhood functon whch s untractabl. For th cas x Bx 2, th complt path can b formulatd by th mbddd Markov chan of vstd stats and th sojourn tms ; ;...; m ;...; m 2 ð m 2 ¼ m þ Þ; s ; s ;...; s m ;...; s m2 ðs m2 ¼Þ; whr m s th numbr of jumps untl httng G and m 2 s th numbr of jumps untl httng th absorbng stat m. Gvn a ralzd obsrvaton (x, x 2 ) of th BPH dstrbuton, a complt obsrvaton of th procss J t on th ntrval (, x 2 ] s rprsntd by y ¼ ;...; m ;...; m 2 ; s ;...; s m ;...; s m 2 ; whr, x s...s m and x 2 s...s m2. To gt th probablty dnsty functon of y, on nds th probablty p jk of jumpng from j to k whch s gvn by ( t jk ; j 6¼ k; j; k ¼ ;...; m; t p jk ¼ Pð nþ ¼ kj n ¼ jþ ¼ jj t j ; k ¼ m þ ; j ¼ ;...; m: t jj Th dnsty of y can b drvd by Markov chan proprts and consdrng that th tm spnt n ach stat has an xponntal dstrbuton wth man /l, whr l t,asn Asmussn t al. (996). Thus, n o n o gðyjhþ ¼p xp k s t ;...xp k s m2 m 2 t ; m2 whr u(p, T) s th paramtr. Lt J ½Š t ;...; J ½nŠ t b n ndpndnt ralzatons of th procss. Ths gvs n mbddd Markov chans wth corrspondng holdng tms s ½vŠ ½vŠ ;...; s½vš m ½vŠ ;...; ½vŠ m ½vŠ ;...; s½vš m ½vŠ 2 ;...; ½vŠ m ½vŠ 2 ; ; v ¼ ;...; n:

8 Fttng bvarat losss Th complt data bcom y(y [],...,y [n] ), whr y ½vŠ ¼ ;...; ½vŠ ;...; s½vš ½vŠ m ½vŠ ;...; ½vŠ m ½vŠ s½vš ; 2 m ½vŠ ;...; s½vš m ½vŠ 2 ; v ¼ ;...; n: Th obsrvd ncomplt data ar th followng functon of th complt data x ½vŠ ¼ x ½vŠ ; x½vš 2 ¼ s ½vŠ þ...þ s½vš m ½vŠ s½vš ; þ...þ s½vš : m ½vŠ 2 Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 Dfn Lt B ½vŠ Z ½vŠ N ½vŠ j n o ¼ ½vŠ ¼ ¼ Xm ½vŠ 2 k¼ ¼ Xm ½vŠ 2 k¼ n ½vŠ k n ½vŠ k B ¼ Xn v¼ Z ¼ Xn v¼ N j ¼ Xn v¼ o ¼ s ½vŠ k ; ¼ ; ½vŠ B ½vŠ ; Z ½vŠ ; N ½vŠ j ; o kþ ¼ j : b th numbr of Markov procsss startng from stat, th total tm spnt n ach stat, and th numbr of jumps from stat to stat j, rspctvly. Thn, th dnsty of th complt data y s th product of n dnsts as n Equaton (7) gðyjhþ ¼ Ym ¼ " # p B xpft Z g Ymþ t N j j j¼;j6¼ ; (8) whr t,m t. Th dnsty Equaton (8) s a mmbr of a curvd mult-paramtr xponntal famly wth suffcnt statstcs B ; Z ; N j ; whr,...,m, j,...,m, "j. For th complt data, maxmum lklhood stmats of th unknown paramtrs wr obtand by Asmussn t al. (996). Thy can b usd n our contxt and, thrfor, th M-stp s gvn by ^p ¼ B n ;^t j ¼ N j ; 6¼ j;^t ¼ N ;mþ Z Z ;^t ¼ ^t þ Xm j¼;j6¼ ^t j!:

