Consistent non-parametric Bayesian estimation for a time-inhomogeneous Brownian motion Gugushvili, S.; Spreij, P.J.C.

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1 UvA-DARE Digital Academic Repository) Cosistet o-parametric Bayesia estimatio for a time-ihomogeeous Browia motio Gugushvili, S.; Spreij, P.J.C. Published i: ESAIM-Probability ad Statistics DOI:.5/ps/2339 Lik to publicatio Citatio for published versio APA): Gugushvili, S., & Spreij, P. 24). Cosistet o-parametric Bayesia estimatio for a time-ihomogeeous Browia motio. ESAIM-Probability ad Statistics, 8, DOI:.5/ps/2339 Geeral rights It is ot permitted to dowload or to forward/distribute the text or part of it without the coset of the authors) ad/or copyright holders), other tha for strictly persoal, idividual use, uless the work is uder a ope cotet licese like Creative Commos). Disclaimer/Complaits regulatios If you believe that digital publicatio of certai material ifriges ay of your rights or privacy) iterests, please let the Library kow, statig your reasos. I case of a legitimate complait, the Library will make the material iaccessible ad/or remove it from the website. Please Ask the Library: or a letter to: Library of the Uiversity of Amsterdam, Secretariat, Sigel 425, 2 WP Amsterdam, The Netherlads. You will be cotacted as soo as possible. UvA-DARE is a service provided by the library of the Uiversity of Amsterdam Dowload date: 23 Ja 29

2 ESAIM: PS 8 24) DOI:.5/ps/2339 ESAIM: Probability ad Statistics CONSISTENT NON-PARAMETRIC BAYESIAN ESTIMATION FOR A TIME-INHOMOGENEOUS BROWNIAN MOTION Shota Gugushvili ad Peter Spreij 2 Abstract. We establish posterior cosistecy for o-parametric Bayesia estimatio of the dispersio coefficiet of a time-ihomogeeous Browia motio. Mathematics Subject Classificatio. 62G2, 62M5. Received October 8, 22. Revised April 23, 23.. Itroductio Cosider a simple liear stochastic differetial equatio dx t = σt)dw t, X = x, t [, ],.) where W is a Browia motio o some give probability space ad the iitial coditio x ad the square itegrable dispersio coefficiet σ are determiistic. We iterpret equatio.) as a short-had otatio for the itegral equatio t X t = x + σs)dw s, t [, ], where the itegral is the Wieer itegral of σ with respect to the Browia motio W. The process X is thus a time-ihomogeeous Browia motio. The fuctio σ ca be viewed as a sigal trasmitted through a oisy chael, where the oise modelled by the Browia motio) is multiplicative. Note that X is a Gaussia process with mea mt) =x ad covariace ρs, t) = s t. By P σ we will deote the law of the solutio X to.). Assume for simplicity that x = ad deote t i, = i/, i =,...,. Suppose that correspodig to the true dispersio coefficiet σ = σ, oe has a sample X,,...,, from the process X at his disposal. Assumig that σ belogs to some o-parametric class X of dispersio coefficiets, our goal is to estimate σ. This problem for a similar model was treated i [5, 9, 9] usig a frequetist approach. However, a o-parametric Keywords ad phrases. Dispersio coefficiet, o-parametric Bayesia estimatio, posterior cosistecy, time-ihomogeeous browia motio. The research of the first author was supported by The Netherlads Orgaisatio for Scietific Research NWO). Mathematical Istitute, Leide Uiversity, P.O. Box 952, 23 RA Leide, The Netherlads. shota.gugushvili@math.leideuiv.l 2 Korteweg-de Vries Istitute for Mathematics, Uiversity of Amsterdam, P.O. Box 94248, 9 GE Amsterdam, The Netherlads. spreij@uva.l Article published by EDP Scieces c EDP Scieces, SMAI 24

