Luhm, Pruscha: Semi-parametric Inference for Regression Models Based on Marked Point Processes

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1 Luhm, Pruscha: Sem-paramerc Inference for Regresson Models Based on Marked Pon Processes Sonderforschungsberech 386, Paper 78 (1997) Onlne uner: hp://epub.ub.un-muenchen.de/ Projekparner

2 Sem-paramerc Inference for Regresson Models Based on Marked Pon Processes by A. Luhm and H. Pruscha Unversy of Munch SUMMARY. We sudy marked pon processes (MPP's) wh an arbrary mark space. Frs we develop some sascally relevan opcs n he heory of MPP's admng an nensy kernel (dz), namely marngale resuls, cenral lm heorems for boh he number n of objecs under observaon and he me endng o nny, he decomposon no a local characersc ( (dz)) and a lkelhood approach. Then we presen sem-paramerc sascal nference n a class of Aalen (1975)-ype mulplcave regresson models for MPP's as n!1, usng paral lkelhood mehods. Furhermore, consderng he case!1,we sudy purely paramerc M-esmaors. KYWORDS: Marked pon process, nensy kernel, (locally square negrable) marngale, local characersc, paral lkelhood, M-esmaor. 1 Inroducon and Basc Denons The monography by Andersen e al. (1993) presens a knd of canoncal approach o he sascal analyss of pon process models. I deals wh mulvarae pon processes where each random even carres nformaon on he occurrence me and he ype of even, he laer beng from a ne se of alernaves. The heorecal fundamen o mulvarae pon processes was lad - among ohers - by Jacod (1975), Bremaud (1981) and Dellachere & Meyer (1982). There are applcaons, however, where an uncounable se (e.g., he se of real numbers) of alernaves - now called marks - s more approprae, see Scheke (1994a,b), Murphy (1995) and Pruscha (1997). A mahemacal foundaon of marked pon processes (MPP's) s gven by Las and Brand (1995), bu hs work does no conan all ools necessary for sascal analyss. The rs goal of he presen paper s o ll hs gap. We presen () resuls on MPPnegrals, () lkelhood funcons of an MPP observaon, () cenral lm laws for wo deren suaons denoed as I and II below. These ools are hen used for he asympoc sascal nference n he case of wo deren knds of daa schemes. Scheme I conans n realsaons over a xed me nerval [,T], wh n endng o nny. Here he sem-paramerc analyss of a wde class of Aalen (1975)-ype models can be presened. Scheme II has one sngle realsaon over a longer me nerval [,T], wh T endng o nny. Here we presen a purely paramerc analyss only, he work on he nonparamerc par of he problem s sll n progress. The proofs are skeched only, for complee versons we refer o a forhcomng paper by Luhm. Mahema. Ins. der Ludwg-Maxmlans-Unversa Munchen, Theresensr.39, D-8333 Munchen 1

3 Unless menoned oherwse, we suppose all random elemens and hus all processes o be dened on some probably space ( F ) beng equpped wh a complee, rgh connuous lraon F. Followng Bremaud (1981), we x an arbrary measurable space ( ) and dene an MPP o be a double sequence ( k k ) k2in such ha () ( k ) k2in s a pon process (wh =) () ( k ) k2in s a sequence of random elemens n : The double sequence ( k k ) k2in shall be dened wh s assocaed counng measure N(d dz) whch s dened by N(] ] A) :=N (A) = 1X k=1 1( k )1( k 2 A) A 2: The lraon F N =(F N ) ha consss of he sgma algebras F N := (N s (A) : s A 2) s called nernal hsory of he MPP. Now le P(F ) denoe he F -predcable -algebra on ] 1[ and ~ P(F ):=P(F ) he F -predcable -algebra. ach mappng H : [ 1[! IR such ha () () Hj fg s F B-measurable, Hj ] 1[ s ~P(F ) B-measurable, we call F-predcable -ndexed process or shorly F-P. Fnally, we dene an nensy kernel (dz) o be a ranson measure from ( [ 1[ FB + )no ( ) such ha 8 A 2 he pon process (N (A)) adms he F -predcable nensy ( (A)).We pu 8 A 2 (A) := s (A) ds = s (dz) ds seng especally := () and := (). Furhermore, we suppose ha Denng < 1 ; a:s: 8 : (dz) := (dz) we oban a probably measure on he mark space gven he hsory unl he occurrence me (cf. Jacod (1975), Bremaud (1981) and Las & Brand (1995)). The par ( (dz)) s hen called local characersc. Throughou he paper, we denoe by jaj and jaj he ukldan norm of a vecor a and a marx A, respecvely a 2 s he marx wh ( j)-enry a a j. For a counable se I, we wre jij for he number of elemens of I, whle I d sands for he d-dmensonal un marx, d 2 IN. A 2