9 8 A. Hassan Zadh and M. Blodau 248 Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 Howvr, th E-stp dffrs consdrably snc th obsrvd and ncomplt data ar now bvarat. As notd n Scton 3, th E-stp for an xponntal famly conssts of computng th condtonal xpctaton of th suffcnt statstcs, gvn th complt data and th currnt paramtr stmats. If th currnt paramtr stmats at stp k of th algorthm s u (k), th complt suffcnt statstcs at th k E-stp consst n th valuaton of th followng condtonal xpctatons B ðkþþ Z ðkþþ N ðkþþ j ¼ Xn v¼ ¼ Xn v¼ ¼ Xn v¼ ðkþ ðkþ ðkþ h h B ½vŠ x ½vŠ Z ½vŠ x ½vŠ h N ½vŠ j x ½vŠ ; (9) ; () ; () for,...,m, j,...,m, "j. Th most complcatd part s th E-stp whch s drvd n th Appndx. All calculatons ar for th cas x Bx 2 ; th othr cas s smlar. Th fnal rsults ar gvn hr. For convnnc, th sam notatons ar usd as n th unvarat cas n Asmussn t al. (996). To smplfy formulas w dfn th sts C k ¼ C k nfm þ g; k ¼ ; 2; th matrx E j wth a on n poston (, j) and zros lswhr, and th vctor wth a on n poston and zros lswhr. Also, lt C ða; b; ; j; TÞ ¼ ð b a Tðu aþ E j Tðb uþ du: Th condtonal xpctatons n Equatons (9) ar gvn as follows: ðkþ h ¼ pðkþ B ½vŠ x ½vŠ whr p ðkþ s th th componnt of p (k), 8 ðkþ ðkþ h h Z ½vŠ x ½vŠ N ½vŠ j x ½vŠ >< ¼ T TðkÞ x ½vŠ G TðkÞ ðx ½vŠ 2 x½vš Þ T ðkþ ; f x ½vŠ h ðkþ p ðkþt C ð;x ½vŠ ;;;TðkÞ ÞG TðkÞ ðx ½vŠ 2 x½vš Þ T ðkþ fðx ½vŠ jh ðkþ Þ p ðkþt TðkÞ x ½vŠ G C ðx ½vŠ ;x½vš 2 ;;;TðkÞ ÞT ðkþ f x ½vŠ h ðkþ >: ð j ; othrws; 8 p t ðkþt C ð;x ½vŠ ;;j;tðkþ ÞG TðkÞ ðx ½vŠ 2 x½vš Þ T ðkþ j fðx ½vŠ jh ðkþ Þ p t ðkþt TðkÞ x ½vŠ E j TðkÞ ðx ½vŠ 2 x½vš Þ T ðkþ j f x ½vŠ h ðkþ >< ¼ ð j p t ðkþt TðkÞx ½vŠ G C ð x ½vŠ j f >: ð j ; othrws; Þ Þ ;x½vš 2 ;;j;tðkþ ÞT ðkþ x ½vŠ h ðkþ Þ ; 2 C c \ Cc 2 ; ; 2 C ; ; ; j 2 C c \ Cc 2 ; ; 2 C c \ Cc 2 ; j 2 C ; ; ; j 2 C ;

10 Fttng bvarat losss ðkþ h N ½vŠ ;mþ x½vš ¼ t pðkþ T TðkÞ x ½vŠ G TðkÞ ðx ½vŠ 2 x½všþ : f x ½vŠ h ðkþ Th functon C (a, b,, j, T) can b valuatd by numrcal mthods n ordnary or partal dffrntal quatons such as th RungKutta mthod of ordr four. Not that for abb, whr C ða; b; ; j; TÞ ¼C ð; b a; ; j; TÞ Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 and C ð; r; ; j; TÞ ¼ ð r Tðr uþ E j Tu du; dc ð; r; ; j; TÞ ¼ E j Tr þ TC ð; r; ; j; TÞ dr wth ntal condtons C ð; ; ; j; TÞ ¼. W nd ths scton wth an ntrstng proprty that holds whn on fts a BPH dstrbuton by th EM algorthm: at ach traton of th EM algorthm, th man of th fttd BPH dstrbuton quals th sampl man. Ths proprty was gvn by Asmussn t al. (996) for PH dstrbuton and t also holds n th bvarat cas as s now shown. Th obsrvatons ar lnar functons of th suffcnt statstcs, X ½vŠ ; X½vŠ 2 ¼ X Z ½vŠ ; X! Z ½vŠ : 2CnC 2CnC 2 For th componnt X, ths mpls n X ¼ Xn X Z ½vŠ : v¼ 2CnC Takng condtonal xpctatons, gvn x, on both sds ylds n X ¼ Xn X ½Š ðkþ Z v x ½Š v v¼ 2CnC X ¼ Xn E Z ½Š v h ð kþ Þ v¼ 2CnC ¼ Xn v¼ h ðkþþ½x ½vŠ Š ¼ n ðkþþ½x Š: h