3 BAYESIAN ESTIMATION FOR A TIME-INHOMOGENEOUS BROWNIAN MOTION 333 Bayesia approach to estimatio of σ is also possible. The likelihood correspodig to the observatios X is give by L σ) = 2π ψ X t i, X ti, t i, σ 2 u)du,.2) where ψu) =exp u 2 /2). For a prior Π o X, Bayes formula yields the posterior measure ΠΣ X t,...,x, )= Σ L σ)πdσ) X L σ)πdσ) of ay measurable set Σ X. I the Bayesia paradigm, the posterior ecodes all the iformatio required for iferetial purposes. Oce the posterior is available, oe ca proceed to computatio of other quatities of iterest i Bayesia statistics, such as Bayes poit estimates, Bayes factors ad so o. It has bee recogised sice log that Bayesia procedures should be theoretically grouded through establishig posterior cosistecy, see e.g. [3]. I our cotext posterior cosistecy will mea that for every eighbourhood U σ of σ i a suitable topology) ΠUσ c X t,,...,x t, ) Pσ.3) P σ as the otatio ξ ξ i.3) ad below stads for covergece of a sequece of radom variables ξ to a radom variable ξ i P σ -probability). I other words, a cosistet Bayesia procedure asymptotically puts posterior mass equal to oe o every fixed eighbourhood of the true parameter. This is similar to the study of cosistecy of frequetist estimators. A method that does ot appear to work i the idealised settig whe a ifiite amout of data is available formalised by assumig that the sample size ) should also be uattractive i the fiite sample settig. Hece the importace of a study of posterior cosistecy. The situatio is typically quite subtle i the ifiite-dimesioal Bayesia settig: it is kow that a careless choice of the prior might reder a Bayes procedure icosistet. For a itroductio to cosistecy issues i Bayesia o-parametric statistics see e.g. [6, 25]. Our task i this work is to establish.3) uder suitable assumptios o the class of dispersio coefficiets σ ad the prior Π. Asymptotic properties of Bayesia procedures i estimatio problems for stochastic differetial equatios have bee already cosidered uder various setups i [8, 2 4, 7], primarily i the cotext of oparametric Bayesia estimatio of the drift coefficiet of a stochastic differetial equatio. Computatioal approaches to o-parametric Bayesia iferece for stochastic differetial equatios were studied i [, 5]. Coveiet overviews of the available results are give i [6, 26]. However, i the above works dealig with Bayesia asymptotics it is assumed that either a cotiuous record of observatios is available o the solutio to a stochastic differetial equatio, or that the solutio is observed at equispaced time poits Δ, 2Δ,...,Δ, with asymptotics treated i the latter case uder the assumptio that Δ is idepedet of ad. Our problem, o the other had, requires a differet approach due to a differet samplig scheme ad the fact that ergodicity of the solutio to a stochastic differetial equatio, that played a promiet role i most of the previous works o o-parametric Bayesia approach to statistical iferece for stochastic differetial equatios, is irrelevat i our case. Although the setup we cosider looks simple, to the best of our kowledge our work is the first oe to treat a iferece problem for a stochastic differetial equatio i the so called high-frequecy data case whe Δ = Δ as usig a o-parametric Bayesia approach. The highfrequecy data settig is particularly relevat i fiacial mathematics, where asset prices are ofte modelled through stochastic differetial equatios ad where huge amouts of observatios o them separated by very short time istaces are available. Perhaps the most iterestig feature of the preset work is the method of proof of posterior cosistecy, which differs i certai respects from the curretly used techiques. See Sectio 4 for a discussio. Also the simplicity of our model should ot ecessarily be cosidered a disadvatage: ideed,

