Efficiency of Some Estimators for a Generalized Poisson Autoregressive Process of Order 1

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1 Ope Joural of Saisics Published Olie Augus 6 i SciRes hp://wwwscirporg/joural/ojs hp://dxdoiorg/436/ojs66454 Efficiecy of Some Esimaors for a Geeralized Poisso Auoregressive Process of Order Louis G Doray Adrew Luog El-Halla Najem Dépareme de Mahémaiques e de Saisique Uiversié de Moréal Moréal Caada École d Acuaria Uiversié Laval Québec Caada Received 7 May 6; acceped Augus 6; published 3 Augus 6 Copyrigh 6 by auhors ad Scieific Research Publishig Ic This work is licesed uder he Creaive Commos Aribuio Ieraioal Licese (CC BY hp://creaivecommosorg/liceses/by/4/ Absrac Various models have bee proposed i he lieraure o sudy o-egaive ieger-valued ime series I his paper we sudy esimaors for he geeralized Poisso auoregressive process of order a model developed by Alzaid ad Al-Osh [] We compare hree esimaio mehods he mehods of momes quasi-likelihood ad codiioal maximum likelihood ad sudy heir asympoic properies To compare he bias of he esimaors i small samples we perform a simulaio sudy for various parameer values Usig he heory of esimaig equaios we obai expressios for he variace-covariace marices of hose hree esimaors ad we compare heir asympoic efficiecy Fially we apply he mehods derived i he paper o a real ime series Keywords Discree Time Series Auoregressive Process Mome Esimaor Quasi-Likelihood Efficiecy Geeralized Poisso Quasi Biomial Disribuio Iroducio Time series are used o model various pheomea measured over ime Successive observaios are ofe correlaed sice hey may deped o some commo exeral facors bu which remai ukow o he aalys I his case auoregressive models will be useful o model his depedece I some siuaios we migh be ieresed i he umber of eves which occur durig a cerai period of ime Such observaios will ecessarily be o-egaive ad ieger-valued Models which have bee used for sequeces of depede discree radom variables iclude he Poisso auoregressive process of order deoed How o cie his paper: Doray LG Luog A ad Najem E-H (6 Efficiecy of Some Esimaors for a Geeralized Poisso Auoregressive Process of Order Ope Joural of Saisics hp://dxdoiorg/436/ojs66454

2 L G Doray e al ( deoed GPAR ( (see Alzaid ad Al-Osh [] The ( margial disribuios is a special case of he GPAR ( PAR iroduced by Al-Osh ad Alzaid [] ad he geeralized Poisso auoregressive process of order PAR process a saioary process wih Poisso The paper is orgaized as follows I Secio for compleeess we review some properies of he geeralized Poisso auoregressive process of order I Secio 3 we derive he expressios for he momes esimaors he quasi-likelihood ad he maximum likelihood esimaors of he 3 parameers of he GPAR ( These mehods have appeared i he lieraure (see Al-Nachawai Alwasel ad Alzaid [3] for he quasilikelihood ad momes mehod ad Brääs [4] for likelihood mehods However asympoic properies such as efficiecies of hese mehods are o discussed i hose papers I his paper (Secios 4 ad 5 we sudy properies of hese esimaors such as bias ad asympoic efficiecy The las secio reaalyzes a real-daa example which ca be modelled wih a ( GPAR process where esig is discussed We hope ha wih his sudy praciioers will have more iformaio o selec oe esimaio mehod versus aoher oe ad o perform ess cocerig values of he parameers GPAR( Process To defie he ( disribuios GPAR process we eed firs o review he geeralized Poisso ad he quasi-biomial A radom variable X has a geeralized Poisso disribuio wih parameers λ ad θ deoed ( if is probabiliy mass fucio (pmf is defied by ( θ x ( λ+ θx λ λ + x x x = [ ] e! if P X = x = for x > m whe θ < GP λθ where max λ m θ m 4 is he greaes posiive ieger for which λ + mθ > whe θ is egaive Noe ha for θ = he radom variable X becomes a Poisso ( λ disribuio I his paper we will resric ourselves o he case where θ Cosul [5] has show ha he expeced value µ ad variace σ of X are give whe θ < by λ > ( < ad ( λ µ = σ = ad 3 θ ( θ so ha for posiive values of θ we have overdispersio (ie Var [ X] > E[ X] The sum X + Y of wo idepede radom variables X ad Y wih disribuios also has a GP disribuio wih parameers ( GP ( λ θ ad GP ( λ + Ambagaspiiya ad Balakrisha [6] have derived he recurrece formula for he probabiliy fucio of he compoud geeralized Poisso disribuio used i risk heory A o-egaive ieger-valued radom variable X has a quasi-biomial disribuio deoed QB ( p θ if is pmf is give by x pq ( p + xθ q + ( x x [ ] ( + θ < p < q = p ad θ is such ha θ < mi ( pq x θ P X = x = x = where Is mea equal o p is idepede of he parameer θ The followig proposiio proved i Alzaid ad Al-Osh [] shows he relaio bewee he QB ad GP disribuios Proposiio : If X ad S( are wo idepede radom variables wih GP ( λθ ad QB ( p θ λ disribuios he S( X follows a GP ( pλ θ disribuio The GPAR ( process geeralizes he PAR ( process iroduced by Al-Osh ad Alzaid [] The PAR ( model where [ X ] = E[ X ] has bee used o model ime series i various fields for example i Var isurace for shor-erm workers' compesaio because of work-relaed ijuries (Freelad ad McCabe [7] ad i medicie for he icidece of ifecious diseases (Cardial Roy ad Lamber [8] Var X is greaer ha I pracice may ieger-valued series will ofe exhibi overdispersio (ie [ ] 638

3 E[ X ] The ( L G Doray e al PAR model would herefore o be appropriae for hose ime series I cases where he exra variaio ca be explaied i a deermiisic way addig regressors would be adequae (see Freelad ad McCabe [7] bu where he exra variaio is of a sochasic aure he GPAR ( model could be used for modellig overdispersed ime series The GPAR ( model iroduced by Alzaid ad Al-Osh [] is defied as X = S ( X + ε = ( where { S ( = } is a sequece of iid radom variables wih a ( { ε } is a sequece of iid radom variables wih a GP ( qλ θ disribuio QB p θ λ disribuio 3 These wo sequeces are idepede of each oher 4 X has a GP ( λθ disribuio idepede of { ε } ad { S ( } Proposiio : The GPAR ( process { X } has a GP margial disribuio Proof: See Alzaid ad Al-Osh [] The GPAR ( process is obaied from he QB ( p θ λ ad GP ( qλ θ disribuios ad o from he QB ( p θ ad GP ( qλ θ disribuios as saed i Al- Nachawai e al [3] X is equal o The auocorrelaio fucio (acf of he GPAR ( process { } k ρ X( k [ X X k] p k The acf of his process is he same as ha of a ( p ( The parial auocorrelaio fucio (pacf of he ( = Corr = for = AR process excep ha i is always o-egaive sice GPAR process is equal o p if k = φkk = if k = 3 The sample acf ad pacf will be useful o ideify he GPAR ( model from a observed ime series 3 Esimaio of he Parameers Esimaig he parameers i a ( disribuio of X = x is he covoluio of a QB ( p θ λ x ad a ( GPAR process will prese some challeges sice he codiioal X give GP qλ θ disribuio I his secio we will review hree esimaio mehods for he parameer vecor Θ= ( p λθ of he GPAR ( process he mehods of momes quasi-likelihood ad codiioal maximum likelihood These mehods have bee proposed i he lieraure see for example Al-Nachawai e al [3] or Brääs [4] However less emphasis is placed o heir asympoic properies such as efficiecy I Secio 4 we sudy he bias of hese esimaors ad i Secio 5 heir efficiecy 3 Mehod of Momes or Yule-Walker The firs auocovariace of he GPAR ( process is equal o Cov [ X X] pvar [ X] By akig he expeced value of boh sides of he equaio give i ( we fid [ ] ( Sice qλ = + θ We also kow ha we obai E[ X ] pe[ X ] ( ( + = ( { } [ ] E S X = EX E S X X x E px = = E X = E S X + µ ε Var [ X ] = ( θ 3 From he observaios x x x we esimae he meas E[ X ] E[ X ] he variace Var [ ] he auocovariace Cov [ X X ] λ + by heir sample aalogs (3 (4 X ad 639