11 A. Hassan Zadh and M. Blodau 25 Th sam argumnt appls to th componnt X Smulatd data from a BPH dstrbuton Consdr a BPH dstrbuton wth stat spac G{, 2, 3, 4}, closd substs G {2, 4} and G 2 {3, 4}, p(,, ) T, and th matrx Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 a pa qa T a 2 A; (2) a 3 whr BpB, qp, a,, 2, 3. Ths s a spcal cas of th MarshallOlkn dstrbuton, s Marshall and Olkn (967). Th jont survval functon can b wrttn wthout matrx xponntal usng th smplfd jont dnsty Equaton (5) ( p a a S h ðx ; x 2 Þ¼ a 2 x ð a a 2Þ x 2 a 2 a2 a x 2 þ q a x 2; x2 x ; q a a a 3 x 2ða a 3 Þ x a 3 a3 a x þ p a x ; x x 2 : Th Parson corrlaton coffcnt, a r h ¼ 2 a 3 pqa 2 ½a 2 2 þð q2 Þa 2 Š =2 ½a 2 3 þð p2 Þa 2 Š ; =2 s obtand from th Laplac transform. It s stmatd from th data usng th usual sampl corrlaton coffcnt ^r. Th maxmum corrlaton of s obtand by lttng a approach, whras th mnmum of /3 s rachd by lttng a 2 and a 3 approach and choosng p/2. Also, f a 2 qa and a 3 pa thn, r. In fact, th last cas corrsponds to two ndpndnt xponntals. A mor approprat masur of dpndnc for dstrbutons wth nonlnar rgrsson s Sparman corrlaton q h ¼ 3 þ 2 ð ð S h ðx ; x 2 ÞdF h; ðx ÞdF h;2 ðx 2 Þ: whr F u,,, 2, ar th two margnal dstrbutons of th jont BPH dstrbuton F u.in th spcal cas a 2 a 3 t s gvn by q h ¼ 7a a2 2 þ 2a3 2 6a3 pq þ a2 a 2ð4p 3Þð4p Þ 2a 3 þ 7a : a 2 ða þ a 2 Þþ2a 3 2

12 Fttng bvarat losss 25 Th mnmum Sparman corrlaton of 3/4 and th maxmum of ar obtand undr th sam crcumstancs as for Parson corrlaton. Anothr approprat masur s Kndall corrlaton s h ¼ þ 4 Z Z S h ðx ; x 2 ÞdF h ðx ; x 2 Þ; whch for ths modl s Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 s h ¼ a 2a 3 2a 2 pq þ a ða 3 q 2 þ a 2 p 2 Þ : ða þ a 2 Þða þ a 3 Þ Kndall t u for ths modl vars btwn /2 and. For any bvarat dstrbuton, Sparman r u and Kndall t u can b consstntly stmatd from th data usng thr mprcal vrsons 2 ^q ¼ nnþ ð Þðn Þ ^s ¼ X n v¼ 4 nðn Þ P n ; R v S v 3 n þ n whr (R v,s v ) ar thpars ofranks and P n s th numbr of concordant pars. Hr, two pars x ½Š v ; x ½Š v 2 and x ½Š c ; x ½Š c 2 ar sad to b concordant whn x ½Š v x ½Š c x ½Š v 2 x ½Š c 2 > : Th EM algorthm was run on, data sts of sz n5,, 2 and 4 gnratd from a BPH dstrbuton wth sub-ntnsty matrx T as n Equaton (2) wth p.5, a.5, a 2., and a 3., whch gvs corrlaton coffcnts of r u.7836, r u.675, and t u.5. For th modl Equaton (2), t can b obsrvd that snc ach of th condtonal xpctatons of th E-stp dos not dpnd on u thn, th EM algorthm convrgs n on traton. Ths rmark gvs also an xplct xprsson for th EM stmator 2 ^T ¼ 6 4 P n n mnðx½vš v¼ ;x½vš 2 Þ P n f v¼ x½vš Bx½vŠ 2 P g n mnðx½vš v¼ ;x½vš 2 P Þ n f v¼ x½vš Bx½vŠ 2 P g n maxð;x½vš v¼ 2 x½vš Þ P n f v¼ x½vš >x½vš 2 P n mnðx½vš v¼ ;x½vš 2 P n f v¼ x½vš >x½vš 2 P n ð v¼ max ;xv xv 2 3 g Þ 7 g 5 Þ

