STK4080/9080 Survival and event history analysis

Transcription

1 STK48/98 Survival ad eve hisory aalysis Marigales i discree ime Cosider a sochasic process The process M is a marigale if Lecure 3: Marigales ad oher sochasic processes i discree ime (recap) where (formally a σ-algebra) deoes "he hisory" by ime, i.e. he iformaio coaied i M, M,..., M Marigales ad sochasic iegrals i coiuous ime Couig processes ad couig process marigales Wieer processes, Gaussia marigales ad he marigale ceral limi heorem Aleraive formulaios of he marigale propery are: E( M ) = I geeral he hisory may coai more iformaio ha wha is coaied i M, M,..., M We will assume ha where F M =. The E( M ) = for all M = M M Variaio processes Two processes describe he variaio of a marigale The variaio processes have he followig impora propery The predicable variaio process M is give by for while M = [ ] The opioal variaio process M is give by From his we obai: [ ] Var( M ) = E( M ) = E M Var( M ) = E( M ) = E M for while [ M ] = 3 4

2 Trasformaios Cosider a marigale Le be a predicable process, i.e. a sequece of radom variables where each H is kow oe sep ahead of ime (a ime we kow H ) Noe ha (usig M =) Z = ( H M ) = H ( M M ) s s s s= = H M s= s s We may defie a ew process Z = { Z, Z, Z,...} by Z = HM + H( M M ) H( M M ) The variaio processes obey he followig rules uder rasformaio If M =, we have ha Z is a mea zero marigale The process Z is deoed he rasformaio of M by H ad i is wrie Z = H M 5 6 Sochasic processes i coiuous ime: some geeral properies Uless oherwise saed, we will cosider ime-coiuous sochasic processes defied o a fiie ime ierval [, τ ] A sochasic process is adaped o a hisory { F } if a ime we kow he value of X(s) for all s (possibly apar from ukow parameers) A realizaio of X is a fucio of ad is called a sample pah Uless oherwise saed, we will cosider sochasic processes wih sample pahs ha are righ-coiuous ad have lef-had limis (cadlag) Marigales i coiuous ime A sochasic process is a marigale relaive o he hisory { F } if i is adaped o he hisory ad saisfies he marigale propery: Example: Le N() be a Poisso process wih iesiy λ Deoe by F he iformaio o all eves ha happe i [, ] The is a marigale (cf. page 53) Heurisically, M is a marigale provided ha A radom variable T > is a soppig ime if we a ime kow wheher T or T > 7 where dm() is he icreme of M over [, + d ), ad F is he hisory "jus before" ime 8

3 The properies of coiuous ime marigales parallel hose of discree ime marigales We will assume hroughou ha M()= The, ad M is a mea zero marigale A marigale has ucorrelaed icremes, i.e. for all Variaio processes The predicable variaio process M ad he opioal variaio process [ M ] of a ime-coiuous marigale M are obaied as limis of he discree ime variaio processes Pariio [,] io ime iervals each of legh / : M k The 9 Iformally, he las expressio gives: Example: N() is a Poisso process wih iesiy λ F is he iformaio o all eves ha happe i [, ] is a marigale As for discree ime marigales we have ha We have Hece dm = dn λ d From his we obai: d M Var( dm ) = F By iegraio his gives = Var( dn F ) = E( dn F ) = λ d M = λ = Var( dn λ d F ) For wo marigales M ad M, we may defie he predicable covariaio process M, M ad he opioal covariaio process M, M, see exercise.5 ad page 5 [ ]