4 334 S. GUGUSHVILI AND P. SPREIJ the model is somewhat similar to the Gaussia white oise model see e.g. Chap.7,Sect.4i[]),which,as is kow, has triggered some importat developmets i mathematical statistics. The paper is orgaised as follows: i the ext sectio we formulate our mai result dealig with posterior cosistecy for o-parametric estimatio of the dispersio coefficiet. Sice posterior cosistecy is closely liked with properties of a prior Π, i Sectio 3 we provide a example of a reasoable prior satisfyig the assumptios made i Sectio 2. Sectio 4 cotais a brief discussio o the obtaied result. The proof of our mai theorem is deferred util Sectio 5, while the Appedix cotais two techical lemmas used i Sectio Results The o-parametric class of dispersio coefficiets we will be lookig at is give i the followig defiitio. Defiitio 2.. Let X be the collectio of dispersio coefficiets σ : [, ] [κ, K], such that σ X is Lipschitz with Lipschitz costat M. Here <κ<k< ad <M< are three costats idepedet of the particular σ X. Remark 2.2. Note that for a costat σ we have P σ = P σ. A positivity assumptio o σ X i Defiitio 2. ca hece be viewed as a simple ad atural idetifiability requiremet. Strict positivity assumptio σ κ > allows oe to escape techical complicatios whe maipulatig the likelihood.2) this coditio has already appeared e.g. i [9]), while the upper boud σt) K, t [, ], restricts the size of the o-parametric class X ad is reasoable i light of Defiitio 2.3 give below. Fially, Lipschitz cotiuity of σ comes i hady at various stages of the proof of posterior cosistecy. The otio of posterior cosistecy depeds o a topology o X. Defiitio 2.3. The topology T o X is the topology iduced by the L 2 -orm 2. We ow formalise the cocept of posterior cosistecy. Defiitio 2.4. Let the prior Π be defied o X. We say that posterior cosistecy holds, if for ay fixed σ X ad every eighbourhood U σ of σ i the topology T from Defiitio 2.3 we have as. We summarise our assumptios. ΠUσ c X t,...,x, ) Pσ Assumptio 2.5. Assume that a) the model.) isgivewithx =adσ X, where X is defied i Defiitio 2., b) σ X deotes the true dispersio coefficiet, c) a discrete-time sample {X } from the solutio to.) correspodig to σ is available, where t i, = i/, i =,...,. Let V σ,ε = {σ X : σ σ <ε}, where deotes the L -orm. The followig is the mai result of the paper. Theorem 2.6. Uder Assumptio 2.5 posterior cosistecy as i Defiitio 2.4 holds, provided the prior Π o X satisfies ΠV σ,ε ) > 2.) for ay ε> ad at ay σ X.

5 BAYESIAN ESTIMATION FOR A TIME-INHOMOGENEOUS BROWNIAN MOTION Example of a prior I this sectio we provide a example of a prior satisfyig coditio 2.). Fix <κ<k< ad take a fixed Lipschitz cotiuous fuctio f : R [,K κ] with Lipschitz costat N>adsetσt) =κ + t fhs))ds, where h :[, ] R rages over the set of Hölder cotiuous fuctios of order β, /2) o [, ] for some fixed β. The each σ maps the iterval [, ] ito the iterval [κ, K] adσ is also Lipschitz with Lipschitz costat K, because t σt) σs) = fhu))du K t s. s We take the collectio of these fuctios σ as the collectio X from Assumptio 2.5a). We will ow costruct apriorπox. Let W = W t ) t be a stadard Browia motio over the time iterval [, ] ad let Z be a stadard ormal radom variable idepedet of W. Defie the Browia motio W =W t ) t iitialised at Z by W t = Z + W t ad itroduce the process Y =Y t ) t, where t [,] Y t = κ + t fw s )ds. Our prior Π o X will be the law of the process Y. We have to check that the prior Π satisfies 2.). To that ed take a fixed σ t) =κ + t fh s))ds ad let w be a geeric realisatio of the process W, so that y t = κ + t fw s)ds is the correspodig geeric realisatio of the process Y. We have ΠV σ,ε) =Πy : y σ <ε) Π w : w h < ε ), N because t sup [fw s ) fh s))]ds fw) fh ) N w h. By Lemma 5.3 i [22], Π w : w h < ε ) exp N if g: g h <ε/2n) ) 2 g 2 H Π W < ε ) 2N Here H deotes the Reproducig Kerel Hilbert Space RKHS) of the process W,g H, while H is the RKHS orm see [22] for a detailed treatmet of these cocepts with a view towards o-parametric Bayesia statistics). I our case H cosists of absolutely cotiuous fuctios g :[, ] R, such that g 2 <, while the RKHS orm is give by g H = g 2 ) + g 2 2, see page 446 i [2]. Note that the set {g H : g h <ε/2n)} is ot empty, as h ca be approximated arbitrarily closely i the L -orm by the covolutio h k b of h with asmoothkerelk b ) =/b)k /b) cf. p. 446 i [2]; we assume that k 2 < ad b ). Furthermore, Π W <ε/2n)) >, because W has a strictly positive desity. Coditio 2.) easily follows. I case oe is iterested i a smoother class of dispersio coefficiets σ tha what we have just costructed, oe ca simply take i the above costructio of the prior Π a smoother, say β times differetiable fuctio f, ad replace the Browia motio W with a Riema Liouville process R =R t ) t with Hurst parameter β, R t = β Z k t k + k= t t s) β /2 d W t, where Z k s are stadard ormal radom variables, W is a stadard Browia motio ad Z,Z,...,Z β, W are idepedet. See Sectio 4.2 i [2] for more iformatio o the Riema Liouville processes. Argumets similar to the oes give above yield that i this case as well 2.) is satisfied.

