Divergences Test Statistics for Discretely Observed Diffusion Processes

Di v er gences T est Statistics for Discretely Observ ed Dif fusion Processes Alessandro De Gre g orio Department of Statistic s, Probabil ity a nd Applied Statistic s P .le Aldo Moro 5, 0018 5 R ome-Italy alessa ndro.de gregor io@uniroma1. it Stefano M. Iacus Department of Economics, Business and Statistics V ia Conse rv atorio 7, 20 124 Mlan- Italy stef ano.iacus@unimi.it October 27, 2018 Abstract In this paper we prop ose the u se of φ -di ver gences as te st stati stics to v er- ify s imple hyp otheses about a one-dime nsiona l parametric dif fusion proce ss d X t = b ( X t , θ )d t + σ ( X t , θ )d W t , from discre te obs ervatio ns { X t i , i = 0 , . . . , n } with t i = i ∆ n , i = 0 , 1 , . . . , n , under the asympto tic scheme ∆ n → 0 , n ∆ n → ∞ and n ∆ 2 n → 0 . The cl ass of φ -d i ver gences is wide a nd includ es se veral specia l members like Kull back-Leibler , R ´ eny i, po wer and α -di ver gences. W e deriv e the asymptoti c distrib ution of the test statistics based on φ -di ver gences. T he limiting law tak es differ ent forms depend ing on the regularit y of φ . These con ver gence dif fer from the classica l results for indepe ndent and identicall y distrib uted rand om v ariables. Numerical analy- sis is use d to s ho w the small sa mple propert ies of the test stat istics in terms of estimated le vel and power o f the test . keyw ords: diffusion processes, empirical lev el , hypotheses testin g, φ -div er gences, α -diver gences 1 1 Introduction W e con sider t he problem of parametric t esting using φ -div ergences. Let X be a r .v . and f ( X , θ ) and g ( X , θ ) , θ ∈ Θ two families of probabi lity densities on the same m easurable sp ace. The φ -dive rgences are defined as D φ ( f , g ) = E θ φ ( f ( X ) /g ( X )) , where E θ is the expected v alue with respect to P θ , the true law of the observations. Because we focus the attention on the use of di vergences for hypotheses test ing, we will us e a simplified n otation: let θ and θ 0 two po ints in the interior of Θ and define the di verge n ce as D φ ( θ , θ 0 ) = E θ 0 φ  p ( X , θ ) p ( X , θ 0 )  (1.1) In equation (1.1) the density { p ( X , θ ) , θ ∈ Θ } is a same family of probability densities and φ ( · ) is a functio n with t he m inimal property that φ (1) = 0 . Examples of div ergences of the form D α ( θ , θ 0 ) = D φ α ( θ , θ 0 ) are the α -dive rgences, de fined by means of the following function φ α ( x ) = 4(1 − x 1+ α 2 ) 1 − α 2 , − 1 < α < 1 Note that D α ( θ 0 , θ ) = D − α ( θ , θ 0 ) . The class of α -div ergences has been widely studied in statisti cs (see, e.g., Csisz ´ ar , 1967 and Amari, 1985) and it is a family o f div ergences which includes sev eral members o f particular interest. For example, in the limit as α → − 1 , D − 1 ( θ , θ 0 ) reduces to the well-known Kullback-Leibler measure D − 1 ( θ , θ 0 ) = − E θ 0 log  p ( X , θ ) p ( X , θ 0 )  while as α → 0 , the Hellinger dis tance (see, e.g., Ber an, 19 77, Simpson, 19 89) emer g es D 0 ( θ , θ 0 ) = 1 2 E  p p ( X , θ ) − p p ( X , θ 0 )  2 As not iced in Chandra and T aniguchi (2006), the α -div ergence is also equ iv alent to the R ´ enyi’ s div ergence ( R ´ enyi, 1961) defi n ed, for α ∈ (0 , 1) , as R α ( θ , θ 0 ) = 1 1 − α log E θ 0  p ( X , θ ) p ( X , θ 0 )  α from which i s easy to see that in th e limi t as α → 1 , R α reduces to the Kullback- Leibler dive rgence. The transformation ψ ( R α ) = (exp { ( α − 1) R α − 1 } / (1 − α ) returns the po wer-di vergence st udied i n Cressie and Read (1984). Liese and V a- jda (1987) provide extensi ve s tudy of a modified version of R α and Morales et al. 2 (1997) consider diver gences with con vex φ ( · ) for independent and identically dis- tributed (i.i.d) observations; for e xampl e the po wer-di vergences D φ λ ( θ , θ 0 ) with φ λ ( x ) = x λ − λ ( x − 1) − 1 λ ( λ − 1) (1.2) and λ ∈ R − { 0 , 1 } . In thi s paper we focus our att ention on the φ -div ergences D φ ( θ , θ 0 ) , defined as in (1.1), for one-dimensional dif fusio n process { X t , t ∈ [0 , T ] } , solution of the following stochastic differential e qu ation dX t = b ( α , X t ) dt + σ ( β , X t ) dW t , X 0 = x 0 , (1.3) where W t is a Brownian motion, θ = ( α, β ) ∈ Θ α × Θ β = Θ , where Θ α and Θ β are respecti vely com pact con vex subset of R p and R q . W e assume that t he process X t is er g odic for e very θ with in var i ant law µ θ . Furthermore X t is observed at discrete t imes t i = i ∆ n , i = 0 , 1 , 2 , ..., n, where ∆ n is the length o f the steps. W e indicate t he observations with X n = { X t i } 0 6 i 6 n . T he asymp totic is ∆ n → 0 , n ∆ n → ∞ and n ∆ 2 n → 0 as n → ∞ . W e stu dy t he prop erties o f the estimated φ -div ergence D φ ( ˜ θ n ( X n ) , θ 0 ) , for discretely observed dif fusi on processes, defi ned as D φ ( ˜ θ n ( X n ) , θ 0 ) = φ f n ( X n , ˜ θ n ( X n )) f n ( X n , θ 0 ) ! where f n ( · , · ) is the approximated likelihood proposed by Dacunha-Castelle and Florens-Zmirou (1986) and ˜ θ n ( X n ) is any consistent, asymptotically normal and ef ficient estimator of θ . W e prove that, for φ ( · ) functions whi ch satisfying three diffe rent regularity cond itions, the stati stic D φ con ver ge weakly t o three different functions of the χ 2 p + q random variable. This result differs from the case of i.i.d. setting. Up to our knowledge the onl y result concerning the use of diver gences for discretely observed diffusion process is due to Riv as et al . (2005) where they consider the model of Brownian motion wit h drift d X t = a d t + b d W t where a and b are two scalars. In that case, th e exact likelihood of the observations is av ailable in e xp licit form and is the gaussian law . Con versely , in the general setup of thi s paper , t he likelihood of the process in (1.3) is known only for three particular stochastic differe n tial equations, namely the Ornstein-Uhl embeck dif- fusion, the geometric Brownian motion and the Cox-Ingersol l-Ross model. In all other cases, the likelihood has to be approximated. W e choos e the approxima- tion due to Dacunha-Castelle and Florens-Zmirou (1986) and, to derive a proper estimator , we use the local gaussian approximation proposed by Y oshida (199 2) 3 although our result h olds for any consistent and asym ptotically Gauss ian estima- tor . Th is approach has been suggested by the work on Akaike Information Criteria by Uchida and Y oshi da (2005). For continuous time observations from dif fus ion processes, V ajda (1990) con- sidered the model d X ( t ) = − b ( t ) X t d t + σ ( t )d W t ; K ¨ uchler and Sørensen (1997) and Morales et al. (2004) c o ntain se veral r esu lts on the likelihood ratio test statis- tics and R ´ enyi statistics for exponential family of diffusions. Explicit deriv a- tions of the R ´ enyi information on the in variant la w of ergodic diffusion processes hav e been presented i n De Gre gorio and Iacus (2007). For small d iffusion pro- cesses, with contin uous time observ ation s, information criteria ha ve been deriv ed in Uchida and Y oshi da (2004) using Mallia v in calculus. The problem of testing s tatistical hypotheses from gene ral diffusion p rocesses is s till a developing stream of research. Kutoyants (2004) and Dachian and Ku- toyants (2008 ) consider t he problem of testi ng statist ical hypotheses for er god ic diffusion models in continuous time; Kutoyants (1984) and Iacus and Kutoyants (2001) consider parametric and semiparametric hypotheses testing for small dif- fusion processes; Negri and Nishi yama (2007a, b) propose a non parametric test based on score marked empirical process for both continuous and discrete time observation from small diffusion processes further extended to the ergodic case in Masuda et al. (2008). Lee and W ee (2008) considered the parametric version o f the same test statisti cs for a simplified model. A ¨ ıt-Sahalia (1996, 2008), Giet and L ubrano (2008) and Chen et al. (2008) proposed test s based on the se veral di stances between parametric and nonpara- metric estimation of the in v ariant density of discretely observed er godi c diffusion processes. The present paper com plements the abov e references. The paper is organized as follows. Section 2 introduces not ation and regular- ity assumptions. Section 3 states the main result. Section 4 contains numerical experiments to test the s mall sample pe rformance of the proposed test statistics in terms of empirical le vel and empirical power under some alternati ves. T he proofs are contained in Section 5. 