4 2 Resuls on Marked Pon Processes 2.1 Marngale Theory As a connuaon of Bremaud (1981), we consder d-dmensonal sochasc processes (d 2 IN) whch are generaed by henegraon of a d-dmensonal F-P H( z) w.r.. he measure M(d dz) :=N(d dz) ; (dz)d.e., we deal wh erms of he form M = Obvously, all hese processes sasfy M =. Theorem 1. The followng mplcaons hold: (a) (1) ) M s a local marngale, (b) (2) ) M s a marngale, H(s z) M(ds dz) : (c) (3) ) M s a unformly negrable marngale, where I I jh (s z)j s (dz) ds < 1 ; a:s: 8 1 d 8 (1) 1 jh (s z)j s (dz) ds jh (s z)j s (dz) ds < d 8 (2) < d: (3) Proof: As o (a),(b), see Bremaud (1981, VIII Corollary 4). Par (c) can be shown by applyng (b), Bremaud (1981, VIII Theorem 3) and Kopp (1984, Theorem 3.3.8). 2 Observe ha all processes we deal wh are of bounded varaon on ne nervals. For square negrable marngales we oban a resul smlar o Theorem 1, connung Bremaud (1972) and Boel e al. (1975). Theorem 2. (a) M s a square negrable marngale () 1 I H 2 (s z) s (dz) ds < d: (4) (b) M s a marngale whch s locally square negrable (= I H 2 (s z) s(dz) ds < d 8 : (5) (c) M s a locally square negrable marngale () H 2 (s z) s(dz) ds < 1 ; a:s: 8 1 d 8 : (6) = R R Proof: Par (a) can be obaned usng he oponal covaraon process [M] H 2 (s z) N(ds dz),, whch s derved accordng o Dellachere & Meyer (1982, VII Theorem 36), whle (c) follows from (a) generalzng he lnes of Lpser & Shryayev 3

5 (1978, Theorem 18.8). Par (b) s a drec consequence of (c) and Theorem 1(b) and was rs formulaed by Scheke (1994a). 2 Noce ha a locally square negrable marngale need no be a marngale as such. To close hs secon, we gve an explc formula for he characersc hm of M. Theorem 3. Le (6) hold. Then he characersc of M s gven by hm = H 2 (s z) s (dz) ds : (7) Proof: Apply Theorem 1 (a) o he oponal varaon process [M] and use he unqueness of he compensaor Cenral Lm Theorems Frs we formulae a Rebolledo (198)-ype cenral lm heorem for he number n of objecs under observaon endng o nny. Theorem 4. For each n 2 IN, le( ) be ameasurable space andn (ddz) be an MPP on ( F F ) wh he nensy kernel (dz). Le furher H be a d-dmensonal F -P for each n 2 IN, fulllng (s z)) 2 s (dz) ds < 1 ; a:s: 8 1 d 8 : (8) (H Fnally, le he followng wo condons hold 8 ">, (n!1) jh (s z)j 2 1(jH (s z)j ") s (dz) ds (H (s z)) 2 s (dz) ds ;! (9) ;! G (1) where G =(g j ) 1 jd s 8 > aposve dene d d-marx, connuous n, and g = 8 1 d. Then we have he followng convergence n dsrbuon n he space D d [ 1[: M D ;! M (n!1) := R R H (s z) M (ds dz),, andm are locally square negrable where M marngales. Furhermore, M s Gaussan wh characersc hm = G. Proof: Apply Meser (1991, Theorem 2.6) n combnaon wh Lndvall (1973, Theorem 3 ), usng Theorems 1,2,3. 2 In he nex heorem, we consder lms of he form!1nsead of n!1. We wll use a famly of non-sngular d d-marces (; ), fulllng (T) () ;! (elemen-wse) as!1 () 9 a famly of d d-marces (C ) beng non-sngular for each > and connuous n such ha for each xed s we have ; ; ;1 s! C s as!1: 4