6 336 S. GUGUSHVILI AND P. SPREIJ 4. Discussio I the preset work we established posterior cosistecy for a statistical model obtaied from a simple liear stochastic differetial equatio. Geeral techiques for provig posterior cosistecy for a wide rage of statistical models are by ow well-developed. I the i.i.d. settig, broadly speakig, two mai approaches exist i the literature: a classical approach as epitomised e.g. by [, 8] we combie these two papers ito oe category, because they i some sese make use of assumptios of similar type, although their actual assertios are differet), ad a martigale approach developed more recetly i [23,24]. The first approach was exteded to the settig of idepedet o-idetically distributed observatios i [2], see i particular Theorem A. there. The secod approach was exteded to the case of discretely observed Markov processes i [7]. A geeral theorem for posterior cosistecy i [2] makes two requiremets: firstly, the prior must put sufficiet mass i arbitrarily small eighbourhoods of the true parameter i a appropriate topology), ad secodly, a sequece of sieves icreasig sequece of subsets of the parameter set) guarateeig existece of certai expoetially cosistet tests has to be exhibited; see page 56 i [2] for additioal details. Although i our settig the observatios X,i =,...,, are ot idepedet, the icremets X X ti, are, ad it appears coceivable that Theorem A. i [2] could be used to establish posterior cosistecy i our model as well. However, we opted for a differet approach, see the proof of our posterior cosistecy result, Theorem 2.6. A similarity shared by Theorem A. i [2] adtheorem2.6 is that both theorems require that the prior puts sufficiet mass i arbitrarily small eighbourhoods of the true parameter i appropriate topologies). A differece is that due to the special structure of our model we do ot eed to make ay referece to tests ad sieves, but ca establish posterior cosistecy by directly maipulatig the posterior; see the Proof of Theorem 2.6 for details. I this sese our approach to provig posterior cosistecy appears to be more direct ad more elemetary tha the oe that would employ Theorem A. i [2]. Neither do we make referece to etropy argumets as doe e.g. i []. As far as the martigale approach to posterior cosistecy for ergodic Markov processes is cocered, we ca be brief here: ergodicity is irrelevat i our settig ad i fact our special samplig scheme seems to make geeralisatio or modificatio of the argumets from [7], [23, 24] impossible. Next a brief remark o coditio 2.) o the prior Π is i order. Although it is formulated i terms of eighbourhoods i the L -orm, the assertio retured by Theorem 2.6 employs the topology iduced by the L 2 -orm. A discrepacy betwee orms used is however ot ucommo i posterior cosistecy results. See for istace []. 5. Proofs ProofofTheorem2.6. Let U σ be a arbitrary, but fixed eighbourhood of σ i the topology T ad let Ũσ,ε = {σ X : σ σ 2 <ε}. There exists ε>, such that Ũσ,ε U σ, ad hece U c σ Ũ c σ,ε. Fix such a ε. I order to prove the theorem, it thus suffices to show that as. Write ΠŨ σ c X,ε t,...,x t, ) Pσ 5.) L ΠŨ σ c Ũσ,ε X t,...,x t, )= c σ)πdσ) R,ε Ũσ X L = c σ)πdσ),ε σ)πdσ) X R, 5.2) σ)πdσ) where R σ) =L σ)/l σ ) deotes the likelihood ratio. We will separately boud the umerator ad deomiator o the right-had side of the last equality i 5.2) we will use the otatio D for the deomiator ad N for the umerator) ad the combie the bouds to establish 5.). As we will see, the left-had side of 5.) i fact decays expoetially fast to zero. Note that whe establishig posterior cosistecy, [] ad[24] also treat the umerator ad deomiator i the expressio for the posterior separately, but similarity of our approach to the oe i those papers largely eds here.

