MOGENS BLADT ABSTRACT

A REVIEW ON PHASE-TYPE DISTRIBUTIONS AND THEIR USE IN RISK THEORY BY MOGENS BLADT ABSTRACT Phase-tye dstrbutons, defned as the dstrbutons of absorton tmes of certan Markov jum rocesses, consttute a class of dstrbutons on the ostve real axs whch seems to strke a balance between generalty and tractablty. Indeed, any ostve dstrbuton may be aroxmated arbtrarly closely by hase-tye dstrbutons whereas exact solutons to many comlex roblems n stochastc modelng can be obtaned ether exlctly or numercally. In ths aer we ntroduce hase-tye dstrbutons and retreve some of ther basc roertes through aealng robablstc arguments whch, ndeed, consttute ther man feature of beng mathematcally tractable. Ths s llustrated n an examle where we calculate the run robablty for a rather general class of surlus rocesses where the remum rate s allowed to deend on the current reserve and where clams szes are assumed to be of hase-tye. Fnally we dscuss ssues concernng statstcal nference for hase-tye dstrbutons and related functonals such as e.g. a run robablty. KEYWORDS Phase-tye dstrbuton, run robablty, EM-algorthm, Markov chan Monte Carlo. 1. INTRODUCTION The am of ths aer s to serve as an ntroducton to the use of hase-tye dstrbutons n rsk theory and at the same tme to outlne a recent lne of research whch ncludes statstcal nference for hase-tye dstrbutons and related functonals such as run robabltes. We start wth a short bblograhc revew. Though hase-tye dstrbutons can be traced back to the oneerng work of Erlang (1909) and to Jensen (1953), t was not untl the late seventes that Marcel F. Neuts and co-workers establshed much of the modern theory avalable today (see Neuts (1981), Neuts (1989), Neuts (1995)). See also Asmussen (2003) for a more recent account. Most of the orgnal alcatons were n the area of queueng theory but many ASTIN BULLETIN, Vol. 35, No. 1, 2005,. 145-161

146 M. BLADT alcatons to rsk theory can be found n Asmussen (2000). O Cnnede (1990) studes theoretcal roertes of hase-tye dstrbutons, such as ther characterzaton. Asmussen and Bladt (1996) generalzes rsk models to stuatons wth Markov modulated arrvals and to stuatons where the remum deends on the current reserve. Asmussen et al. (2002) rovdes an elegant algorthmc soluton to the fnte tme-horzon run robablty. In Aalen (1995) a roosal of alcaton to survval analyss s outlned. For a more elementary textbook, whch manly draws examles from queueng theory, we refer to Latouche and Ramaswam (1999). Many results usng hase-tye methodology have been generalzed nto the broader class of matrx-exonental dstrbutons (dstrbutons wth a ratonal Lalace transform), ether by analytc methods (see Asmussen and Bladt (1997)) or more recently usng a flow nterretaton (see Bladt and Neuts (2003)). Statstcal nference for hase-tye dstrbutons s of more recent date, where lkelhood estmaton was frst roosed by Asmussen et al. (1996) usng the EM-algorthm whereas a Markov chan Monte Carlo (MCMC) based aroach was suggested n Bladt et al. (2003). For an overvew of earler attemts to the estmaton of hase-tye dstrbutons by methods other than maxmum lkelhood or MCMC, see Asmussen et al. (1996). The rest of the aer s organzed as follows. In Secton 2 we rovde the necessary background on the theory of Markov jum rocesses n order to ntroduce the concet of hase-tye dstrbutons n Secton 3. In Secton 4 we outlne the method of robablstc reasonng wth hase-tye dstrbutons by establshng some of ther basc roertes. Secton 5 ntroduces hase-tye renewal theory and a frst alcaton n rsk theory. In Secton 6 we consder a model for the surlus n nsurance where the remum s allowed to deend on the current reserve. Probablstc arguments nvolvng hase-tye dstrbutons allow us to establsh a system of couled dfferental equatons, the soluton of whch s the run robablty. The soluton yelds the exact soluton to the roblem though n ractce the dfferental equatons are solved by numercal methods. The last two Sectons 7 and 8 are dedcated to the estmaton and statstcal nference for hase-tye dstrbutons and related functonals (such as e.g. the run robablty n a surlus rocess). The style of the aer s exostory and techncal detals wll occasonally be omtted but can be traced through arorate references. 2. MARKOV JUMP PROCESS Before defnng hase-tye dstrbutons we shall recall some basc roertes of Markov rocess wth fnte state saces (also called contnuous tme Markov chans or Markov jum rocesses). Let {X(t)} t 0 be a Markov rocess whch takes values n the state sace E {1,2,...,n}. Then {X(t)} t 0 behaves n the followng way. Let T 1,T 2,... denote the tmes where {X(t)} t 0 jums from one state to another. Defne for convenence T 0 0. Then the dscrete tme rocess {Y n } n, where Y n X(T n ), s a Markov chan that kees track of whch states have been vsted. Let Q {q j },j E denote ts transton matrx. Furthermore,