6 Theorem 5. Le N(d dz) be an MPP wh nensy kernel (dz). Le H be a d-dmensonal F-P sasfyng (6) and (; ) be a famly of non-sngular d d-marces fulllng (T). Le furher 8 "> and for!1he followng wo condons hold j; H(s z)j 2 1(j; H(s z)j ") s (dz) ds (; H(s z)) 2 s (dz) ds where G s a posve dene d d-marx. Then ; M D ;! N d ( G) (!1): ;! (11) ;! G (12) Proof: Generalze he lnes of Pruscha (1984, Theorem 2.4.9), where Rebolledo (198) was used Lkelhood Followng Jacod (1975), Lpser & Shryayev (1978), Bremaud (1981), Pruscha (1984) and Las & Brand (1995), we presen a process L, whch can be nerpreed as he Radon- Nkodym dervave of an MPP w.r.. anoher MPP, especally a marked Posson process, and whch opens he door o he lkelhood approach. Le wo probably measures on ( F) be gven sasfyng. If N(d dz) adms an (F )-nensy kernel (dz), hen here exss a unque (up o - ndsngushably) F-P h = h( z), called Jacod-process, such ha N(d dz) adms he (F )-nensy kernel (dz) =h( z) (dz) z 2 (13) (see Jacod (1975, Theorem 4.1) and Las & Brand (1995, Theorem 1.2.1)). Consderng he decomposons ( (dz)) and ( (dz)) as local characerscs,.e., (dz) = (dz) (dz) = (dz) we ge an F-P g = g( z) sasfyng (dz) =g( z) (dz) by denng g( z) := h( z) z2 : (14) Now le be he probablyonf admng a local characersc (1 (dz)) ndependen of, sasfyng (dz) (dz) 8. Then N(d dz) s a marked sandard Posson process (see Bremaud (1981), VIII xercse 3), and denng ~ := ~ j F for any probably measure ~ and any soppng me, we formulae Theorem 6. Le h denoe he Jacod process nroduced n(13) and L () be he densy of w.r... If s an F -soppng me wh <1 -and-a.s., hen wh d d = L 5

7 where he process (L ) s dened by L := L Y 1kN h( k k ) 1 A exp (1 ; h(s z)) (dz) ds : Proof: The Theorem can be derved from Jacod (1975, Proposon 4.3), Bremaud (1981, VIII Theorem 1) and Las & Brand (1995, Theorems and 1.2.6), followng he lnes of Lpser & Shryayev (1978, Theorem 19.9) and Pruscha (1984, Theorem 2.3.1). 2 Le he nensy kernel depend on some - no necessarly ne dmensonal - unknown parameer of he form ( dz) =h( z ) (dz). Then L L() can be wren as L () = exp log h(s z ) N(ds dz) ; ()+R 2 (15) where R := log L () +, does no depend on 2. 3 Sem-paramerc Mulplcave Models 3.1 A Class of Mulplcave Models As a generalzaon of Andersen e al. (1993, sec.vii.2), we consder a wde class of mulplcave regresson models. Le I be a counable se and N (d dz) for each 2 I be an MPP wh nensy kernel (dz) of he ype ( dz) = ( 1 ) ( 2 dz) ( 1 ) = () r( 1 X ) Y 2 I z 2 where (wh c d 1 d 2 2 IN, d := d 1 + d 2, B = B 1 B 2, B j open IR d j, j =1 2) Y 2 I s an observable, bounded, (ofen f 1g-valued) nonnegave F-P, : IR +! ] 1[ s R an unknown baselne hazard funcon sasfyng (s) ds < 1 8, r : IR d 1 IR c! ] 1[ s a known regresson funcon, =( 1 2 ) T 2 B s an unknown d-dmensonal parameer, X 2 I s an observable, c-dmensonal F -predcable process of covaraes. By N we denoe he superposon P 2I N. Assume ha here exss a probably measure (dz) on such ha ( 2 dz) (dz) 8 2 I : Then (dz) nduces a probably measure on ( F) such ha he MPP N(d dz) on ( F F ) wh he (F )-local characersc (1 (dz)) s a marked sandard Posson process. Consequenly, here exss 8 2 I an F-P g as n (14) sasfyng 6

