MOGENS BLADT ABSTRACT


 Linda Townsend
 3 years ago
 Views:
Transcription
1 A REVIEW ON PHASETYPE DISTRIBUTIONS AND THEIR USE IN RISK THEORY BY MOGENS BLADT ABSTRACT Phasetye 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 hasetye dstrbutons whereas exact solutons to many comlex roblems n stochastc modelng can be obtaned ether exlctly or numercally. In ths aer we ntroduce hasetye 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 hasetye. Fnally we dscuss ssues concernng statstcal nference for hasetye dstrbutons and related functonals such as e.g. a run robablty. KEYWORDS Phasetye dstrbuton, run robablty, EMalgorthm, Markov chan Monte Carlo. 1. INTRODUCTION The am of ths aer s to serve as an ntroducton to the use of hasetye dstrbutons n rsk theory and at the same tme to outlne a recent lne of research whch ncludes statstcal nference for hasetye dstrbutons and related functonals such as run robabltes. We start wth a short bblograhc revew. Though hasetye 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 coworkers 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,
2 146 M. BLADT alcatons to rsk theory can be found n Asmussen (2000). O Cnnede (1990) studes theoretcal roertes of hasetye 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 tmehorzon 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 hasetye methodology have been generalzed nto the broader class of matrxexonental 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 hasetye dstrbutons s of more recent date, where lkelhood estmaton was frst roosed by Asmussen et al. (1996) usng the EMalgorthm 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 hasetye 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 hasetye dstrbutons n Secton 3. In Secton 4 we outlne the method of robablstc reasonng wth hasetye dstrbutons by establshng some of ther basc roertes. Secton 5 ntroduces hasetye 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 hasetye 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 hasetye 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 hasetye 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,
3 A REVIEW ON PHASETYPE 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. PHASETYPE DISTRIBUTIONS We now let {X(t)} t 0 be a Markov jum rocess on the fnte statesace 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.
4 148 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 hasetye dstrbuton and we wrte t ~ PH(,T). The set of arameters (,T) s sad to be a reresentaton of the hasetye dstrbuton. The dmenson of,, s sad to be the dmenson of the hasetye dstrbuton. In the followng we gve three examles of hasetye dstrbutons. Examle 3.2. Let X 1,...,X n be ndeendent wth X ~ ex(l ). Then S X X n has a hasetye dstrbuton wth reresentaton (1,0,...,0) (dmenson n), and Jl l N 1 1 K T 0 l l O 2 2 K O, K l O n L P J0 N K t 0 O K O.... K l O n L P Indeed the sum X 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 hyerexonental. 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
5 A REVIEW ON PHASETYPE DISTRIBUTIONS 149 FIGURE 3.1. A flow dagram leadng to a Coxan dstrbuton. n where a >0,! a Then f s hasetye wth reresentaton (a 1,...,a n ), Jl N 1 K 0 l O 2 T K O K 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) Jl t N 1 12 K 0 l t O 2 21 K O K l O L P 4. PROPERTIES OF PHASETYPE DISTRIBUTIONS In ths secton we derve some basc roertes of hasetye 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
6 150 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
7 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 hasetye dstrbuton decays exonentally whch makes hasetye dstrbutons thntaled. We have the ntegraton rule as a drect consequence of A REVIEW ON PHASETYPE 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 subexonental 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 hasetye dstrbutons may easly be verfed. Theorem The nth (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, hasetye 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 hasetye dstrbutons whch converges weakly to m (see (2003) for detals). In other words, hasetye dstrbutons may aroxmate arbtrarly closely any dstrbuton wth suort on the ostve reals. Ths means that for thntaled dstrbutons one may assume wthout (too much) loss of generalty that dstrbutons are of hasetye. For heavy taled dstrbutons more care should be taken. Though n rncle hasetye 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. PHASETYPE RENEWAL THEORY Consder a hasetye renewal rocess {N(t)} t 0, that s, a renewal rocess where the nterarrval tmes have a hasetye 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