4 L G Doray e al x = x = x = x = Var [ X ] ( = x x = Cov [ X+ X] = ( x x( x+ x = Solvig he sysem of Equaios ( (3 (4 wih E[ X ] E[ X ] Var [ X ] ad Cov [ X X] by heir sample values we obai he momes esimaors Θ= ( p λθ of parameer vecor ( p λθ p = ( x x( x+ x = λ = ( x x = θ = ( px 3 x 3 q ( x x = λ q ( x px + replaced Θ= where q = p We have correced here mispris i he formulas for he mome esimaors of he parameers λ ad θ give by Al-Nachawai e al [3] 3 Quasi-Likelihood Mehod This mehod proposed iiially by While [9] replaces he rue likelihood by he oe which assumes ha he observaios come from a ormal disribuio wih he same codiioal mea ad variace Al-Nachawai e al [3] obaied he quasi-likelihood esimaors Θ= ( p λθ by maximizig ( x µ ( σ L p λθ = e = πσ where ad µ ad σ are give by [ ] ( µ ε λq ( θ µ = E X X = x = E S x + = px + [ X X x ] S ( X X x [ ε ] σ = Var = = Var = + Var x j x! ( θλ 3 = pq x λq ( θ j + j= ( x j! + x ( θλ We have used he expressio i Sheo [] for he formula of he variace of a quasi-biomial disribuio 64

5 which is a bi differe from he oe give i Al-Nachawai e al [3] Sice he ( L G Doray e al GPAR process is resriced o o-egaive iegers ad herefore o symmerical oe migh suspec ha he esimaors are less efficie ha he maximum likelihood esimaors which is ideed he case (see Secio 5 for umerical resuls 33 Codiioal Maximum Likelihood Mehod To obai he codiioal maximum likelihood esimaors (MLE s ˆ ( pˆ ˆ λθ ˆ Θ= we eed he codiioal disribuio of ( X X = x which is he covoluio of a QB ( p θ λ x disribuio ad a GP ( qλ θ disribuio Give he observaios x x x x we have o maximize he fucio ( λθ = ( = = L p P X x X x = mi ( x x = P S ( x = r P[ ε = x r] = r= x mi r ( x x r x pqλ pλ + rθ qλ + ( x r θ = = r= r λ + θx λ + θx λ + θx x ( ( x r λq λq+ θ x r r! e λq θ( x r We will work wih he loglikelihood fucio l( p λθ equal o l ( L p λθ which will have o be maximized umerically o obai he MLE ˆΘ Uder ormal regulariy codiios usig likelihood heory (see Gouriéroux ad Mofor [] or Hamilo [] he vecor ˆΘ has a asympoic muliormal disribuio ie where I( Θ = E l( Θ θi θ j 4 Bias of Esimaors ( Θ Θ ( ( Θ ˆ N I deoes covergece i law is he vecor of zeros of dimesio 3 ad is Fisher s expeced iformaio marix of dimesio 3 3 Wih simulaios we will sudy he bias of he momes esimaors ad he MLE s Seig he values of he 3 parameers o hose i Table wo series of 5 ad observaios were geeraed from model ( i C++ This experime was repeaed imes For each series he momes esimaors were calculaed as well as heir average ad he bias The codiioal MLE's were calculaed usig he ieraive Dowhill Simplex mehod (see Press Teukolsky Veerlig ad Flaery [3] which does o require he calculaio of he derivaives of he fucio o be maximized As iiial values we used he momes esimaors The resuls of he simulaios appear i Figures -3 From Figures -3 we see ha he bias of he MLE s is smaller ha ha of he momes esimaors ad ha Table Values of parameers p λ θ