8 ( 2 dz)=g ( z 2 ) (dz) 2 I z 2 leadng o an nensy kernel of he form ( dz) = () r( 1 X ) Y g ( z 2 ) (dz) 2 I z 2 : Thus, he Jacod process s gven by h ( z )=() r( 1 X ) Y g ( z 2 ) 2 I z 2 : Smlarly o Andersen e al. (1993, p.482), we dene S ( 1 )= X 2I r( 1 X ) Y and by vrue of formula (15), wh =( ), we ge log L ( ) = ; [log (s) + log r( 1 X s ) + log g (s z 2 )] N (ds dz) ; (16) S s ( 1 ) (s) ds + R where R neher depends on nor on. In case of a mulvarae pon process (N ), 2 I (.e. when jj = 1), wh c = d 1 = d and r( X ):=exp( T X ) 2 I we oban he classc Cox' regresson model. 3.2 Paral Log-Lkelhood Guded by he Nelson-Aalen esmaor (cf. Andersen e al. (1993, p.482)), we subsue 1 log () by log and () S d by 1 ( 1 ) S dn ( 1 ) n formula (16), obanng he paral loglkelhood l () = X 2I ; [log r( 1 X s )+logg (s z 2 )] N (ds dz) ; log S s ( 1 ) dn s + R whch only depends on he unknown parameer 2 B. We wll suppose ha all processes are connuously derenable as ofen as needed, and ha he order of summaon, negraon and derenaon may always be changed. For sake of smpler noaon, we dene for 2 I z 2 he followng processes whch weassumeobef-p's. 7

9 r (1) ( 1 X ):= d d 1 r( 1 X ), (d 1 -dmensonal vecor), r (2) ( 1 X ):= d2 d 1 d T 1 r( 1 X ), (d 1 d 1 -marx), g (1) ( z 2 ):= d d 2 g ( z 2 ), g (2) ( z 2 ):= d2 g d 2 d T ( z 2 ), 2 S (1) ( 1 ):= d d 1 S ( 1 ) S (2) ( 1 ):= d2 S d 1 d T ( 1 ) 1 (d 2 -dmensonal vecor), (d 2 d 2 -marx), (d 1 -dmensonal vecor), (d 1 d 1 -marx). Furhermore, S (1) () :=((S (1) ( 1 )) T ::: ) T (d-dmensonal vecor), () := S(2) ( 1 ) S (2)! (d d-marx). Fnally, wenroduce for j =1 2hevecors and marces, respecvely, (j) ( 1 X ):= r(j) ( 1 X ), r( 1 X ) (j) ( z ( z 2 ):= g(j) 2 ) g ( z 2 ). Usng mehods of purely paramerc nference (cf. Pruscha (1984), Andersen e al. (1993, sec. VI.2.2)), we ge he followng hree Lemmas. Lemma 1. Denng U () := d d l (), we oban! U () = X 2I = X 2I (1) ( 1 X s ) (s z 2 ) (1) N(ds dz) ; K (s z ) M ( ds dz) where M ( d dz) =N (d dz) ; ( dz) d and K ( z ):= (1) ( 1 X ) (1) ( z 2 ) R! ; S(1) S (1) () s S s () dn s = () 2 I z 2 : S () R K (s z ) s ( dz) ds = 8. 2 Proof: Observe ha P 2I Lemma 2. Assume ha condon X (A) K 2 j (s z ) s( dz) ds < 1 -a.s. 8 1 j d 2I holds. Then hu( ) = X 2I K 2 (s z ) s ( dz) ds : Proof: Apply Theorems 2 (c) and

10 For he nex Lemma, we pu for 2 I z 2 2 B C z () := D z () := (2) ( 1 X ) ; (1) 2 ( 1 X ) (1) (1) ( 1 X ) 2 (2) ( z 2 ) ; (1) 2 ( z 2 ) ( z 2 )( (1) ( 1 X )) T (1) (1) ( 1 X )( (1) ( z 2 )) T 2 ( z 2 ) Lemma 3. For he d d-marces W () := d d (U ()) T, we ge under (A) X 2! 2 3 W () = C s z () N (ds dz) ; 4 S(2) s () 2I S s () ; S(1) s () 5 dns = S s () where for = w ( ) ; w ( ) w ( ) = X 2I 8 2 < : C s z() ; () S s () ; 4 S(2) s X w ( ) = D s z () s ( dz) ds ; 2I = hu( ) S(1) s () S s ()! 2 39 = 5 1 C A : 1 C A M ( ds dz) (s) (S(1) s ()) 2 S s () ds = Noce ha, n conras o hu( ), neher U() nor W () depend on he funcon. Inroducng he furher abbrevaon R () := X 2I D z () r( 1 X ) Y g ( z 2 ) (dz) we oban w ( ) = hu( ) = ()) 2 R s () ; (S(1) s S s ()! (s) ds : Observe ha n he purely mulvarae case (where jj =1and = 1 ) R () = X 2I (r (1) ( X )) 2 r( X ) Y : In he even more specal Cox' case we have R() S (2) () andc z =. 3.3 Asympocal Inference as n!1 Nowwe subsue he counable se I of he las secons by a sequence (I ) n2in of counable ses, where I mgh label he objecs under observaon (.e., I = f1 ::: ng). Accordngly,we consder sequences ((N (ddz) Y X ), 2 I, z 2, ) n2in 9