8 152 M. BLADT FIGURE 5.1. The samle aths of the Markov jum rocess constructed by concatenatng the Markov rocesses underlyng the hasetye 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 hasetye rocess or as a result of an arrval, where the hasetye rocess exts from one state and a new hasetye 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 hasetye 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 nterarrval 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 hasetye 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
9 A REVIEW ON PHASETYPE 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 nondelayed 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 hasetye 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
10 154 M. BLADT that s underlyng the concatenated descendng ladder heghts s a termnatng hasetye 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 hasetye 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. hasetye 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 nonconstant 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
11 A REVIEW ON PHASETYPE DISTRIBUTIONS 155 FIGURE 6.1. A surlus rocess wth nonconstant remum functon and hasetye clams. At jum tmes the Markov rocesses underlyng the hasetye 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 Taylorexand 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 nonlnear 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
12 156 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 nontrval 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 nonlnear system of couled dfferental equatons, whch may be effectvely solved by a numercal rocedure lke for nstance a fourth order RungeKutta 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 Matrxexonental 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. hasetye dstrbuted random varables of dmenson and reresentaton PH(,T). All we observe are the
13 A REVIEW ON PHASETYPE 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 EMalgorthm. 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 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 EMalgorthm s an teratve rocedure that maxmzes n each ste the condtonal exected value of the loglkelhood functon gven ncomlete data. Hence there are essentally two stes nvolved n each teraton: the calculaton of the condtonal exectaton of the loglkelhood gven absorton tmes (the Este) and the maxmzaton (the Mste). Takng the logarthm of the lkelhood functon gves j! log^ j, Th B log] g + N log] tg  t Z !! Nj log^tjh!! tj Z. 1 1 j! j! + !!! Hence the loglkelhood 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
14 158 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 Este above essentally nvolves matrxexonentals and an ntegral of matrxexonentals whch cannot be reduced further. Effectve methods for calculatng matrxexonentals are gven n Moler and Van Loan (1978) ncludng the RungeKutta method, where the matrxexonental 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 RungeKutta method. The EMalgorthm 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 byroduct of the EM algorthm. It nvolves dervatves wth resect to the arameters of the condtonal exectaton of the loglkelhood 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 loglkelhood n the neghborng onts. To whch extent t makes sense at all to talk about confdence ntervals for hasetye dstrbuton s a controversal ssue due to the roblems of dentfablty and overarameterzaton. 8. MARKOV CHAIN MONTE CARLO BASED INFERENCE In ths secton we resent an alternatve method for fttng hasetye dstrbutons based on Bladt et al. (2003). Ths method can only estmate functonals of hasetye dstrbutons whch are nvarant under dfferent reresentatons.
15 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 statesace 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 socalled 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 (burnn) the Markov chan wll settle nto a statonary mode. Suose that g(,t) s a functon of the hasetye arameters whch s nvarant under dfferent reresentatons of the same dstrbuton, that s, f (,T) and (,T ) are two dfferent reresentatons of the same hasetye 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 gvalues 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 socalled ror dstrbuton: ƒ(,t ) A REVIEW ON PHASETYPE DISTRIBUTIONS b  1 nj 01 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 + n01 ^ j, Th % % t ex^ t] Z + z 1 1 N j + nj 1 # % % t ex  t Z z. j ^ j ] + gh 1 j! gh
16 160 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 nontrval task. It turns out, however, that we may emloy a MetroolsHastngs (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 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 (burnn) 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 MetroolsHastngs algorthm. 2. Draw (,T) from osteror 3. Calculate run robabltes (or other reresentatonnvarant functonals of nterest) usng the current arameters (, T). 4. GO TO 1 After a number of ntal teratons (burnn), 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 hyerror and a concrete statstcal analyss of run robabltes, we refer to Bladt et al. (2003).