A REVIEW ON PHASE-TYPE DISTRIBUTIONS 147 f Y n, then T n +1 T n s exonentally dstrbuted wth a certan arameter l. Also gven Y 0 0,Y 1 1,...,Y n n, the holdng tmes T 1 T 0, T 2 T 1,..., T n +1 T n are ndeendent. The latter roerty s referred to as condtonal ndeendence gven {Y n } n. Snce the holdng tmes T n +1 T n are exonentally dstrbuted wth arameter l gven that Y n, the condtonal robablty that there wll be a jum n the rocess {X(t)} t 0 durng the nfntesmal tme nterval [t,t + dt) s l dt. Gven a jum at tme t out of state, the robablty that the jum leads to state j s by defnton q j. Hence for j, l dtq j s the robablty of a jum from to j durng [t,t + dt). Thus for j, l j l q j! j s nterreted as the ntensty of jumng from state to j. Defne l j! l and L {l j }, j E. The matrx L s called the ntensty matrx or nfntesmal generator of the rocess. Let the transton robabltes of the Markov jum rocess be j t (X(t)j X(0) ) and the corresondng transton matrx P t {j} t, j E. Then we have the followng mortant relaton between P t and L, P t ex (Lt), where ex(a) denotes the exonental of a matrx A defned n the usual way by seres exanson, ex(a) 3! n 0 n A n!. Classfcaton of states for Markov jum rocesses s as follows: a state s recurrent (res. transent) f s recurrent (res. transent) for the chan {Y n } n. A state s absorbng f t s mossble to jum out of t agan, that s f q j 0 for all j mlyng l j 0 for all j. 3. PHASE-TYPE DISTRIBUTIONS We now let {X(t)} t 0 be a Markov jum rocess on the fnte state-sace E {1,2,...,, + 1} where states 1,..., are transent and state + 1 s absorbng. Then {X(t)} t 0 has an ntensty matrx on the form L c T t 0 0 m, (3.1) where T s dmensonal matrx, t s a dmensonal column vector (or 1 dmensonal matrx) and 0 s the dmensonal row vector of zeros. We shall make the followng conventon: unless otherwse stated matrces are denoted by boldface catal letters (Latn or Greek), boldface lowercase Latn letters refer to column vectors and lowercase boldface Greek letters refer to row vectors.

148 M. BLADT Snce the ntenstes of rows must sum to zero, we notce that t Te where, e (1,1,...,1). The ntenstes t are the ntenstes by whch the rocess jums to the absorbng state and are referred to as ext rates (ext from the transent subset of states). Let (X 0 ), 1,...,, (X 0 + 1) 0 denote the ntal robabltes. Notce that we are not allowed to ntate n state + 1 (ths would cause an atom at zero; extenson to ths case s, however, straghtforward). Let ( 1,..., ) denote the ntal dstrbuton of {X(t)} t 0 over the transent states only. Defnton 3.1. The tme untl absorton t nf{t 0 X t +1} s sad to have a hase-tye dstrbuton and we wrte t ~ PH(,T). The set of arameters (,T) s sad to be a reresentaton of the hase-tye dstrbuton. The dmenson of,, s sad to be the dmenson of the hase-tye dstrbuton. In the followng we gve three examles of hase-tye dstrbutons. Examle 3.2. Let X 1,...,X n be ndeendent wth X ~ ex(l ). Then S X 1 +... + X n has a hase-tye dstrbuton wth reresentaton (1,0,...,0) (dmenson n), and J-l l 0... 0 0 N 1 1 K T 0 -l l... 0 0 O 2 2 K O,.................. K 0 0 0 0 0 -l O n L P J0 N K t 0 O K O.... K l O n L P Indeed the sum X 1 +... + X n may be nterreted as the tme untl absorton by a Markov jum rocess wth n transent states whch ntates n state 1 and always jums to the next state n the sequence, u to state n from whch t jums to the absorbng state. Snce the sum S n can be obtaned by summng the X s n any order we see that reresentatons are by no means unque. Indeed we mght ermute the above states and obtan an alternatve reresentaton of the same dstrbuton. Examle 3.3. The next dstrbuton s known as hyer-exonental. Let X 1,...,X n be ndeendent and X ~ ex(l ) and let f denote the corresondng exonental densty. Let f n! 1 a f