11 of he correspondng processes such ha for each 2 I he MPP N (d dz) adms he local characersc ( ( 1 ) ( 2 dz)) wh ( 1 ) = () r( 1 X ) Y ( 2 dz) = g ( z 2 ) (dz) z2 n 2 IN: Whn hs seng we denoe condon (A) by (A n ). Observe ha he funcons r and he unknown parameer do no depend on n 2 IN. For he res of hs secon, we x an arbrary T> o descrbe he end of he observaon nervall [ T]. All lms n hs secon are aken as n ends o nny. Nowwenroduce a sequence of non-sngular d d-normng marces (; n ) n2in sasfyng (N) () ; n! (elemen-wse) () 9 C ; 2 ] 1[ such ha j; n j 2 ji jc ; 8 n 2 IN: Adopng he erms of he prevous secons and equppng hem wh an addonal ndex n 2 IN f necessary, we presen a furher se of condons: There exs mappngs s (1) (2) R (1) : [ T] B! IR IR d IR dd IR dd,respecvely, such ha as n!1 (Bn ) () () () (v) (v) sup 2[ T ] 2B sup 2[ T ] 2B sup 2[ T ] 2B sup 2[ T ] 2B sup 2[ T ] 2B j; n j 2 S S (1 n) () S () ; s () () ; (1) () S (2 n) () ; n S () ; (2) () (S (1 n) ()) 2 S () ; n R (); T n ; R (1) ;! ;! ;! ; T n ; s () ( (1) ()) 2 ;! () ;! (C n ) (D n ) () s (1) (2) and R (1) are bounded n [ T] B () s (1) (2) and R (1) are connuous funcons n 2 B unformly n 2 [ T]: The d d-marx () := s posve dene 8 2 ] T]: R (1) s () ; s s () ( (1) s ()) 2 (s) ds ( n ) There exss a > such ha sup 2I 2[ T ] 1 r( 1 X ; n )g (1) ( 1 X (1 n) ) ( z 2 ) ( z 2 ) > exp ( ;! Y (dz) 1 (1) ( 1 X (1 n) ) ( z 2 ) ;! :! )!

12 In case of ; n = 1 p an I d wh <a n!1, he followng condon s sucen for (B n )()-(v) (B n) () () sup 2[ T ] 2B sup 2[ T ] 2B 1 S (m n) () ; s (m) () a n 1 R () ; R (1) () a n ;! ;! (m = 1 2) where s (m) () s() (m) () m=1 2, demandng addonally ha s () : [ T]! IR s bounded away from. In he purely mulvarae Cox' case, (B n)() conans (B n)() wh R 1 () s (2) (), see Andersen e al. (1993 p.497, condon VII.2.1). A sequence ( ^ ) n2in of d-dmensonal random vecors s called conssen paral lkelhood esmaor or shorly conssen PML, wehaveforn!1 (j ^ ; j < U ( ^ )=)! 1 8 > where U s gven by Lemma1. Proposon 1. Under (A n ),(B n )(),(),(v),(v),(c n ),(D n ),( n ) we have (U n ) ; n U () D ;! N d ( ()) 8 2 [ T]: Proof: One can show ha(b n )(),(), (C n ) and ( n ) mply (9), whle (B n )(v),(v) and (D n ) yeld (1). Condon (A n )allows he applcaon of Theorem 4 whch complees he proof. 2 Now we consder sequences of d-dmensonal random vecors ( n) n2in fulllng (B n) ; ;T n ( n ; ) n 2 IN s -sochascally bounded: Proposon 2. Under (A n ),(B n ),(C n ) we have (W n) ;; n W ; T n ;! () 8 2 [ T] and 8 sequences of d-dmensonal random vecors ( ) n n2in sasfyng (B ): n Proof: Decompose j;; n W ; T n ; ()j smlar o Andersen e al. (1993, p.5-1). 2 Theorem 7. Le (A n ),(B n ),(C n ),(D n ) hold. Then here exss for each 2 [ T]a conssen PML ( ^ ) n2in for fulllng (B ). n Proof: See Pruscha (1996 sec.vi, Saz 1.2). The basc deas are due o Achson & Slvey (1958), Bllngsley (1971) and Fegn (1975). 2 Theorem 8. (A n ),(B n ),(C n ),(D n ) and ( n ) mply (U n), (W n), and under (U n), (W n) we have for any conssen PML ( ^ ) n2in sasfyng (B n) ; ;T n ( ^ ; ) D ;! Nd ( ;1 ()) 8 2 [ T]: Proof: Apply Proposons 1,2 and Pruscha (1996 sec.vi, Saz 1.6). 2 11