6 L G Doray e al Figure Bias of he esimaors of p (Mome: MLE: Figure Bias of he esimaors of λ (Mome: MLE:

7 L G Doray e al Figure 3 Bias of he esimaors of θ (Mome: MLE: i decreases whe he size of he series icreases Figure shows ha he bias of ˆp is much smaller ha ha of p excep whe = ad θ = where hey are almos equal o The bias of he wo esimaors is egaive I Figure we see ha he bias of ˆλ ad λ is close o whe λ = ; as λ icreases ˆλ ad λ are more biased I all cases he bias of he esimaor of λ is posiive The bias of he esimaor of θ behaves like ha of p (Figure 3; for he wo esimaio mehods i is similar for λ = 5 or Sice he momes esimaors ad he codiioal MLE s are almos ubiased for large we sudy heir asympoic efficiecy i he ex secio 5 Asympoic Efficiecy of Esimaors We will firs discuss he echiques by which we ca obai he asympoic variace-covariace marix of he esimaors uder he hree esimaio mehods To sudy efficiecies we calculae i subsecio 54 he raios of he variaces of he esimaors ad he raio of he deermias of heir variace-covariace marix usig observaios simulaed from a GPAR ( process for various values of he parameers The resuls are summarized i Table ad Table 3 of his secio 643

8 L G Doray e al Table Efficiecy of momes esimaors λ p θ Var ( Var ( ˆ p p Var ( λ Var ( ˆ λ Var ( Var ( ˆ θ θ Θ Θ ˆ Mehod of Momes By usig a asympoically equivale facor of isead of i Equaio (3 momes esimaors Θ= p λθ are give as soluios of he sysem of equaios ( ( ( X X( X+ X = ( X X = p = 644

9 L G Doray e al Table 3 Efficiecy of quasi-likelihood esimaors λ p θ Var ( p Var ( pˆ Var ( λ Var ( ˆ λ Var ( θ Var ( ˆ θ Θ Θ ˆ Le us defie he fucios = = X ( λθ ( p ( X X ( p λ = θ λ = 3 ( θ ( ( X X( X+ X ( X X f p = p = 645

10 L G Doray e al ad he vecor ( λθ ( g p = X p ( λθ ( h p = X X ( p θ λ ( θ F( Θ = f ( p λθ g ( p λθ h ( p λθ = = = The expeced values E f ( p λθ E g ( p λθ ad E h ( p λθ Usig a Taylor series expasio aroud Θ = ( p λ θ F( F( F( ( ε p where ε wih Sice Θ is a soluio of ( Usig Slusky s heorem we fid ha where wih probabiliy λ 3 are asympoically equal o he rue parameer value we obai Θ = Θ + Θ Θ Θ + (5 p deoig covergece i probabiliy F Θ = Equaio (5 ca rewrie as ( ( ( F Θ Θ Θ = F Θ ε Marix A evaluaed a Θ ca be esimaed by If or F ( Θ ( Θ Θ = F ( Θ + o ( p F Θ Θ Θ F Θ ( ( ( or ( N A ( Y( A Θ Θ Var A= lim F( Θ ad Y = lim F( Θ + λ p p A = x + θ θ θ 3λ ( θ ( θ ( λ( ( ˆ 3 4 θ λ ad p are ukow hey ca be replaced by appropriae esimaes The variace-covariace marix of Y is equal o Var ( f Cov ( f g Cov ( f h i= i= i= i= i= ( g f ( g i i i ( g i h = = = = i= ( h i f i ( h i g i ( h = = = = i= Cov Var Cov Cov Cov Var Le us cosider he firs eleme of his marix: Var f E[ ff k] = = = k= (6 646