A REVIEW ON PHASE-TYPE DISTRIBUTIONS 149 FIGURE 3.1. A flow dagram leadng to a Coxan dstrbuton. n where a >0,! a 1 1 1. Then f s hase-tye wth reresentaton (a 1,...,a n ), J-l 0 0... 0 0 N 1 K 0 -l 0... 0 0 O 2 T K O................... K 0 0 0 0 0 -l O n L P Examle 3.4. Coxan dstrbutons arse from the convoluton of exonental dstrbutons wth a random (yet bounded) number of terms (called hases or stages). Ths can be nterreted as the tme untl absorton of a Markov jum rocess as reresented by the flow dagram n Fgure 3.1: startng from state 1, there s a total jum rate out of state 1 of t 1 + t 12. The robablty of a jum to state 2 s t 12 /(t 1 + t 12 ) and the robablty of a jum to the absorbng state s t 1 /(t 1 + t 12 ). Ths s equvalent to the jum rate from state 1 to state 2 beng t 12 whle the jum rate to the absorbng state (the ext rate) s t 1. All other states j,,..., 1 behave smlarly, whle the robablty of jumng from state to the absorbng state s 1. If we let l t + t, +1 for 1,..., 1 and l t then the followng choce of arameters yelds a reresentaton for the Coxan dstrbuton dscrbed by the flow dagram n Fgure 3.1. T (1,0,0,,0) J-l t 0... 0 0 N 1 12 K 0 -l t... 0 0 O 2 21 K O................... K 0 0 0 0 0 -l O L P 4. PROPERTIES OF PHASE-TYPE DISTRIBUTIONS In ths secton we derve some basc roertes of hase-tye dstrbutons by robablstc arguments whch, due to ther mortance later on, wll be selled out n detals. The followng result s of man mortance. Wth the notaton (3.1), we have ex] Tsg e - ex] Tsge ex(ls) d. 0 1 n

150 M. BLADT The roof uses the defnton of the matrx exonental and the facts that t Te. We recall that ex(ls) s the transton matrx P s of the Markov jum rocess {X(t)} t 0 and the mortant thng to notce here s that the restrcton of P x to the transent states s gven by ex(tx). Hence we are able to comute transton robabltes j s (X(s) j X(0)) ex(ts) j for,,...,. We now take t ~PH(,T) and derve the densty f of t. The quantty f(s)ds may be nterreted as the robablty (t [s, s + ds)). If t [s, s + ds), then the underlyng Markov jum rocess {X(t)} t 0 must be n some transent state,..., at tme s. If the rocess ntates n a state 1,..,, the robablty that X(s)j s then j s ex(ts) j. The robablty that the rocess X(t) starts n state s by defnton. If X(s)j, the robablty of a jum to the absorbng state + 1 durng [s, s + ds) s t j ds (cfr. Secton 2). Condtonng on the ntal state of the rocess and ts state by tme s we then get that f ] sgds ^t! 6s, s + dsgh! 1!!! 1 s tds ex] Tsgtds. j j ex] Tsg t ds j j We have thus roved the followng theorem: Theorem 4.1. If t ~ PH(,T) the densty f of t s gven by where t Te. f(s) ex(ts)t, We could now obtan an exresson for the dstrbuton functon by ntegratng the densty, but we shall retreve ths formula by an even smler argument. If F denotes the dstrbuton functon of t, then 1 F(s) s the robablty that {X(t)} t 0 has not yet been absorbed by tme s (.e. t > s). But the event {t > s} s dentcal to that of {X(s) {1,2,...,}}. Hence, by a smlar condtonng argument as above, we get that 1- F] sg ^X] sg! " 1, 2,...,, h! 1!! 1 s j ex] Tsg ex] Tsge.! j

Thus we have roved: Theorem 4.2. If t ~ PH(,T), the dstrbuton functon F of t s gven by where t Te. F(s) 1 ex(ts)e, As a corollary we may observe that the tal of a hase-tye dstrbuton decays exonentally whch makes hase-tye dstrbutons thn-taled. We have the ntegraton rule as a drect consequence of A REVIEW ON PHASE-TYPE DISTRIBUTIONS 151 # ex(ts)ds T 1 ex(ts) ex(ts)t 1 (4.1) d ex(ts) Tex(Ts) ex(ts)t (4.2) ds and that T s nvertble, beng a sub-exonental matrx. Of course (4.2) s also vald for L but (4.1) s not snce L s sngular. Usng (4.1) the followng analytc roertes of hase-tye dstrbutons may easly be verfed. Theorem 4.3. 1. The n-th (n 1) moment of t s gven by (t n )( 1) n n! T n e. 2. The moment generatng functon of t s gven by (e st )( si T) 1 t, where I denotes the dentty matrx of dmenson. Aart from beng mathematcally tractable, hase-tye dstrbutons have the addtonal aealng feature of formng a dense class of dstrbutons wthn the class of dstrbuton on the ostve real axs, that s, for any dstrbuton m on the ostve real axs there exsts a sequence of hase-tye dstrbutons whch converges weakly to m (see (2003) for detals). In other words, hase-tye dstrbutons may aroxmate arbtrarly closely any dstrbuton wth suort on the ostve reals. Ths means that for thn-taled dstrbutons one may assume wthout (too much) loss of generalty that dstrbutons are of hase-tye. For heavy taled dstrbutons more care should be taken. Though n rncle hase-tye dstrbutons are able to aroxmate also heavy taled dstrbutons arbtrarly well, the aroxmatons wll always be bad n the tals, and the tal behavor s of crucal mortance n many stuatons. 5. PHASE-TYPE RENEWAL THEORY Consder a hase-tye renewal rocess {N(t)} t 0, that s, a renewal rocess where the nterarrval tmes have a hase-tye dstrbuton ~ PH(,T ). For the subsequent alcatons we are n artcular nterested n the renewal densty u of {N(t)} t 0, whch has the nterretaton that u(s)ds s the robablty of a