13 Now we can esmae as n he classcal pon process heory (see, e.g., Andersen e al. (1993, sec.vii.2)). Frs we approxmae A() := R (s) ds by he Breslow esmaor ^A ( ^ ):= 1(Y S s s > ) ( ^ ) dn s 2 [ T] n2 IN where Y := P 2I Y. Fnally, we oban he kernel esmaor ^ = 1 b N X k=1 ; k b where K s a kernel funcon and b he bandwdh. 1 A 1 S k 2 [ T] n2 IN 4 Paramerc Inference as!1 In hs chaper, we deal wh purely paramerc problems as me ends o nny when here s jus one objec under observaon, develloppng opcs from Pruscha (1984), Hjor (1985) and Andersen e al. (1993). 4.1 M-esmaors We consder an MPP N(d dz) wh nensy kernel ( dz) admng he decomposon ( dz) =h( z ) (dz) z2 where 2 open IR d d2 IN, (dz) s a probably measure on sasfyng ( dz) (dz) 8 2, and h s he accompanyng Jacod process whch we assume o be connuously derenable w.r.. An M-esmaor ^ of 2 s a soluon of he equaons U () =, where U () := K(s z ) M( ds dz) and M( d dz) =N(d dz) ; ( dz) d. For he d-dmensonal process K, we assume jk(s z )j s ( dz) ds < 1 ; a:s: 8 : If K( z ) = d h( z ) d,we refer o he lkelhood case, and^ h( z ) s called maxmum lkelhood esmaor (ML). Now we sae a rs se of basc condons (A ) () K has connuous rs order dervaves w.r.. he processes K K (1) := d d K T and h (1) := d h are F-P's. d () K and h have connuous second order dervaves w.r.. he processes K j K (1)! 1jd 12 and h (2) := d2 h are F-P's. ddt

14 Noe ha he Jacod-process h s by denon an F-P. The followng Lemma can easly be proven usng derenaon rules. Lemma 4. (a) Assumng (A )(), we have for where W () := d d (U ()) T w () := w () := = w () ; w () K (1) (s z ) M( ds dz) K(s z ) (h(1) (s z )) T h(s z ) s ( dz) ds: (b) In he lkelhood case, (A )() mples for wh W () = v () ; v () v () := v () := h (2) (s z ) h(s z ) (h (1) (s z )) 2 h 2 (s z ) M( ds dz) N(ds dz): ven n he lkelhood case, we have w () 6= v () andw() 6= v() however, w() she compensaor of v(). Boh w () andv () have marngale srucure. We furher dene he d d-marx and he d 2 d 2 -marx I () := I () := For hese marces we sae a furher condon: K 2 (s z ) s ( dz) ds (K (1) (s z )) 2 s ( dz) ds : (B ) () Le he dagonal elemens of I () be ne -a.s. 8 : () Le he dagonal elemens of I () be ne -a.s. 8 : Noe ha (B ) mples va Cauchy-Schwarz nequalyhaall he elemens of he marces are ne -a.s. Now Theorems 2(c) and 3 yeld Lemma 5. Le (A )() hold. Then we have (a) (B )() ) U() s a locally square negrable marngale wh characersc hu() = K 2 (s z ) s ( dz) ds : (b) (B )() ) w () s a locally square negrable marngale wh characersc hw () = (K (1) (s z )) 2 s ( dz) ds : 13