13 2 A. Hassan Zadh and M. Blodau 252 Tabl. Bas and standard dvaton of th EM stmator. n Bas Standard dvaton Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr : :5 :4 A :42 :5 :2 :2 A :9 :2 : : A : :99 :46 :53 :@ :66 A :565 :74 :53 :28 A :28 :5 :36 :49 A :47 :36 :25 :4 A :3 :25 :8 :72 A :7 In ths spcal cas, t can b asly sn that th EM stmator concds wth th maxmum lklhood stmator (MLE). To ths nd, not that th probablty dnsty functon quals f h ðx ; x 2 Þ ¼ pa a 2 x a a 2 ðx 2 x Þ ; x 2 x ð pþa a 3 x 2a a 3 ðx x Þ 2 : ; x x 2 Th rmanng calculatons of th MLE ar straghtforward. Tabl rports absolut valu of bass and standard dvatons of. th EM stmat computd ovr th, Pn rplcats. For xampl, for n5, ^T ¼ n v¼ mn x ½Š v ; x ½Š v 2 has a bas of. and a standard dvaton of.74, whras ^T 2 has a bas of.5 and a standard dvaton of.53. As for any rgular maxmum lklhood stmator, bas dcrass as n and bcoms nglgbl compard to standard dvaton whch dcrass as n /2. 5. Goodnss-of-ft tst Th statstc usd for tstng goodnss-of-ft s V 2 n ¼ Xn v¼ h x ½vŠ ; x½vš 2 ^S 2; n x ½vŠ ; x½vš 2 (3) S^hn

14 whr ^S n s th mprcal survval functon,.. ^S n ðx ; x 2 Þ ¼ X n n v¼ Fttng bvarat losss n o X ½vŠ > x ; X ½vŠ 2 > x 2 ; Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 and s th paramtrc survval functon Equaton (2) wth u stmatd by th EM S^hn algorthm. Larg valus of Vn 2 ar vdnc aganst th paramtrc modl. For rgular paramtrc modls, Stut t al. (993) stablshd that th paramtrc bootstrap of ð h W 2 n ¼ n ðxþ ^F n ðxþ ðxþ F^hn 2dF^hn s consstnt for tstng goodnss-of-ft. An argumnt smlar to th Lmma n Scton 2 of Kfr (959) can b usd to show that th followng vrson Z h 2d W 2 n ¼ n ðxþ ^F n ðxþ ^F n ðxþ ¼ Xn v¼ F^hn F^hn h ^F 2 n x ½Š v x ½Š v can also b bootstrappd consstntly. Th proposd Vn 2 s obtand by rplacng th dstrbuton functon by th survval functon whch has a smplr xprsson for BPH dstrbutons. Th goodnss-of-ft bootstrap tst of sgnfcanc lvl a s prformd as follows. Gvn a sampl of sz n, ðx ½Š v ; x ½Š v 2 Þ; v ¼ ;...; n; stmat u by ^h n usng th EM algorthm for BPH dstrbutons, and calculat th goodnss-of-ft statstc Vn 2 n Equaton (3). Thn, rpat a larg numbr, say B, of tms th followng 3 stps. () Gnrat a bootstrap sampl of sz n from th BPH dstrbuton wth paramtr ^h n dnotd ð~x ½vŠ ; ~x½vš 2 Þ; v ¼ ;...; n: (2) Fnd th EM stmat ~ h n from th bootstrap sampl. (3) Comput th goodnss-of-ft statstc ~V 2 n ¼ Xn v¼ h S ~hn ~x ½vŠ ; ~x½vš 2 S ~ 2; n ~x ½vŠ ; ~x½vš 2 whr ~ S n s th mprcal survval functon computd from th bootstrap sampl. Aftr rpatng th prvous loop B tms, ths Mont Carlo smulaton producs B (ordrd) valus: ~V 2 n;ðþ ~V 2 n;ð2þ ~V 2 n;ðbþ : Th bootstrap tst rjcts th modl whn Vn 2 whn Vn 2 > ~V n 2 ;ðdð aþbþ : xcds th (a)b ordr statstc,..