4 Sochasic iegrals Sochasic iegrals are he coiuous ime aalogue o rasformaios for discree ime marigales Le be a marigale ad a predicable process, which iuiively meas ha for ay ime we kow he value of H() "jus before" (a sufficie codiio for predicabiliy is ha H is adaped ad lef-coiuous) We will defie he sochasic iegral We pariio [,] io ime iervals each of legh / Defie H k M k I geeral i is quie iricae o defie a sochasic iegral (cf. he Iô iegral), bu for he siuaios we will cosider i may be defied as a limi of rasformaios of discree ime marigales 3 The 4 A sochasic iegral Couig processes has similar properies as a rasformaio for discree ime marigales: A couig process is a sochasic process wih sample pahs ha are righ-coiuous sep fucios wih seps of size + is a mea zero marigale I = HdM = H s d M s [ ] [ ] I = HdM = H ( s) d M ( s) For resuls o covariace processes for wo sochasic iegrals, see exercise.8 ad page 5 We assume ha he couig process is adaped o he hisory { } F The iesiy process λ() w.r. he hisory is give iformally by { } F 5 6

5 The, as see earlier, he process is a mea zero marigale The opioal variaio process of M() becomes = lim k = ( N ) k Whe cosiderig wo couig processes N () ad N () adaped o he same hisory { F }, we will assume ha hey do o jump a he same ime The oe may show ha he correspodig marigales M () ad M () are orhogoal, which meas ha For he predicable variaio process, oe ha This moivaes he impora relaio (exercise.) 7 8 Sochasic iegrals for couig process marigales For a sochasic iegral relaive o a couig process marigale, he variace processes ake he form Mos sochasic iegrals we will ecouer are relaive o couig process marigales A sochasic iegral relaive o a couig process marigale is simple o udersad: a sum over he jump imes T <T <.. a ordiary iegral (pahwise) 9 Example: For he Nelso-Aale esimaor, we have: dm ( s) Aˆ( ) A ( ) = Thus A ˆ A is a mea zero marigale wih opioal variaio process ˆ dn ( s) A A = Noe ha ˆ A A is a ubiased esimaor for he variace of he Nelso-Aale esimaor

6 We wa o sudy he large sample disribuio of he Nelso-Aale esimaor We suppose ha our couig process N() is a aggregaed process obaied from observaio of idividual processes (e.g. as o pages 3-3) We will sudy he disribuio of as Aˆ( ) A ( ) = dm ( s ) To his ed we eed a asympoic heory for marigales Wieer process A Wieer process (or Browia moio) W() is a sochasic process wih: W() = W() W(s) ormally disribued wih mea ad variace s idepede icremes coiuous sample pahs Gaussia marigales Le V() be a sricly icreasig coiuous fucio wih V() = ad cosider he ime rasformed Wieer process U() = W(V()) The process U() iheris he properies of he Wieer process, so U() is a sochasic process wih U() = U() U(s) ormally disribued wih mea ad variace V() V(s) idepede icremes coiuous sample pahs Oe may show ha U() is a mea zero marigale wih predicable variaio U = V (exercise.) U() is called a Gaussia marigale 3 Rebolledo's marigale ceral limi heorem Uder fairly geeral assumpios, a sequece of marigales { Mɶ } will coverge i disribuio o a Gaussia marigale U() as defied o he previous slide Wha is eeded, is ha as (i) ɶ ( ) M V for all [, τ ] ɶ (ii) The sizes of he jumps of M go o zero (covergece i probabiliy) A precise formulaio of he codiios is give i secio.3.3 4

7 We will cosider marigales of he form = Mɶ H ( s) dm ( s) ( where H ) is a predicable process ad M N ( s) ds = λ is a couig process marigale For he Nelso-Aale esimaor we have ha Aˆ( ) A ( ) = dm ( s ) wih H ( s) = Assume ha = H ( s) dm ( s) y( s) > as for all s [, τ ] Assume ha we may wrie The sufficie codiios for Rebolledo's heorem are (i) = H s λ s v s s V v( s) ds for all [, τ ] The Rebolledo's codiios are saisfied: H ( s) λ( s) = α( s) α( s) = / α s y( s) (ii) ( H ) ( s) for all s [, τ ] 5 H ( s) = = / 6 By he marigale ceral limi heorem, we have ( ) A ˆ( ) A ( ) U ( ) (i disribuio) where U() is a mea zero Gaussia marigale wih variace fucio σ = α( s) ds y( s) I paricular for a fixed value of he Nelso-Aale esimaor A ˆ is approximaely ormally disribued 7