17 A REVIEW ON PHASETYPE 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 hasetye assumton. REFERENCES AALEN, O.O. (1995) Phase tye dstrbutons n survval analyss. Scand. J. Statst., 22, 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. SrngerVerlag, New York. ASMUSSEN, S. and BLADT, M. (1996) Phasetye dstrbuton and rsk rocesses wth statedeendent remums. Scand. Actuar. J., 1, ASMUSSEN, S. and BLADT, M. (1997) Renewal theory and queueng algorthms for matrxexonental dstrbutons. In S. Chakravarthy and A. Alfa, edtors, Matrxanalytc methods n stochastc models, volume 183 of Lecture Notes n Pure and Al. Math., ages Dekker, New York. ASMUSSEN, S., NERMAN, O. and OLSSON, M. (1996) Fttng hasetye dstrbutons va the EMalgorthm. Scand. J. Statst., 23, ASMUSSEN, S., AVRAM, F. and USABEL, M. (2002) Erlangan aroxmatons for fntehorzon run robabltes. Astn Bulletn, 32(2), BLADT, M. and NEUTS, M.F. (2003) Matrxexonental dstrbutons: calculus and nterretatons va flows. Stochastc Models, 19(1), BLADT, M.,GONZALEZ, A.andLAURITZEN, S.L. (2002) The estmaton of hasetye related functonals through Markov chan Monte Carlo methods. Scand. Actuar. J., 4, ERLANG, A. (1909) Sandsynlghedsregnng og telefonsamtaler. Nyt tdsskrft for Matematk, 20, 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. ASASIAM 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, NEUTS, M.F. (1981) Matrxgeometrc 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, O CINNEIDE, C.A. (1990) Characterzaton of hasetye dstrbutons. Comm. Statst. Stochastc Models, 6, PRESS, W., TEUKOLSKY, S., VETTERLING, W. and FLANNERY, B. (1992) Numercal Reces n Fortran. Cambrdge Unversty Press. MOGENS BLADT IIMASUNAM A.P Mexco, D.F. Mexco Emal:
Recurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationGraph Theory and Cayley s Formula
Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll
More informationA New Technique for Vehicle Tracking on the Assumption of Stratospheric Platforms. Department of Civil Engineering, University of Tokyo **
Fuse, Taash A New Technque for Vehcle Tracng on the Assumton of Stratosherc Platforms Taash FUSE * and Ehan SHIMIZU ** * Deartment of Cvl Engneerng, Unversty of Toyo ** Professor, Deartment of Cvl Engneerng,
More informationOptimal maintenance of a productioninventory system with continuous repair times and idle periods
Proceedngs o the 3 Internatonal Conerence on Aled Mathematcs and Comutatonal Methods Otmal mantenance o a roductonnventory system wth contnuous rear tmes and dle erods T. D. Dmtrakos* Deartment o Mathematcs
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More informationApplied Research Laboratory. Decision Theory and Receiver Design
Decson Theor and Recever Desgn Sgnal Detecton and Performance Estmaton Sgnal Processor Decde Sgnal s resent or Sgnal s not resent Nose Nose Sgnal? Problem: How should receved sgnals be rocessed n order
More informationPortfolio Loss Distribution
Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets holdtomaturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment
More informationA Note on the Decomposition of a Random Sample Size
A Note on the Decomposton of a Random Sample Sze Klaus Th. Hess Insttut für Mathematsche Stochastk Technsche Unverstät Dresden Abstract Ths note addresses some results of Hess 2000) on the decomposton
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationSTATIONARY DISTRIBUTIONS OF THE BERNOULLI TYPE GALTONWATSON BRANCHING PROCESS WITH IMMIGRATION
Communcatons on Stochastc Analyss Vol 5, o 3 (2 45748 Serals Publcatons wwwseralsublcatonscom STATIOARY DISTRIBUTIOS OF THE BEROULLI TYPE GALTOWATSO BRACHIG PROCESS WITH IMMIGRATIO YOSHIORI UCHIMURA
More informationLecture 3: Force of Interest, Real Interest Rate, Annuity
Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annutymmedate, and ts present value Study annutydue, and
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More information2.4 Bivariate distributions
page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationA Study on Secure Data Storage Strategy in Cloud Computing
Journal of Convergence Informaton Technology Volume 5, Number 7, Setember 00 A Study on Secure Data Storage Strategy n Cloud Comutng Danwe Chen, Yanjun He, Frst Author College of Comuter Technology, Nanjng
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σalgebra: a set
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More information1 Example 1: Axisaligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationDynamic Load Balancing of Parallel Computational Iterative Routines on Platforms with Memory Heterogeneity
Dynamc Load Balancng of Parallel Comutatonal Iteratve Routnes on Platforms wth Memory Heterogenety Davd Clare, Alexey Lastovetsy, Vladmr Rychov School of Comuter Scence and Informatcs, Unversty College
More informationThe Application of Fractional Brownian Motion in Option Pricing
Vol. 0, No. (05), pp. 738 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qngxn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com
More informationChapter 7. RandomVariate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 RandomVariate Generation
Chapter 7 RandomVarate Generaton 7. Contents Inversetransform Technque AcceptanceRejecton Technque Specal Propertes 7. Purpose & Overvew Develop understandng of generatng samples from a specfed dstrbuton
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationL10: Linear discriminants analysis
L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationLecture 2: Absorbing states in Markov chains. Mean time to absorption. WrightFisher Model. Moran Model.