15 4 A. Hassan Zadh and M. Blodau 254 A smulaton was conductd to vrfy th sgnfcanc lvl of th bootstrap tst. It consstd n gnratn, sampls of sz n2 from th BPH dstrbuton Equaton (2), ach wth a dffrnt sub-ntnsty matrx T gnratd at random. Th paramtrs wr ndpndntly gnratd from th followng dstrbutons: a, a 2, a 3 unform on th ntrval (,) and p unform on (,). Each bootstrap tst was don at lvl a.5 wth B, bootstrap sampls. Th proporton of sgnfcant bootstrap tsts obtand, out of 2,, was.54, whch s clos to th ntndd.5 lvl. Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr Fttng ALAE data wth a BPH modl Ths scton conssts of fttng a BPH dstrbuton to nsuranc company ndmnty clams. Th data st contans,5 b-dmnsonal obsrvatons. Th varabls ar LOSS or ndmnty paymnt and ALAE, allocatd loss adjustmnt xpnss, whch covrs xpnss attrbutd to th sttlmnt of ndvdual clams such as clam nvstgaton xpnss. S Klugman t al. (24) for mor nformaton on th data. Frs and Valdz (998) and Klugman and Parsa (999) fttd ths data st usng copulas. Th varabl LOSS tratd n ths two paprs s th loss ncurrd to th nsurd so that ths varabl s cnsord whn th clam xcds th polcy lmt. Hr, LOSS s always th ndmnty paymnt so that thr s no cnsorng. It can b obsrvd from Fgur that thr s a modrat postv sampl Parson corrlaton of ^r.422. Th dscrptv statstcs ar gvn n Tabl 2. ALAE 95 th 75 th 5 th 25 th 5 th LOSS Fgur. ALAE vrsus LOSS wth curvs for condtonal quantls and man. Th dottd curv s th condtonal man of ALAE gvn LOSS.

16 Fttng bvarat losss Tabl 2. Dscrptv statstcs of ALAE data. LOSS ALAE Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 Man 4,28 2,588 Mdan 2, 5,47 Standard dvaton 2,748 28,46 Mnmum 5 Maxmum 2,73,595 5, quantl 4, 2, quantl 35, 2,577 Parson ^r.422 Sparman ^q.459 Kndall ^s.354 Th varabls wr rscald as LOSS/, and ALAE/, so that thy ar about th sam ordr of magntud. Ths rscalng also maks th lmnts of th sub-ntnsty matrx T not too clos to zro, thus mprovng th numrcal stablty. A fw chocs wr trd for G and G 2 and t was found that th choc G{,...2}, G {5, 6, 7, 2} and G 2 {8, 9,,, 2} ylds a good ft. Th EM algorthm was tratd 3 tms startng wth random valus of p and T. Th fttd BPH dstrbuton could captur th ssntal charactrstcs of th jont dstrbuton, such as th dpndnc structur. As shown n Tabl 3, th fttd BPH survval functon s clos to th mprcal survval functon ovr th whol doman. Th fttd BPH dstrbuton has standard dvatons for LOSS and ALAE of 2,75 and 28,46, rspctvly. Parson, Sparman, and Kndall corrlaton coffcnts ar r^hn ¼ :3932, q^hn ¼ :452 (.6), and s^hn ¼ :2968 (.42), rspctvly. Ths ar vry clos to th sampl statstcs n Tabl 2. Snc thr s no xplct xprsson for th lattr two coffcnts, thy can b computd numrcally, wth any dsrd dgr of accuracy, through a smulaton. Th valuaton of Kndall s^hn,for xampl, was don by jontly smulatng 5, data ponts from th fttd (jont) BPH dstrbuton, F^hn ; and avragng, ovr ths 5, valus, th survval functon, S^hn : As a Tabl 3. Comparson of survval functons. (LOSS,ALAE) ^Sn S^hn G-H Frank (3,) (83,) (4,) (6,) (7,).... (,3) (,4) (,5) (2,) (2,2) (3,2) (4,5) (3,3) (4,3) (.54,53) (53,.54) (44,.24) (6,5) V 2 n

17 6 A. Hassan Zadh and M. Blodau LOSS Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr Fgur 2. ALAE Margnal fttd dstrbuton functons (smooth) and mprcal dstrbuton functons. masur of accuracy, th standard rror of th man accompans n parnthss ths avrag. Sparman corrlaton can b valuatd smlarly, wth th xcpton that th two componnts ar ndpndntly gnratd from th margnal PH dstrbutons of th fttd BPH dstrbuton. Also, th mans of th fttd BPH dstrbuton always qual th sampl mans as shown n Scton 3.2. Th margnals of th fttd BPH dstrbuton ar plottd n Fgur 2. Th columns G-H and Frank of Tabl 3 gv th two fts obtand n Frs and Valdz (998) usng Parto dstrbutd margnals n both cass wth thr th Gumbl- Hougaard copula or th Frank copula. Th stmats of th paramtrs of ths two fts wr computd by Frs and Valdz (998) and wr smply usd hr to comput th survval functons from th dstrbuton functons. Th last row s th masur of global ft usd for th goodnss-of-ft tst n Scton 5 and computd hr ovr all th n,5 data ponts. Th mnmum valu s obtand for th BPH dstrbuton. Th Gumbl- Tabl 4. Condtonal quantls and man of ALAE gvn LOSS. Quantls LOSS Man 22,24 3, 6,2 4,59 5, ,24 3,2 6,4 4,59 5, ,278 3,88 6,36 5,4 5, ,5 3,529 7,324 9,6 6, ,895 6,36 9,28 25,32 7, ,32 7,364 5,7 36,28 2, ,869 9,592 8,94 45,99 5,98 5,246 5,84 2,567 25,5 89,59 24,34 6 6,25 29,84 6,93,66 8,5 7,72