11 sice asympoically E[ ff k] = Cov ( f fk (because [ ] expressios sice Cov ( f f as k equaliy becomes k L G Doray e al E f = as I pracice we rucae hese If we limi ourselves o a differece of k 5 he las f E ff k = k 5 [ ] Var Usig he law of large umbers we ca esimae his las erm by ff k k 5 The oher elemes of he marix ca be esimaed i he same way 5 Quasi-Likelihood Mehod To deermie he quasi-likelihood esimaor ( Θ we have o maximize Le us defie he quasi-score vecor ( x µ ( 5 l ( π 5 l l p λθ = ( σ + (7 = σ ( ( ( 3 S ( Θ = l( Θ = ( S S S = l( Θ l( Θ l( Θ p λ θ From Hamilo [] usig quasi-likelihood heory we coclude ha ( Θ Θ ( N D SD where wih probabiliy D ad S are limis i probabiliy marices They are defied as ad ( ( ( E ( S X ( E S X E S X = = = ( p λ θ ( ( ( D = lim E ( S X E ( S X E ( S X = = = p λ θ ( 3 ( 3 ( 3 E ( S X E ( S X = = = p λ E ( S X θ ( ( ( ( ( ( 3 S S S S S S = = = ( ( ( ( ( ( 3 S = lim S S S S S S = = = ( 3 ( ( 3 ( ( 3 ( 3 S S S S S S = = = evaluaed a Θ he rue parameer We ca obai esimaes for ˆD ad Ŝ where marix ˆD is defied as ( ( ( ( ( S X ( S X S X = = = p λ θ ˆ ( ( ( D = ( S X ( S X ( S X = = = p λ θ ( 3 ( 3 ( 3 ( S X ( S X ( S X = = = p λ θ ad Ŝ is he fiie versio of S evaluaed a Θ ; he elemes of ˆD are evaluaed umerically usig (8 647

12 L G Doray e al expressio (7 Packages such as MATHEMATICA ca hadle hese derivaives calculaios umerically Cosequely he variace-covariace marix of Θ ca be esimaed by ˆ D SD ˆ ˆ 53 Codiioal Maximum Likelihood Usig he rue loglikelihood fucio from secio 43 we defie he score vecor ( ( ( 3 ( Θ = ( Θ = ( S l S S S From Hamilo [] usig likelihood heory we fid ha ˆ Θ Θ N S (9 ( ( where marix S is defied aalogously as i he previous secio bu wih a differe loglikelihood fucio 54 Numerical Comparisos Table ad Table 4 give he esimae of he asympoic efficiecy of he mome ad he quasi-likelihood esimaors compared o he MLE calculaed from observaios ( series of observaios geeraed from a GPAR ( process wih various parameer values Comparig Table ad Table 3 he quasi-likelihood esimaor for p has a smaller variace ha he momes esimaor; for λ i depeds o he values of he parameers The momes esimaor of θ has a smaller variace ha he quasi-likelihood esimaor excep whe p = where θ is beer ha θ The esimaed deermia of he variace-covariace marix of ˆΘ usig he average of he deermias is always smaller ha ha of Θ ad Θ (las colum of Table ad Table 3 The MLE is more efficie ha he mome or he quasi-likelihood esimaor ad he mome esimaor more efficie ha he quasi-likelihood esimaor i geeral 6 Applicaios: Number of Compuer Breakdows I his secio we perform some ess o a real ime series preseed by Al-Nachawai e al [3] o he umber of weekly compuer breakdows for 8 cosecuive weeks This series is overdispersed sice is mea ad variace are equal o 46 ad 454 I Figure 4 he acf fucio is see o decrease wih he lag while he GPAR model could herefore be appropriae for his series We pacf is high for lag ad low hereafer; a ( use he GPAR ( model i he aalysis Sice he MLE was show o be he bes asympoic esimaor i he previous secio he parameers were esimaed wih his mehod; he esimaes appear i Table 4 wih he esimaed variace-covariace marix Wih he esimaed variace-covariace marices based o expressios (6 (8 ad (9 of Secio 5 Wald ess ca be performed quie easily depedig o which esimaor has bee chose For example o es H : θ = θ usig θ he quasilikelihood esimaor he saisic ca be based o he saisic Z = ( θ θ V ( θ where V ( θ is a esimae of he variace of θ which ca be obaied from he correspodig diagoal eleme of Vˆ Dˆ SD ˆ ˆ = Sice ( θ θ V ( θ is asympoically N ( we rejec H a level α if Z is greaer ha z α To es H : p = p λ = λ θ = θ he es saisic ca be based o ˆ Θ Θ V Θ Θ χ 3 disribuio asympoically I is expeced ha he more efficie he esimaor is he which follows a ( Table 4 MLE s of he parameers ˆp ˆλ ˆ θ Variace-covariace marix