Lecture 2: Absorbng states n Markov chans. Mean tme to absorpton. WrghtFsher Model. Moran Model. Antonna Mtrofanova, NYU, department of Computer Scence December 8, 2007 Hgher Order Transton Probabltes
More informationNONCONSTANT SUM REDANDBLACK GAMES WITH BETDEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia
To appear n Journal o Appled Probablty June 2007 OCOSTAT SUM REDADBLACK GAMES WITH BETDEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate
More information9.1 The Cumulative Sum Control Chart
Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s
More informationNew bounds in BalogSzemerédiGowers theorem
New bounds n BalogSzemerédGowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A
More informationFormula of Total Probability, Bayes Rule, and Applications
1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.
More informationNasdaq Iceland Bond Indices 01 April 2015
Nasdaq Iceland Bond Indces 01 Aprl 2015 Fxed duraton Indces Introducton Nasdaq Iceland (the Exchange) began calculatng ts current bond ndces n the begnnng of 2005. They were a response to recent changes
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationThe Performance Analysis Of A M/M/2/2+1 Retrial Queue With Unreliable Server
Journal of Statstcal Scence and Applcaton, October 5, Vol. 3, No. 9, 6374 do:.765/384/5.9.3 D DAV I D PUBLISHING The Performance Analyss Of A M/M//+ Retral Queue Wth Unrelable Server R. Kalyanaraman
More informationA NEW ACTIVE QUEUE MANAGEMENT ALGORITHM BASED ON NEURAL NETWORKS PI. M. Yaghoubi Waskasi MYaghoubi@ece.ut.ac.ir. M. J. Yazdanpanah Yazdan@ut.ac.
A NEW ACTIVE QUEUE MANAGEMENT ALGORITHM BASED ON NEURAL NETWORKS M. Yaghoub Waskas MYaghoub@ece.ut.ac.r M. J. Yazdananah Yazdan@ut.ac.r N. Yazdan Yazdan@ut.ac.r Control and Intellgent Processng Center
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More informationLatent Class Regression. Statistics for Psychosocial Research II: Structural Models December 4 and 6, 2006
Latent Class Regresson Statstcs for Psychosocal Research II: Structural Models December 4 and 6, 2006 Latent Class Regresson (LCR) What s t and when do we use t? Recall the standard latent class model
More informationMultivariate EWMA Control Chart
Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant
More informationA Prediction System Based on Fuzzy Logic
Proceedngs of the World Congress on Engneerng and Comuter Scence 2008 WCECS 2008, October 2224, 2008, San Francsco, USA A Predcton System Based on Fuzzy Logc Vadeh.V,Monca.S, Mohamed Shek Safeer.S, Deeka.M
More informationGeneralizing the degree sequence problem
Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts
More informationA Comprehensive Analysis of Bandwidth Request Mechanisms in IEEE 802.16 Networks
A Comrehensve Analyss of Bandwdth Reuest Mechansms n IEEE 802.6 Networks Davd Chuck, KuanYu Chen and J. Morrs Chang Deartment of Electrcal and Comuter Engneerng Iowa State Unversty, Ames, Iowa 500, USA
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationEfficient Computation of Optimal, Physically Valid Motion
Vol. xx No. xx,.1 5, 200x 1 Effcent Comutaton of Otmal, Physcally Vald Moton Anthony C. Fang 1 and Nancy S. Pollard 2 1 Deartment of Comuter Scence, Natonal Unversty of Sngaore 2 Robotcs Insttute, Carnege
More informationMAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPPATBDClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More informationRiskbased Fatigue Estimate of Deep Water Risers  Course Project for EM388F: Fracture Mechanics, Spring 2008
Rskbased Fatgue Estmate of Deep Water Rsers  Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn
More informationParametric Survival (Weibull) Regression
Chater 566 Parametrc Survval (Webull) Regresson Introducton Ths module fts the regresson relatonsh between a ostvevalued deendent varable (often tme to falure) and one or more ndeendent varables. The
More informationThe Probit Model. Alexander Spermann. SoSe 2009
The Probt Model Aleander Spermann Unversty of Freburg SoSe 009 Course outlne. Notaton and statstcal foundatons. Introducton to the Probt model 3. Applcaton 4. Coeffcents and margnal effects 5. Goodnessofft
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationResearch Article Competition and Integration in ClosedLoop Supply Chain Network with Variational Inequality