18 Fttng bvarat losss Hougaard and Frank modls wth th cnsord varabl LOSS gv fts qut smlar to th on obtand wth th BPH modl wthout cnsorng. Strctly spakng howvr, such comparsons ar dffcult to ntrprt snc th varabl LOSS has dffrnt manngs. 6.. Condtonal quantls and man Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 Th condtonal survval functon of X 2, gvn X x, can b obtand from Equaton (2), n PðX 2 > x 2 jx Þ ¼ pt Tx G Tðx 2 x Þ =½p T Tx Tg Š; x Bx 2 ; p T Tx 2 Tðx x 2 Þ Tg =½p T Tx Tg Š; x x 2 : For a gvn valu of LOSS, quantls of ALAE wr calculatd basd on th condtonal survval functon Equaton (4) and prsntd n Tabl 4 and Fgur. Th condtonal man of X 2 gvn X x s somwhat tdous. For ths purpos, by usng thr Equaton (4) drctly or th condtonal probablty dnsty functon drvd from Equaton (4), on can wrt a systm of dffrntal quatons of th frst ordr whch can b numrcally solvd by,.g., Rung-Kutta mthods. Th probablty dnsty functon of X 2 gvn X x was usd hr. Aftr som straghtforward calculatons, th condtonal man s gvn by EðX 2 jx Þ ¼ pt Tx G ð T þ x IÞ þ p T C 2 ðx ; G 2 ; TÞTg ; p T Tx Tg whr C 2 (x, G, T) s a functon satsfyng th dffrntal quaton d dx C 2 ðx; G; TÞ ¼C 2 ðx; G; TÞT þ xtx G; wth ntal condtons C 2 (, G, T). Th condtonal man s gvn n th last column of Tabl 4. It s obsrvd n Fgur that th man s always gratr than th mdan whch rflcts th rght-skwnss of ALAE. Rfrncs Åhlström, L., Olsson, M., & Nrman, O. (999). A paramtrc stmaton procdur for rlaps tm dstrbutons. Lftm Data Analyss, 2, 332. Assaf, D., & Lvkson, B. (982). Closur of phas typ dstrbutons undr opratons arsng n rlablty thory. Annals of Probablty,, Assaf, D., Langbrg, N. A., Savts, T. H., & Shakd, M. (984). Multvarat phas-typ dstrbutons. Opraton Rsarch, 32, Asmussn, S., Nrman, O., & Olsson, M. (996). Fttng phas-typ dstrbutons va th EM algorthm. Scandnavan Journal of Statstcs, 23, Asmussn, S. (2). Run probablts. Advancd Srs on Statstcal Scnc & Appld Probablty, 2. Rvr Edg, NJ: World Scntfc Publshng Co., Inc.