152 M. BLADT FIGURE 5.1. The samle aths of the Markov jum rocess constructed by concatenatng the Markov rocesses underlyng the hase-tye dstrbutons. The crosses on the tme axs denote arrval tmes. As we can see there are two tye of jums n the constructed Markov rocess: as a result of a jum n an underlyng hase-tye rocess or as a result of an arrval, where the hase-tye rocess exts from one state and a new hase-tye rocess ntates n a new state (ossbly the same). renewal (an arrval) durng the nfntesmal tme nterval [s, s + ds). Formally, f F and f are the dstrbuton functon and densty of the nterarrval tmes resectvely, then 3 3!! n n Us ] g F * ] sg, us ] g U ] sg f * ] sg n 0 n 1 where * denotes convoluton. By concatenatng the samle aths of the Markov jum rocesses underlyng the hase-tye dstrbutons between arrvals we obtan a new Markov jum rocess {J(s)} s 0 on the state sace {1,2,...,} wth ntensty matrx G {g j },,..., T + t (see Fgure 5.1). More recsely, we let {X (t)} 0 t <T denote the Markov jum rocess generatng the th nter-arrval tme T observed only u to the tme of absorton. Then we let J(s) X (s T 0... T 1 ) f s [T 1,T ). Then {J(s)} s 0 s a Markov jum rocess wth state sace {1,2,...,}. For ths rocess, a transton from to j can take lace n ether of two mutually exclusve ways: ether through a rocess {X (t)} 0 t <T jumng from to j or by such a rocess extng from state (to the absorbng state) and the next rocess, {X +1 (t)} 0 t <T+1, ntatng n state j. The robablty of the former s t j ds whle the latter has robablty t ds j. Hence g j ds t j ds + t ds j, rovng that G T + t. The transton matrx of {J(t)} t 0 s hence gven by P s ex((t + t)s) whch s the key to fndng an exresson for the renewal densty u. At tme s, the rocess {J(t)} t 0 wll develo through some rocess {X (t)} t 0. There s a renewal at tme s f and only f the hase-tye rocess {X (t)} t 0 makes a transton to the absorbng state durng [s, s +ds). Condtonng on the ntal state of {X 1 (t)} 0 t <T1 and the state of the rocess {J(t)} t 0 at tme s we get that u] sgds! 1!! 1! s t ds j ex^] T + tg shtds, j ex^] T + tg sh t ds j j

A REVIEW ON PHASE-TYPE DISTRIBUTIONS 153 whch roves the followng theorem: Theorem 5.1. The renewal densty u of a renewal rocess wth nterarrval tmes whch are PH(,T) s gven by u(s) ex((t + t)s) t, s 0. A delayed renewal rocess s one where the watng tme of the frst arrval has a dfferent dstrbuton than the dstrbuton of the rest of the nterarrval tmes. If the watng tme of the frst arrval s dstrbuted PH(b,T) whle the remanng nterarrval tmes have a dstrbuton PH(,T), then the renewal densty u 0 of the delayed rocess s gven by u 0 (s) bex((t + t)s) t, whch follows mmedately by notng that the ntal dstrbuton of {J(s)} s 0 s now b. The overshoot or the resdual watng tme s another examle of dstrbutons whch are of rme nterest n alcatons. The resdual watng tme z(s) at tme s s defned as the tme untl the next arrval. The dstrbuton of z(s) s easly obtaned by the followng argument. Snce T + t s the ntensty matrx of {J(s)} s 0 then the dstrbuton of J(s) s gven by b s ex((t+ t)s) n the non-delayed case. Snce {J(s)} s 0 moves accordng to T untl the next arrval, we then conclude that z(s) ~PH(b s,t ). A renewal rocess s called termnatng f the nterarrval dstrbuton s 3 defectve, that s, f # df (s)f( ) < 1. Ths s usually nterreted as the dstrbuton havng an atom at +. We notce that all arguments above hold also 0 for termnatng renewal rocesses, and hence all the results revously establshed also hold true. We now consder an mmedate alcaton of the hase-tye renewal theory to the followng model for the surlus. Let R t be the surlus rocess gven by N t! ]g U n n 0 R t u + t, where u s the ntal catal, the remum rate, {N(t)} t 0 a Posson rocesses wth ntensty b > 0 and U 1,U 2,.....d. clams wth dstrbuton PH(,T). We assume that R t drfts toward + whch amounts to bt 1 e < 1 (see Asmussen (2000). 227 for detals). We are nterested n calculatng the robablty of run for an nfnte tme horzon, c(u) ( nf R s <0 R 0 u). (5.1) 0 # s < 3 Fgure 6.1 shows a samle ath of such a rocess, though wth a more general remum ncome (R t ) rather than t. In ths case we see that the rocess