13 L G Doray e al Figure 4 Acf ad pacf more powerful he es will be Wih he esimaed parameers we ca es he GPAR ( model versus he simpler PAR ( model Sice he codiioal MLE ˆ θ equals 47 wih a variace of 6 performig he es H : θ = vs H : θ > gives Z = 47 6 = 94 This leads us o rejec H ad o coclude ha he GPAR ( model is more appropriae: here is overdispersio i he observaios Ackowledgemes The auhors graefully ackowledge he fiacial suppor of he Naural Scieces ad Egieerig Research Coucil of Caada ad of he Fods pour la Coribuio à la Recherche du Québec Refereces [] Alzaid AA ad Al-Osh MA (993 Some Auoregressive Movig Average Processes wih Geeralized Poisso Margial Disribuios Aals of he Isiue of Mahemaical Saisics hp://dxdoiorg/7/bf77589 [] Al-Osh MA ad Alzaid AA (987 Firs-Order Ieger-Valued Auoregressive (INAR( Process Joural of Time Series Aalysis hp://dxdoiorg//j b438x [3] Al-Nachawai H Alwasel I ad Alzaid AA (997 Esimaig he Parameers of he Geeralized Poisso AR( process Joural of Saisical Compuaio ad Simulaio hp://dxdoiorg/8/ [4] Brääs K (994 Esimaio ad Tesig i Ieger-Valued AR( Models Umeå Ecoomic Sudies 335 Umeå Swede [5] Cosul PC (989 Geeralized Poisso Disribuio: Properies ad Applicaios Marcel Dekker Ic New York [6] Ambagaspiiya RS ad Balakrisha N (994 O he Compoud Geeralized Poisso Disribuios ASTIN Bullei hp://dxdoiorg/43/ast4569 [7] Freelad RK ad McCabe B (4 Aalysis of Low Cou Time Series Daa by Poisso Auoregressio Joural of Time Series Aalysis hp://dxdoiorg//j x [8] Cardial M Roy R ad Lamber J (999 O he Applicaio of Ieger-Valued Time Series Models for he Aalysis of Disease Icidece Saisics i Medicie hp://dxdoiorg//(sici97-58(999858:5<5::aid-sim63>3co;-d [9] While P (96 Gaussia Esimaio i Time Series Bullei of he Ieraioal Saisical Isiue 39-6 [] Sheo LR (986 Quasi-Biomial Disribuios Ecyclopedia of Saisical Scieces Joh Wiley & Sos New York Vol

14 L G Doray e al [] Gouriéroux C ad Mofor A (983 Cours de Hamilo JD (994 Time Series Aalysis Priceo Uiversiy Press New Jersey [] Hamilo JD (994 Time Series Aalysis Priceo Uiversiy Press New Jersey [3] Press WH Teukolsky SA Veerlig WT ad Flaery BP ( Numerical Recipies i C++ The Ar of Scieific Compuig Cambridge Uiversiy Press Cambridge Submi or recommed ex mauscrip o SCIRP ad we will provide bes service for you: Accepig pre-submissio iquiries hrough Facebook LikedI Twier ec A wide selecio of jourals (iclusive of 9 subjecs more ha jourals Providig 4-hour high-qualiy service User-friedly olie submissio sysem Fair ad swif peer-review sysem Efficie ypeseig ad proofreadig procedure Display of he resul of dowloads ad visis as well as he umber of cied aricles Maximum dissemiaio of your research work Submi your mauscrip a: hp://papersubmissioscirporg/ 65