19 8 A. Hassan Zadh and M. Blodau 258 Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 Brur, L. (22). An EM algorthm for batch Markovan arrval procsss and ts comparson to a smplr stmaton procdur. Annals of Opratons Rsarch, 2, Ca, J., & L, H. (25a). Condtonal tal xpctatons for multvarat phas-typ dstrbutons. Journal of Appld Probablty, 42, Ca, J., & L, H. (25b). Multvarat rsk modl of phas typ. Insuranc: Mathmatcs and Economcs, 36, Dmpstr, A. P., Lard, N. M., & Rubn, D. B. (977). Maxmum lklhood from ncomplt data va th EM algorthm. Journal of Royal Statstcal Socty Srs B, 39, 38. Drkc, S., Dckson, D. C. M., Stanford, D. A., & Wllmot, G. E. (24). On th dstrbuton of th dfct at run whn clams ar phas-typ. Scandnavan Actuaral Journal, 2, 52. Frs, E. W., & Valdz, E. A. (998). Undrstandng rlatonshps usng copulas. North Amrcan Actuaral Journal, 2, 25. Kfr, J. (959). K-sampl analogus of th Kolmogorov-Smrnov and Cramr-V. Mss tsts. Th Annals of Mathmatcal Statstcs, 3, Klugman, S. A., Panjr, H. H., & Wllmot, G. E. (24). Loss modls: from data to dcson (2nd d). Hobokn, NJ: John Wly & Sons. Klugman, S. A., & Parsa, R. (999). Fttng bvarat loss dstrbutons wth copulas. Insuranc: Mathmatcs and Economcs, 24, Kulkarn, V. G. (989). A nw class of multvarat phas typ dstrbutons. Opraton Rsarch, 37, 558. L, H. (23). Assocaton of multvarat phas-typ dstrbutons, wth applcatons to shock modls. Statstcs & Probablty Lttrs, 64, L, S., & Garrdo, J. (24). On run for th Erlang(n) rsk procss. Insuranc: Mathmatcs and Economcs, 34, Marshall, A. W., & Olkn, I. (967). A multvarat xponntal dstrbuton. Journal of th Amrcan Statstcal Assocaton, 62, 344. Marshall, A. W., & Shakd, M. (986). Multvarat nw bttr than usd dstrbutons. Mathmatcs of Opratons Rsarch,, 6. Nuts, M. F. (98). Matrx-gomtrc solutons n stochastc modls. An algorthmc approach. Baltmor, MD: Johns Hopkns Unvrsty Prss. Nuts, M. F. (994). Matrx-gomtrc solutons n stochastc modls. An algorthmc approach. Nw York: Dovr Publcatons Inc. O Cnnd, C. A. (989). On nonunqunss of rprsntatons of phas-typ dstrbutons. Communcatons n Statstcs. Stochastc Modls, 5, Robrts, W. J. J., & Ephram, Y. (28). An EM algorthm for on-channl currnt stmaton. IEEE Transactons on Sgnal Procssng, 56, Robrts, W. J. J., Ephram, Y., & Dguz, E. (26). On Rydén s EM algorthm for stmatng MMPPs. IEEE Sgnal Procssng Lttrs, 3, Rydn, T. (996). An EM algorthm for stmaton n Markov-modulatd Posson procsss. Computatonal Statstcs & Data Analyss, 2, Stut, W., Gonzáls, M. W., & Wncslao, P. Q. M. (993). Bootstrap basd goodnss-of-ft tsts. Mtrka, 4, Wu, C. F. J. (983). On th convrgnc proprts of th EM algorthm. Annals of Statstcs,, 953. Appndx W assum a BPH dstrbuton for whch PðX > ; X 2 > Þ As a rsult, th ntal stat can b only n C c \ Cc 2 : All calculatons ar for th cas x Bx 2. Th xprssons for th cas x x 2 ar gvn wthout furthr xplanatons.. Calculatons of E ½B jxš :

20 Fttng bvarat losss For x Bx 2 ; 2 C c \ Cc 2 ; ½B jxš ¼ ½f ¼ gjxš ¼ P h ð ¼ jxþ ¼ P h ð ¼ Þ P h ðx 2 dxj ¼ Þ P h ðx 2 dxþ ¼ p T Tx G Tðx 2 x Þ T : Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 For x > x 2 ; 2 C c \ Cc 2 ; 2. Calculatons of E½Z jxš : For x Bx 2 ; 2 C c \ Cc 2 ; For x > x 2 ; 2 C c \ Cc 2 ; For x Bx 2 ; 2 C ; ½Z jxš ¼ ½B jxš ¼ p T Tx 2 G2 Tðx x Þ 2 Tg : 2 ð ½ Š ¼ 4 Z jx ¼ ¼ ð ð P h ðj u ¼ jxþdu fj u ¼ g 3 dujx5 P h ðj u ¼ ÞP h ðx 2 dxjj u ¼ Þ du P h ðx 2 dxþ ¼ pt C ð; x ; ; ; TÞ G Tðx 2 x Þ T : ½Z jxš ¼ pt C ð; x 2 ; ; ; TÞG 2 Tðx x Þ 2 Tg : ð P h ðj u ¼ jxþdu ð P ¼ h ðx 2 dx ÞP h ðj u ¼ jx 2 dx ÞP h ðx 2 2 dx 2 jj u ¼ Þ P h ðx 2 dxþ ¼ pt Tx G C ðx ; x 2 ; ; ; TÞT :