154 M. BLADT that s underlyng the concatenated descendng ladder heghts s a termnatng hase-tye renewal rocess wth nterarrval dstrbuton PH(,T ) for some defectve dstrbuton. It s clear that ( ) s the robablty that a Markov jum rocess underlyng the hase-tye clams downcrosses level u n state when the surlus rocess jums to a level below u for the frst tme. Snce there s a ostve robablty of {R t } t 0 never jumng to a level below u, the dstrbuton s defectve (does not sum to 1). Snce run haens f and only f the termnatng renewal rocess s n some state 1,2,..., by tme u (see Fgure 6.1) we conclude that c(u) e (T + t )u e. Indeed, ( ex((t + t )u)) s the robablty that the defectve renewal rocess s n state by tme u. Summng over all the states then gves the result. It s a bt more nvolved to rove that bt 1 /; see Asmussen (2000) for detals. 6. SURPLUS PROCESSES OF MORE GENERAL KIND In ths secton we shall consder a surlus rocess where arrvals occur accordng to a Posson rocess wth rate b > 0 and the clam szes are..d. hase-tye PH(,T). Between jums the surlus rocess R t moves accordng to the (determnstc) dfferental equaton d Rt (R dt t ) for some well behaved remum functon. Hence we have a model where the remum deends on the current reserve. If (x) constant we are back at the classcal rsk model. We are agan nterested n calculatng the nfnte tme horzon run robablty (5.1). The dea s essentally the same as for the case of a constant remum functon. We consder the rocess obtaned by rojectng the rocesses underlyng the descendng ladder heght on the vertcal axs. The dfference from earler s that ths rocess s no longer a renewal rocess and we aroach the roblem n a slghtly dfferent way. We shall agan be lookng at downcrossng robabltes. We consder the frst tme the surlus rocess jums to a level below ts ntal level u. Such a jum s evdently caused by a clam and we let n (u) denote the robablty that the underlyng Markov jum rocess of such a clam downcrosses level u n state (see Fgure 6.1). We notce that n (u) corresonds to n the case of a constant remum functon, but wth a non-constant remum functon ths robablty wll n general deend on the ntal catal u. By a condtonng argument smlar to the lnear case we obtan that n (u) bdt +(1 bdt)! n (u + (u)dt)(d j + t j (u)dt + t j (u)dtn (u)). j

A REVIEW ON PHASE-TYPE DISTRIBUTIONS 155 FIGURE 6.1. A surlus rocess wth non-constant remum functon and hase-tye clams. At jum tmes the Markov rocesses underlyng the hase-tye clams are ndcated as dashed lnes wth three states movng downwards. Here we condton on the event of an arrval n a small tme nterval [0, dt] the robablty of whch s bdt. In case of an arrval, the robablty of downcrossng level u n state s smly. If there are no arrvals, R dt u + (u)dt and condtonng on ths new level beng downcrossed n state j, the robablty of whch s n j (u+(u)dt), ether the rocess whch downcrosses level u + (u)dt n state j contnues and downcrosses level u n state wth robablty (u)dtt j (f j 1+t (u)dt s the robablty of no change of state), or the downcrossng rocess exts between u + (u)dt and u wth robablty t j (u)dt, n whch case the robablty of downcrossng level u amounts to n (u). Then we Taylor-exand n j (u + (u)dt) n j (u)+n j (u)(u)dt, and nsertng ths exresson and uttng (dt) k 0 for k > 1, we obtan the followng system of non-lnear dfferental equatons: (u)n (u)b + n (u) e! nj] utu g j ] g- bo +! nj (u)t j (u), 1,...,. (6.1) To fnd the run robablty we now look at the termnatng descendng ladder rocess {I t } t 0 whch ntates (tme zero) at level u. Run occurs f and only f

156 M. BLADT I t reaches tme u (.e. level zero). Let g (t) (I t ) and consder g (t + dt). Condtonng on the state of I t at tme t we easly get that g (t + dt)! g (t)(d j + t j dt + t j dtn (u t)) j where the term t j dt corresonds to the rocess gong from state j to generated by one of the ladder rocesses (f j agan 1 + t dt s the robablty that a ladder rocess beng n state wll not change state durng [t, t + dt)). The term t j dtn (u t) corresonds to the robablty that at tme t (level u t) a ladder rocess wll ext from state j and eventually later downcross level u t n state. Agan by Taylor exanson g (t + dt) g (t)+g (t)dt we obtan the followng system of dfferental equatons g (t)! gj (y)n (u t)t j + gj! (t)t j, 1,...,. (6.2) The ntal condton for (6.2) s obvously g (0) n (u), whereas for (6.1) the ntal condton s a non-trval matter. However, f the remum functon was such that t would be constant (equal to c, say) above a certan level v then the corresondng surlus rocess R v t would be lnear above ths level and at least we would know that n v (v) bt 1 /c, where n v (u) denotes the corresondng downcrossng robablty for the modfed surlus rocess. Lettng v then n v (u) n (u). In ractce one would lnearze at e.g. v 2u,3u,4u,... and solve for n v (u) untl convergence s obtaned. Thus (6.1) and (6.2) consttutes a non-lnear system of couled dfferental equatons, whch may be effectvely solved by a numercal rocedure lke for nstance a fourth order Runge-Kutta method (see e.g. Press et al. (1992)). Exlct solutons can n general not be obtaned. At last we are able to calculate the run robablty by notng that run occurs f and only f I u {1,2,..., } yeldng c(u)! ]g. u g 1 An mortant extenson of the above model can be obtaned by generalzng the arrval rocess to a Markov modulated Posson rocess where the rates of the Posson rocess deend on an underlyng Markovan envronment, see Asmussen and Bladt (1996) for detals. Also an extenson of the model to clam szes havng a Matrx-exonental dstrbuton (dstrbutons wth ratonal Lalace transforms) has been roved n Bladt and Neuts (2003). 7. MAXIMUM LIKELIHOOD ESTIMATION We consder data x 1,x 2,...,x n whch we mght thnk of as clam szes. We now suose that these data are generated by..d. hase-tye dstrbuted random varables of dmenson and reresentaton PH(,T). All we observe are the