15 4.2 Asympocal Inference as!1 As a generalzaon of Pruscha (1984) and Andersen e al. (1993, sec.vi.2), we presen a se of condons under whch he asympoc heory of Pruscha (1996, sec.vi.1) can be appled for M-esmaors. Consderng a famly of non-sngular d d-marces (; ) sasfyng (T), we formulae a furher condon whch has an ergodc [(),()] and a Lndeberg-lke par [()]. All he lms n hs secon are aken as!1. (C ) () () () ; hu() ; T ;! () where () s a posve dene, symmerc d d-marx ; w (); T ;! B() where B() s a posve dene d d-marx L ( ") ;! 8 "> where L ( ") := j; K(s z )j 2 1(j; K(s z )j >") s ( dz) ds: In he lkelhood case, condons (C )() and (C )() are dencal, wh B() =(). Proposon 3. Le condons (A )(), (B )(), (C )(),() be sased. Then we have he followng convergence n dsrbuon (U ) ; U () D ;! N d ( ()) : Proof: (B )(), (C )() and (C )() yeld (6), (11) and (12), respecvely, such ha he applcaon of Theorem 5 complees he proof. 2 To sae asympoc resuls for he process W (), we formulae a furher se of condons (D () kg jh hw hg ) () s -sochascally bounded 8 1 g h j k d where ; =( jk ) 1j kd and w () =(wjk ()) 1j kd : () There exss a neghborhood of and 9 M < 1 such ha lm!1 (j; ~ R ( ~ ); T j <M 8 ~ 2 )=1 where ~ R () =( ~ R jk ()) 1j kd ~ R jk () := dx l=1 () R jkl () W jk l R jkl Proposon 4. (A ), (B )(), (C )() and (D )() mply (W ) ; ; W (); T ;! B() Proof: Apply Lemma 5(b) and use Lenglar's nequaly. 2 Now we consder sequences of d-dmensonal random vecors ( ) fulllng (B ) ; ;T ( ; ) s -sochascally bounded: Proposon 5. Under condons (A ), (B )(), (C )() and (D ) we have as!1 ;; W ( );T (W ) ;! B() 8 sequences of d-dmensonal random vecors ( ) sasfyng (B ): 14

16 Proof: xpand W () n a Taylor seres around he rue parameer and apply Proposon 4. 2 Theorem 9. (A ), (B ), (C ) and (D ) mply (U ), (W), and under (U), (W),he followng holds: There exss a conssen M-esmaor (^ ) for fulllng (B ). If here exss an esmaon funcon l : IR +! IR such ha U() = d l(), hen wh d -probably endng o one, l akes a local maxmum a ^. Furhermore, we have for conssen M-esmaors (^ ) fulllng (B ) ; ;T (^ ; ) D ;! N d ( B ;1 ()()B ;T ()): Proof: Apply Proposons 3,5 and Pruscha (1996 sec.vi, Saz 1.2 and Saz 1.6, usng deas due o Achson & Slvey (1958), Bllngsley (1961) and Fegn (1975)). 2 References [1] Aalen, O.O. (1975): Sascal Inference for a Famly of Counng Processes. PhD hess, Unversy of Calforna, Berkeley. [2] Achson, J. and Slvey, S.D. (1958): Maxmum Lkelhood smaon of Parameers subjec o Resrans. Ann. Mah. Sas. 29, [3] Andersen, P.K. and Borgan,. (1985): Counng Process Models for Lfe Hsory Daa: A Revew. Scand. J. Sascs 12, [4] Andersen, P.K., Borgan,., Gll, R.D. and Kedng, N. (1993): Sascal Models Based on Counng Processes. Sprnger, New York. [5] Bllngsley, P. (1961): Sascal Inference for Markov Processes. The Unversy of Chcago Press, Chcago. [6] Boel, R., Varaya, P. and Wong,. (1975): Marngales on Jump Processes I, II. Sam J. Conrol 13, and [7] Bremaud, P. (1972): A Marngale Approach o Pon Processes. lecronc Res. Lab., Memo no. M-345. Unversy of Calforna, Berkeley. [8] Bremaud, P. (1981): Pon Processes and Queues Marngale Dynamcs. Sprnger, New York. [9] Dellachere, C. and Meyer, P.-A. (1982): Probables and Poenal B. Norh- Holland, Amserdam. 15

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