Hndaw Publshng Cororaton Mathematcal Problems n Engneerng Volume 2012, Artcle ID 524809, 21 ages do:10.1155/2012/524809 Research Artcle Cometton and Integraton n ClosedLoo Suly Chan Network wth Varatonal
More informationFINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals
FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant
More informationGas Deliverability Model with Different Vertical Wells Properties
PROC. ITB En. Scence Vol. 35 B, No., 003, 538 5 Gas Delverablty Model wth Dfferent Vertcal Wells Proertes L. Mucharam, P. Sukarno, S. Srear,3, Z. Syhab, E. Soewono,3, M. Ar 3 & F. Iral 3 Deartment of
More informationRing structure of splines on triangulations
www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAMReport 201448 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More informationForecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network
700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School
More informationLoad Balancing of Parallelized Information Filters
IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING, VOL. XXX, NO. XX, XXXXXXX 2001 1 Load Balancng of Parallelzed Informaton Flters Nel C. Rowe, Member, IEEE Comuter Socety, and Amr Zaky, Member, IEEE
More informationThe Analysis of Outliers in Statistical Data
THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate
More informationThe covariance is the two variable analog to the variance. The formula for the covariance between two variables is
Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.
More informationIMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1
Nov Sad J. Math. Vol. 36, No. 2, 2006, 009 IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS Modrag S. Petkovć 2, Dušan M. Mloševć 3 Abstract. A new theorem concerned
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationLogistic Regression. Steve Kroon
Logstc Regresson Steve Kroon Course notes sectons: 24.324.4 Dsclamer: these notes do not explctly ndcate whether values are vectors or scalars, but expects the reader to dscern ths from the context. Scenaro
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationSIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA
SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA E. LAGENDIJK Department of Appled Physcs, Delft Unversty of Technology Lorentzweg 1, 68 CJ, The Netherlands Emal: e.lagendjk@tnw.tudelft.nl
More informationTime Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University
Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton
More informationPrediction of Disability Frequencies in Life Insurance
Predcton of Dsablty Frequences n Lfe Insurance Bernhard Köng Fran Weber Maro V. Wüthrch October 28, 2011 Abstract For the predcton of dsablty frequences, not only the observed, but also the ncurred but
More informationgreatest common divisor
4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no
More informationThe eigenvalue derivatives of linear damped systems
Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by YeongJeu Sun Department of Electrcal Engneerng IShou Unversty Kaohsung, Tawan 840, R.O.C emal: yjsun@su.edu.tw
More informationMethods for Calculating Life Insurance Rates
World Appled Scences Journal 5 (4): 653663, 03 ISSN 88495 IDOSI Pulcatons, 03 DOI: 0.589/dos.wasj.03.5.04.338 Methods for Calculatng Lfe Insurance Rates Madna Movsarovna Magomadova Chechen State Unversty,
More informationOn the Optimal Control of a Cascade of HydroElectric Power Stations
On the Optmal Control of a Cascade of HydroElectrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;
More informationAryabhata s Root Extraction Methods. Abhishek Parakh Louisiana State University Aug 31 st 2006
Aryabhata s Root Extracton Methods Abhshek Parakh Lousana State Unversty Aug 1 st 1 Introducton Ths artcle presents an analyss of the root extracton algorthms of Aryabhata gven n hs book Āryabhatīya [1,
More informationState function: eigenfunctions of hermitian operators> normalization, orthogonality completeness
Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators> normalzaton, orthogonalty completeness egenvalues and
More informationEvaluation of the information servicing in a distributed learning environment by using monitoring and stochastic modeling
MultCraft Internatonal Journal of Engneerng, Scence and Technology Vol, o, 9, 4 ITERATIOAL JOURAL OF EGIEERIG, SCIECE AD TECHOLOGY wwwestngcom 9 MultCraft Lmted All rghts reserved Evaluaton of the nformaton