A REVIEW ON PHASE-TYPE DISTRIBUTIONS 157 tmes untl absorton of the underlyng Markov jum rocesses and not the underlyng trajectores. The data are hence ncomlete and n the followng we shall descrbe a method for calculatng the maxmum lkelhood estmator usng the EM-algorthm. We follow Asmussen et al. (1996) whch may be consulted for further detals. Suose that we observed comlete data such that j 1,..., j n are the samle aths of the underlyng Markov rocesses generatng the absorton tmes X 1 x 1,..., X n x n. Then the Lkelhood functon for j (j 1,..., j n ) s gven by % % % % B N 0 Nj ^ j, Th t ex] -t Zg t ex^-tj Zh, j 1 1 1 j! where B s the number of tmes the rocesses of j ntates n state, N 0 s the number of jums the rocesses erform from state to the absorbng state +1,N j s the total number of jums from state to state j and Z s the total tme the Markov rocesses are n state. The maxmum lkelhood estmators for and T are gven by B ˆ N, t n j j N, j, t Z Z 0, t -! tj - t,,,...,. (7.1) The EM-algorthm s an teratve rocedure that maxmzes n each ste the condtonal exected value of the log-lkelhood functon gven ncomlete data. Hence there are essentally two stes nvolved n each teraton: the calculaton of the condtonal exectaton of the log-lkelhood gven absorton tmes (the E-ste) and the maxmzaton (the M-ste). Takng the logarthm of the lkelhood functon gves j! log^ j, Th B log] g + N log] tg - t Z 0 1 1 1!! Nj log^tjh!! tj Z. 1 1 j! j! + -!!! Hence the log-lkelhood s lnear n the suffcent statstcs B, N 0, N j, Z for,,...,, j so the calculaton of the condtonal exectaton of the loglkelhood, gven X (X 1,...,X n ), reduces to the calculaton of the condtonal exectaton of each of the B, N 0, N j for, j 1,...,, j. Let B (k) be the ndcator for whether the kth Markov jum rocess j k ntated n state, Z (k) the total tme j k sent n state, N (k) j the number of jums from state to j n j k and N (k) 0 the ndcator for whether the ext to the absorbng state n j k was caused by a jum from state. Then B n k 1 B () k!, 1,...,, Z n k 1 Z () k!, 1,...,, n () k N j! k 1Nj, 1,...,, j 0,...,, and t s suffcent to calculate the condtonal exectatons of the statstcs wth suerscrt k. In Asmussen et al. (1996) t s shown that

158 M. BLADT ], T ], T ], T g g g ] kg e ex] Txkg t `B X x, 1,...,, k kj ex] Txkg t ] g k `Z Xk xkj ] g `N Xk xkj k j # 0 xk ex] Tugee ex^t] xk - ugh tdu, 1,...,, ex] Txkg t # x k tj ex] Tugeej ex^t] xk - ugh tdu 0, j, 1,...,,! j, ex] Txkg t and ] kg ex] Txkg et ], Tg`N X x, 1,...,. k 0 kj ex] Txkg t We see that the E-ste above essentally nvolves matrx-exonentals and an ntegral of matrx-exonentals whch cannot be reduced further. Effectve methods for calculatng matrx-exonentals are gven n Moler and Van Loan (1978) ncludng the Runge-Kutta method, where the matrx-exonental s recognzed as a soluton to a lnear system of dfferental equatons. An alternatve effectve method, gven n Neuts (1995). 232, s the unformzaton method whch works well even n hgh dmensons. In Asmussen et al. (1996) a system of dfferental equatons of dmenson ( + 1) s solved usng a fourth order Runge-Kutta method. The EM-algorthm now works as follows. Gven any ntal arameter ( 0,T 0 ) we calculate the condtonal exected values above and lug them n as the suffcent statstcs for comlete data. Then we calculate the maxmum lkelhood estmates (7.1) and reeat the rocedure wth ( 0,T 0 ) relaced by the maxmum lkelhood estmator. Ths goes on untl convergence of the maxmum lkelhood estmates. Whle the EM algorthm always converges, t does not necessarly converge to the maxmum lkelhood estmator as t can get traed n a local maxmum. If that haens a more elaborated search may have to be establshed by a sutable varaton of the ntal arameters 0 and T 0. In Oakes (1999) t s descrbed how to obtan confdence ntervals drectly as a by-roduct of the EM algorthm. It nvolves dervatves wth resect to the arameters of the condtonal exectaton of the log-lkelhood functon. Such dervatves are not readly avalable but can be obtaned numercally by varaton of the arameters n a small neghborhood arround the maxmum lkelhood estmate and a recalculaton of the exected log-lkelhood n the neghborng onts. To whch extent t makes sense at all to talk about confdence ntervals for hase-tye dstrbuton s a controversal ssue due to the roblems of dentfablty and over-arameterzaton. 8. MARKOV CHAIN MONTE CARLO BASED INFERENCE In ths secton we resent an alternatve method for fttng hase-tye dstrbutons based on Bladt et al. (2003). Ths method can only estmate functonals of hase-tye dstrbutons whch are nvarant under dfferent reresentatons.