7 BAYESIAN ESTIMATION FOR A TIME-INHOMOGENEOUS BROWNIAN MOTION 337 Let S σ) = log R σ). The D = X exps σ))πdσ). Now S σ) = t log i, σu)du 2 2 X t i, X ti, ) 2 2 = T, σ)+t 2, σ), X X ti,)2 σ 2 u)du with obvious defiitios of T, σ) adt 2, σ). Let ε > be a costat with its value to be chose appropriately later o. Sice D V σ, ε R σ)πdσ), Lemmas A. ad A.2 from the Appedix ad formula A.) givethat with probability tedig to oe, D V σ, ε exp 4K ) κ 2 ε Πdσ) =exp 4K ) κ 2 ε ΠV σ, ε). By assumptio 2.), ΠV σ, ε) >. Fix a costat β =5K ε/κ 2. The for all large eough, exp 4K ) κ 2 ε ΠV σ, ε) e β. As a cosequece, with probability tedig to oe, D e β 5.3) as. This is our required lower boud for the deomiator D. Usig similar techiques, we will ext treat the umerator N. Firstly, ote that by elemetary argumets oe ca show that for a arbitrary fixed costat C>there exists aother costat c>, such that the iequality log + y) y cy 2, <y C holds oe ca take c =[2C +)] ). Hece t log i, σ 2u)du [σ 2 u) σ2 u)]du c [σ 2 u) σ2 u)]du for some costat c idepedet of σ X,iad. Therefore, after a simple, but legthy computatio employig Assumptio 2.5a), cf. the proof of Lemma A., T, σ) [σu) 2 σ 2 u)]du 2 c t i, [σu) 2 σ 2 2 u)]du 2 = σu) 2 σ 2 u) 2 σ 2 du c σu) 2 σ 2 u)) 2 ) u) 2 σ 4 du + O, u) where the remaider term is of order uiformly i σ X. Hece S σ) c σu) 2 σ 2 u)) 2 2 σ 4 du 5.4) u) + T 2, σ)+ 2 + O σ 2u) σ2 u) σ 2 du 5.5) u) ). 5.6) 2

8 338 S. GUGUSHVILI AND P. SPREIJ To boud from above the term o the right-had side of iequality 5.4), use the fact that c 2 σ 2 u) σ 2 u)) 2 σ 4 u) du 2κ2 c K 4 ε2 for σ Ũ σ c. Furthermore, by Lemma A.2, uiformly i σ Ũ c,ε σ,ε ad with probability tedig to oe as, the term 5.5) is smaller tha ay positive umber fixed beforehad. So is the term 5.6). Therefore, uiformly i σ Ũ σ c,ε ad with probability tedig to oe as, S σ) κ2 c K 4 ε2, say. Thus with probability tedig to oe as, ) N = exps σ))πdσ) exp κ2 c K 4 ε2. 5.7) Ũ c σ,ε This fiishes boudig from above the umerator N. We ow combie bouds 5.3) ad5.7) to coclude that with probability tedig to oe as, κ ΠŨ σ c 2 X c,ε t,...,x t, ) exp K 4 ε2 5K ) ) κ 2 ε. Pickig ε small eough, so that κ 2 c K 4 ε2 5K ε >, κ2 implies 5.) ad completes the proof of the theorem. Appedix A Lemma A.. Uder the same assumptios as i Theorem 2.6, fort, σ) as i the Proof of Theorem 2.6, all σ V σ, ε simulateously ad for large eough, T, σ) 2K ε/κ 2. Proof. The elemetary iequality gives that y log + y), y + y t log i, σu)du 2 > [σu) 2 σ 2 u)]du σ 2u)du Next, employig Assumptio 2.5a) ad c), by a simple computatio oe ca show that [σ 2u) σ2 u)]du σ 2u)du = σ2 ) σ 2 ) ) σ 2t + O, i,) where the remaider term is of order uiformly i σ X. Therefore, T, σ) 2 σ 2 ) σ 2 ) σ 2 ) + O )