21 2 A. Hassan Zadh and M. Blodau 26 For x > x 2 ; 2 C 2 ; ½Z jxš ¼ pt Tx 2 G2 C ðx 2; x ; ; ; TÞTg h 3. Calculatons of E N j jx : For vry small o, N j can b approxmatd by Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 N j ¼ X k¼ n o J k ¼ ; J ðkþþ ¼ j : h For ach cas w hav calculatd Nj jx : Th xact valu can b obtand by lttng o. For x Bx 2 ; ; j 2 C c \ Cc 2 ; ¼ ¼ h " X n o # N jjx ¼ J k ¼ ; J ðkþþ ¼ j jx ½ x =Š ½ x =Š k¼ k¼ X P h ðj k ¼ ; J ðkþþ ¼ j; X E dxþ k¼ P h ðx E dxþ X P h ðj k ¼ Þ P h ðj ðkþþ ¼ jjj k ¼ Þ P h ðx E dxj ðkþþ ¼ jþ : P h ðx E dxþ Snc Tu s a contnuous functon and thn, and T I! T; as! ; P h ðj ðkþþ ¼ jj k ¼ Þ!t j ; h N j jx p T C ð; x ; ; j; TÞG Tðx 2 x ÞTg2! t j : For x > x 2 ; ; j 2 C c \ Cc 2 ; h p T C ð; x 2; ; j; TÞG 2 Tðx x 2Þ Tg N j jx ¼ t j :

22 Fttng bvarat losss 2 26 Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 For x Bx 2 ; 2 C c \ Cc 2 ; j 2 C ; h N jjx ¼ P h ðj x ¼ ; J x ¼ jjxþ ¼ P h ðj x ¼ ÞP h ðj x ¼ jjj x ¼ ÞP h ðx 2 2 dx 2 jj x ¼ jþ P h ðx 2 dxþ p T Tðx Þ E T j P h ðx 2 2 dx 2 J x ¼ jþ ¼! P h ðx 2 dxþ p T Tx Ej Tðx t 2 x Þ T j : For x > x 2 ; 2 C c \ Cc 2 ; j 2 C 2 ; h p T Tx 2 Ej Tðx x Þ 2 Tg N j jx ¼ t j : For x Bx 2 ; ; j 2 C ; ¼ ¼ For x > x 2 ; ; j 2 C 2 ; h ½x ¼ X 2=Š N j jx X ½x 2=Š k¼½x =Š ½xX 2=Š k¼½x =Š k¼½x =Š P h ðj k ¼ ; J ðkþþ ¼ jjxþ P h ðx 2 dx ; J k ¼ ; J ðkþþ ¼ j; X 2 2 dx 2 Þ P h ðx 2 dxþ P h ðx 2 dx ÞP h ðj k ¼ ; J ðkþþ ¼ jx j 2 dx Þ : P h ðx 2 2 dx 2 jj ðkþþ ¼ jþ P h ðx 2 dxþ p T Tx G C! t ðx ; x 2 ; ; j; TÞT j : h N j jx p T Tx 2 G2 C ¼ t ðx 2 ; x ; ; j; TÞTg j : For x Bx 2 ; 2 C ; j ¼ m þ ; h N j jx ¼ P h ðx 2 dx ÞP h ðj x 2 ¼ jx 2 dx ÞP h ðj x 2 ¼ jjj x2 ¼ Þ P h ðx 2 dxþ! t p T Tx G Tðx 2 x Þ For x > x 2 ; 2 C 2 ; j ¼ m þ ; h N j jx p T Tx 2 G2 Tðx x Þ 2 ¼ t :

23 22 A. Hassan Zadh and M. Blodau 262 Th BPH dstrbuton wth a mass on X X 2 occurs rarly n applcaton but s stll worthy of mnton. Basd on th ntrprtaton of th sub-ntnsty matrx T, ths cas happns whn thr s a postv probablty of movng from som stat n C c \ Cc 2 drctly to th absorbng stat m. Indd, all calculatons wll b on th st C c \ Cc 2 bcaus th only stat vstd on G S G 2 s th absorbng stat m. Ths cas s vry smlar to th unvarat cas whch s tratd n dtals n Asmussn t al. (996). Th xpctatons of th E-stp ar Downloadd by [Bblothèqus d l'unvrsté d Montréal] at 8:9 23 Octobr 25 ½B jxš ¼ p Tx g T p T Tx g T ; 2 Cc \ Cc 2 ; ½Z jxš ¼ pt C ð; x; ; ; TÞg T 2 C c p T Tx g T \ Cc 2 ; h p T C N j jx ¼ t ð; x; ; j; TÞg T j ; ; j 2 C c p T Tx g T \ Cc 2 ; h p T Tx g N j jx ¼ t T p T Tx g T ; 2 Cc \ Cc 2 ; j ¼ m þ :