Ths means that we shall not be able to estmate the arameters themselves, but only quanttes such as densty functon, quantles or run robabltes. The method we resent may ether be vewed as Bayesan or as enalzed lkelhood. The key dea s to roduce a statonary sequence of measures (dstrbutons) whch has a statonary dstrbuton beng that of a condtonal hasetye dstrbuton gven the data. Let J denote a Markov jum rocess wth ntensty matrx L of the form (3.1) and let a (,0) denotes ts ntal dstrbuton. We then construct a statonary Markov chan of robablty measures (general state-sace tye of chan) whch has as statonary dstrbuton that of (,T,J ) gven the data x (x 1, x 2,..., x n ). To ths end we use a so-called Gbbs samler whch wll roduce the statonary dstrbuton by alternately drawng (,T) gven (J,x) and J gven (,T,x). After a certan number of ntal teratons (burn-n) the Markov chan wll settle nto a statonary mode. Suose that g(,t) s a functon of the hase-tye arameters whch s nvarant under dfferent reresentatons of the same dstrbuton, that s, f (,T) and (,T ) are two dfferent reresentatons of the same hase-tye dstrbuton, then g(,t) g(,t ). Ths s for examle the case for the densty, quantles or the run robabltes n the models above. If (,T ), 1,..., m, denote m arameter sets obtaned from the statonary dstrbuton above, then g( 1,T 1 ),..., g( m,t m ) s agan a statonary sequence of numbers. Ergodcty condtons are n general not avalable but wll be merely assumed and emrcally verfed by the behavor of the samle means. Hence, assumng ergodcty, one may average the numbers n order to roduce an estmator for the true value and one may calculate quantles for the arameters from the emrcal dstrbuton of g( 1,T 1 ),..., g( m,t m ). If g s not nvarant under dfferent reresentatons, t s not ossble to average over the dfferent g-values to obtan an estmator because the tye of reresentaton may swtch through the teratons. As a consequence the resent method cannot estmate the arameter values themselves. For arameter estmaton we refer to the EM algorthm n the revous secton. We now outlne the detals for the alternate drawngs of (,T) gven (J,x) and of J gven (,T,x). We need to mose a robablty structure on (,T ), the so-called ror dstrbuton: ƒ(,t ) A REVIEW ON PHASE-TYPE DISTRIBUTIONS 159 1 b - 1 nj 0-1 nj - 1 t ex t t ex t, ]- g% j ^- j h 1 1 j! % % % z z where b, z, n 0, n j for, j 1,..., are constants. Hence has a Drchlet dstrbuton whereas t and t j are Gamma dstrbuted. Furthermore they are all ndeendent. Ths makes t easy to samle arameters from ths ror. The osteror dstrbuton, whch s the roduct of the ror and the lkelhood, s hence gven by b1+ B -1 N 0 + n0-1 ^ j, Th % % t ex^- t] Z + z 1 1 N j + nj -1 # % % t ex - t Z z. j ^ j ] + gh 1 j! gh

160 M. BLADT Thus (,T) gven (J,x) (here x s of course unmortant because J contans nformaton about x n artcular) s smly drawn from the osteror dstrbuton, whch amounts to drawng from the ror wth arameters b, z, n 0, n j for,,..., where b b + B, n 0 n 0 + N 0, n j n j + N j and z z + Z. Drawng J gven (,T,x) s much more nvolved. Gven arameters and T and absorton tmes x 1,...,x n we must roduce realzatons of Markov jum rocesses wth the secfed arameters whch get absorbed exactly at tmes x 1,..., x n. Snce the robablty of ths event s zero t s a non-trval task. It turns out, however, that we may emloy a Metrools-Hastngs (MH) algorthm to smulate such Markov jum rocesses (see Bladt et al. (2003) for techncal detals). The MH algorthm amounts to the followng smle rocedure for smulatng a Markov jum rocess j whch gets absorbed exactly at tme x: 0. Draw a Markov jum rocess j whch s not absorbed by tme x. Ths s done by smle rejecton samlng: f a Markov jum rocess s absorbed before tme x t s thrown away and a new Markov jum rocess s tred. We contnue ths way untl we obtan the desred Markov jum rocess. 1. Draw a new Markov jum rocesses j as n 0. 2. Draw a unform random number U n [0,1]. 3. If U mn (1,t j x /t jx ) then j : j, otherwse kee j. 4. GO TO 1. Here x denotes the lmt from the left so j x s the state of the realzaton { j s } s 0 just ror to ext. We terate ths rocedure a number of tmes (burn-n) n order to get t nto statonary mode. After ths ont and onwards, any j roduced by the rocedure may be consdered as a draw from the desred condtonal dstrbuton and hence as a realzaton of a Markov jum rocess whch gets absorbed exactly at tme x. The full rocedure (Gbbs samler) s then as follows. 0. Draw (,T) from ror. 1. Draw j (j 1,..., j n ) underlyng Markov trajectores gven (,T) usng the Metrools-Hastngs algorthm. 2. Draw (,T) from osteror 3. Calculate run robabltes (or other reresentaton-nvarant functonals of nterest) usng the current arameters (, T). 4. GO TO 1 After a number of ntal teratons (burn-n), the rocedure wll stablze nto a statonary mode and from ths ont onwards we may may roduce a samle of e.g. run robabltes to be analyzed. In order to dslay the qualty of actual ft to the data t may be desrable also to roduce a sequence of densty values at each draw. Averagng these values through the teratons then roduces an estmate for the densty whch may be comared to a hstogram of the orgnal data. For ractcal detals on mlementaton, extensons, choce of ror or hyer-ror and a concrete statstcal analyss of run robabltes, we refer to Bladt et al. (2003).