9 BAYESIAN ESTIMATION FOR A TIME-INHOMOGENEOUS BROWNIAN MOTION 339 By aother simple computatio, σt 2 i, ) σ 2 ) σ 2) = σu) 2 σ 2 ) u) σ 2u) du + O, where the remaider term is of order uiformly i σ X. For σ V σ, ε we have uder Assumptio 2.5a) that σu) 2 σ 2 u) σ 2u) du 2K κ 2 ε. A.) This implies the statemet of the lemma. Lemma A.2. Deote Q σ) = T 2,σ)+ σu) 2 σ 2 u) 2 σ 2 du u), where T 2, σ) is defied i the Proof of Theorem 2.6. The uder the same assumptios as i Theorem 2.6, sup σ X Q σ) Pσ as. Furthermore, for ay fixed ε >, for all σ V σ, ε simultaeously, with probability tedig to oe as,t 2, σ) 2K ε/κ 2. Proof. The first statemet of the lemma will be derived from a applicatio of Theorem 8.4 from [2]. I particular, viewig Q as a process o X with bouded sample paths, we will show that it coverges i distributio to a zero process o X. The first statemet of the lemma will the be a cosequece of equivalece of covergece i distributio ad i probability for costat limits. Note that i order to circumvet possible o)-measurability issues, outer probability is employed i the formulatio of Theorem 8.4 from [2] see Sect. 8.2 i [2] for more iformatio o outer probability). Sice o such problems will arise i our settig, we ca istead directly work uder probability P σ. Ideed, the summads i T 2, σ) are of the form F i, σ)x X ti, ) 2,withF i, the obvious fuctioal of σ. Hece takig the supremum over σ does ot affect the measurability property of T 2, σ). I order to apply Theorem 8.4 from [2], we eed to verify its coditios. I our settig they reduce to the followig oes: firstly, margial vectors of Q must coverge i distributio to zero vectors, i.e. Q σ ),...,Q σ l )) Dσ,...,), }{{} l N. A.2) l Secodly, the tightess coditio must be satisfied: for arbitrary costats η>adξ>, oe must be able to fid a partitio of X ito fiitely may X,...,X l, such that ) lim sup P σ sup sup Q σ ) Q σ 2 ) ξ η. A.3) k l σ,σ 2 X k Deote F i, = σx tj,,j =,...,i) ad χ i, σ) = 2 X t i, X ti, ) 2 [σ 2u) σ2 u)]du σ 2u)du t i, Also let E σ be the expectatio operator with respect to measure P σ. Note that E σ χ i, σ) F i, )= 2 [σ 2u) σ2 u)]du = 2 σ 2u) σ2 u) σ 2 du + O u) ),

10 34 S. GUGUSHVILI AND P. SPREIJ where the remaider term is of order uiformly i σ X. Furthermore, Assumptio 2.5a) yields that E σ χ 2 i, σ) F i,) = where the order boud is uiform i σ X. It follows that [σ 2 u) σ2 u)]du E σ χ 2 i, σ) F i,). 2 ) = O 2, Lemma 9 i [4] the implies that Q σ) Pσ. This verifies A.2). We will ow check A.3). Fix ξ ad η i A.3). By a legthy, but simple computatio employig Assumptio 2.5a) ad the triagle iequality, Q σ ) Q σ 2 ) K κ 4 σ σ 2 X X ti,) 2 + K3 κ 4 σ σ 2. A.4) By the Arzelà Ascoli theorem, uder Assumptio 2.5a) the family X is totally bouded for the supremum metric. By defiitio this meas that for every ζ>there exists a fiite set X X, such that for ay σ X there is some σ X with σ σ <ζ/2. This ad the triagle iequality imply existece of a fiite partitio X,...,X l of X, such that sup sup σ σ 2 <ζ. A.5) k l σ,σ 2 X k Furthermore, by the defiitio of the quadratic variatio of the process X, X X ti,) 2 Pσ σ 2 u)du. A.6) Combiatio of A.4) A.6) yields A.3) forζ small eough, ad cosequetly the first statemet of the lemma too. The secod statemet of the lemma is a cosequece of the first oe, the fact that σ V σ, ε, a aalogue of iequality A.), σ 2u) σ2 u) 2 σ 2 du u) K κ 2 ε, ad of a simple rearragemet T 2, σ) = { T 2, σ)+ 2 This completes the proof of the lemma. σu) 2 σ 2 } u) σ 2 du u) 2 σu) 2 σ 2 u) σ 2 du. u) Ackowledgemets. The authors would like to thak the referees for a careful readig of the mauscript ad useful suggestios. Refereces [] A. Barro, M.J. Schervish ad L. Wasserma, The cosistecy of posterior distributios i oparametric problems. A. Statist ) [2] N. Choudhuri, S. Ghosal ad A. Roy, Bayesia estimatio of the spectral desity of a time series. J. Amer. Statist. Assoc ) 5 59.

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