More informationPrediction of Disability Frequencies in Life Insurance
1 Predcton of Dsablty Frequences n Lfe Insurance Bernhard Köng 1, Fran Weber 1, Maro V. Wüthrch 2 Abstract: For the predcton of dsablty frequences, not only the observed, but also the ncurred but not yet
More informationProductForm Stationary Distributions for Deficiency Zero Chemical Reaction Networks
Bulletn of Mathematcal Bology (21 DOI 1.17/s11538195174 ORIGINAL ARTICLE ProductForm Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationA Structure Preserving Database Encryption Scheme
A Structure Preservng Database Encryton Scheme Yuval Elovc, Ronen Wasenberg, Erez Shmuel, Ehud Gudes BenGuron Unversty of the Negev, Faculty of Engneerng, Deartment of Informaton Systems Engneerng, Postfach
More informationAn Overview of Financial Mathematics
An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take
More informationESCI 341 Atmospheric Thermodynamics Lesson 9 Entropy
ESCI 341 Atmosherc hermodynamcs Lesson 9 Entroy References: An Introducton to Atmosherc hermodynamcs, sons Physcal Chemstry (4 th edton), Levne hermodynamcs and an Introducton to hermostatstcs, Callen
More informationFinite Math Chapter 10: Study Guide and Solution to Problems
Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount
More informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? ChuShu L Department of Internatonal Busness, Asa Unversty, Tawan ShengChang
More informationNonlinear data mapping by neural networks
Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal
More informationStochastic epidemic models revisited: Analysis of some continuous performance measures
Stochastc epdemc models revsted: Analyss of some contnuous performance measures J.R. Artalejo Faculty of Mathematcs, Complutense Unversty of Madrd, 28040 Madrd, Span A. Economou Department of Mathematcs,
More informationThis paper concerns the evaluation and analysis of order
ORDERFULFILLMENT PERFORMANCE MEASURES IN AN ASSEMBLE TOORDER SYSTEM WITH STOCHASTIC LEADTIMES JINGSHENG SONG Unversty of Calforna, Irvne, Calforna SUSAN H. XU Penn State Unversty, Unversty Park, Pennsylvana
More informationRisk Model of LongTerm Production Scheduling in Open Pit Gold Mining
Rsk Model of LongTerm Producton Schedulng n Open Pt Gold Mnng R Halatchev 1 and P Lever 2 ABSTRACT Open pt gold mnng s an mportant sector of the Australan mnng ndustry. It uses large amounts of nvestments,
More informationTraffic State Estimation in the Traffic Management Center of Berlin
Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,
More informationEE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN
EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson  3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson  6 Hrs.) Voltage
More informationWorking Paper Testing weak crosssectional dependence in large panels. CESifo working paper: Empirical and Theoretical Methods, No.
econstor www.econstor.eu Der OenAccessPublkatonsserver der ZBW LebnzInformatonszentrum Wrtschaft he Oen Access Publcaton Server of the ZBW Lebnz Informaton Centre for Economcs Pesaran, M. Hashem Workng
More informationA hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm
Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):18841889 Research Artcle ISSN : 09757384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel
More informationBinomial Link Functions. Lori Murray, Phil Munz
Bnomal Lnk Functons Lor Murray, Phl Munz Bnomal Lnk Functons Logt Lnk functon: ( p) p ln 1 p Probt Lnk functon: ( p) 1 ( p) Complentary Log Log functon: ( p) ln( ln(1 p)) Motvatng Example A researcher
More informationHYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION
HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR Emal: ghapor@umedumy Abstract Ths paper
More informationA DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION. Michael E. Kuhl Radhamés A. TolentinoPeña
Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATIONBASED OPTIMIZATION
More informationI. SCOPE, APPLICABILITY AND PARAMETERS Scope
D Executve Board Annex 9 Page A/R ethodologcal Tool alculaton of the number of sample plots for measurements wthn A/R D project actvtes (Verson 0) I. SOPE, PIABIITY AD PARAETERS Scope. Ths tool s applcable
More informationLoop Parallelization
  Loop Parallelzaton C52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I,J]+B[I,J] ED FOR ED FOR analyze
More information