A REVIEW ON PHASE-TYPE DISTRIBUTIONS 161 The EM and MCMC aroaches should not be seen as comettors but rather as comlementary methods. Whle the objectve of the EM algorthm s to obtan a maxmum lkelhood estmator of the arameters, the MCMC aroach focuses on nference for related functonals whch may be useful when addressng ssues concernng more comlex models whch artly deend on a hase-tye assumton. REFERENCES AALEN, O.O. (1995) Phase tye dstrbutons n survval analyss. Scand. J. Statst., 22, 447-463. ASMUSSEN, S. (2000) Run Probabltes, volume 2 of Advanced Seres on Statstcal Scence & Aled Probablty. World Scentfc Publshng Co. Inc., Rver Edge, N.J. ASMUSSEN, S. (2003) Aled Probablty and Queues. Srnger-Verlag, New York. ASMUSSEN, S. and BLADT, M. (1996) Phase-tye dstrbuton and rsk rocesses wth state-deendent remums. Scand. Actuar. J., 1, 19-36. ASMUSSEN, S. and BLADT, M. (1997) Renewal theory and queueng algorthms for matrx-exonental dstrbutons. In S. Chakravarthy and A. Alfa, edtors, Matrx-analytc methods n stochastc models, volume 183 of Lecture Notes n Pure and Al. Math., ages 313-341. Dekker, New York. ASMUSSEN, S., NERMAN, O. and OLSSON, M. (1996) Fttng hase-tye dstrbutons va the EMalgorthm. Scand. J. Statst., 23, 419-441. ASMUSSEN, S., AVRAM, F. and USABEL, M. (2002) Erlangan aroxmatons for fnte-horzon run robabltes. Astn Bulletn, 32(2), 267-281. BLADT, M. and NEUTS, M.F. (2003) Matrx-exonental dstrbutons: calculus and nterretatons va flows. Stochastc Models, 19(1), 113-124. BLADT, M.,GONZALEZ, A.andLAURITZEN, S.L. (2002) The estmaton of hase-tye related functonals through Markov chan Monte Carlo methods. Scand. Actuar. J., 4, 280-300. ERLANG, A. (1909) Sandsynlghedsregnng og telefonsamtaler. Nyt tdsskrft for Matematk, 20, 33-39. JENSEN, A. (1953) A dstrbuton model alcable to economcs. Munksgaard, Coenhagen. LATOUCHE, G. and RAMASWAMI, V. (1999) Introducton to matrx analytc methods n stochastc modelng. ASA-SIAM Seres on Statstcs and Aled Probablty. Socety for Industral and Aled Mathematcs (SIAM), Phladelha, PA. MOLER, C. and VAN LOAN, C. (1978) Nneteen dubous ways to comute the exonental of a matrx. SIAM Rev., 20, 801-836. NEUTS, M.F. (1981) Matrx-geometrc solutons n stochastc models., volume 2 of John Hokns Seres n the Mathematcal Scences. Johns Hokns Unversty Press, Baltmore, Md. NEUTS, M.F. (1989) Structured stochastc matrces of the M/G/1 tye and ther alcatons, volume 5 of Probablty: Pure and Aled. Marcel Dekker Inc., New York. NEUTS, M.F. (1995) Algorthmc robablty. Stochastc Modelng Seres. Chaman & Hall, London. OAKES, D. (1999) Drect calculaton of the nformaton matrx va the EM algorthm. J. Royal Statst. Soc. B, 61, 479-482. O CINNEIDE, C.A. (1990) Characterzaton of hase-tye dstrbutons. Comm. Statst. Stochastc Models, 6, 1-57. PRESS, W., TEUKOLSKY, S., VETTERLING, W. and FLANNERY, B. (1992) Numercal Reces n Fortran. Cambrdge Unversty Press. MOGENS BLADT IIMAS-UNAM A.P. 20-726 01000 Mexco, D.F. Mexco E-mal: bladt@sgma.mas.unam.mx