2 Assumpti ons on diffusion model W e consi der the family of one-dimensional diff u sion processes { X t , t ∈ [0 , T ] } , solution to dX t = b ( α , X t ) dt + σ ( β , X t ) dW t , X 0 = x 0 , (2.1) where W t is a Brownian motion. Let θ = ( α, β ) ∈ Θ α × Θ β = Θ , where Θ α and Θ β are respecti vely compact con vex subs et of R p and R q . Furthermore we assume that the dri ft function b : R × Θ α → R and the dif fusi on coefficient σ : R × Θ β → R are known apart from the parameters α and β . W e assume that 4 the p rocess X t is ergodic for every θ with in variant law µ θ . The process X t is observed at d iscrete times t i = i ∆ n , i = 0 , 1 , 2 , ..., n, where ∆ n is the length of the steps. W e indicate the observ ations with X n = { X t i } 0 6 i 6 n . The asymptotic is ∆ n → 0 , n ∆ n → ∞ and n ∆ 2 n → 0 as n → ∞ . In th e definiti on of the φ -di ver g ence (1.1) the likelihood of the process is need, but as noted in the Introduction, this is usu ally not know . There are seve ral ways to approximate th e likelihood of a discretely ob served d iffusion process (for a re view see, e.g., Chap. 3, Iacus, 2008). In this paper , we use the approximation proposed by Dacunha-Castelle and Florens-Zm irou (1986) althoug h ou r result hold true (with some adaptations of the p roofs) for ot her approximations, like, e.g. the one based on Hermit e polynomial expansion by A¨ ıt-Sahalia (2002). T o write it in explicit way , w e use the sam e setup as in Uchida and Y oshida (200 5). W e introduce the following functions s ( x, β ) = Z x 0 du σ ( β , u ) , B ( x, θ ) = b ( α, x ) σ ( β , x ) − σ ′ ( β , x ) 2 e B ( x, θ ) = B ( s − 1 ( β , x ) , θ ) , e h ( x, θ ) = e B 2 ( x, θ ) + e B ′ ( x, θ ) The following set of assumptio ns ensure the good behaviour of the approxim ated likelihood and the existence of a we ak s olution of (2.1) Assumption 2.1. [Re gulari ty on the pr ocess] i) Ther e exists a constant C s uch that | b ( α 0 , x ) − b ( α 0 , y ) | + | σ ( β 0 , x ) − σ ( β 0 , y ) | ≤ C | x − y | . ii) inf β ,x σ 2 ( β , x ) > 0 . iii) The pr ocess X is er godic for e very θ with in variant pr o bability measur e µ θ . All polynomial moments of µ θ ar e finite. iv) F or all m ≥ 0 and for all θ , sup t E | X t | m < ∞ . v) F or every θ , the coefficients b ( α, x ) and σ ( β , x ) ar e twice di ffer enti able with r espect to x and the derivatives ar e polynomi al gr owth in x , uniformly in θ . vi) The coefficients b ( α, x ) and σ ( β , x ) and all their partial derivatives r espect to x up to or der 2 ar e three times differ entiable re s pect to θ for all x in the state space. Al l d erivatives r espect to θ ar e pol ynomial growth in x , uniformly in θ . Assumption 2.2. [Re gulari ty for the appr oximation] 5 i) e h ( x, θ ) = O ( | x | 2 ) as x → ∞ . ii) inf x e h ( x, θ ) > −∞ fo r all θ . iii) sup θ sup x | e h 3 ( x, θ ) | ≤ M < ∞ . iv) Ther e e xists γ > 0 suc h t hat f or every θ and j = 1 , 2 , | e B j ( x, θ ) | = O ( | e B ( x, θ ) | γ ) a s | x | → ∞ . Assumption 2.3. [Identifiabilit y] The coefficients b ( α , x ) = b ( α 0 , x ) a nd σ ( β , x ) = σ ( β 0 , x ) for µ θ 0 a.s. a ll x then α = α 0 and β = β 0 . Under Assumption s 2.1 and 2.2 Dacunha-Castelle and Florens-Zmirou (1986) introduced the following approximati on of transiti on dens ity f of the process X from y to x at lag t f ( x, y , t, θ ) = 1 √ 2 π tσ ( y , β ) exp  − S 2 ( x, y , β ) 2 t + H ( x, y , θ ) + t ˜ g ( x, y , θ )  (2.2) and its logarithm l ( x, y , t, θ ) = − 1 2 log(2 π t ) − log σ ( y , β ) − S 2 ( x, y , β ) 2 t + H ( x, y , θ ) + t ˜ g ( x, y , θ ) where S ( x, y , β ) = Z y x du σ ( u , β ) H ( x, y , θ ) = Z y x  b ( α, u ) σ 2 ( β , u ) − 1 2 σ ′ ( β , u ) σ ( β , u )  du ˜ g ( x, y , θ ) = − 1 2  C ( x, θ ) + C ( y , θ ) + 1 3 B ( x, θ ) B ( y , θ )  C ( x, θ ) = 1 3 B 2 ( x, θ ) + 1 2 B ′ ( x, θ ) σ ( x, β ) The approximated likelihood and log-l ikelihood functions of the observa t ions X n become respectiv ely f n ( X n , θ ) = n Y i =1 f (∆ n , X t i − 1 , X t i , θ ) l n ( X n , θ ) = n X i =1 l ( ∆ n , X t i − 1 , X t i , θ ) 6 3 Construc tion of the test statistic s and r esults Consider the div ergence defined in (1.1) and let φ ( · ) be such that φ (1) = 0 and, when they exist, define C φ = φ ′ (1) and K φ = φ ′′ (1) . W e cons ider three different setup Assumption 3.1. C φ 6 = 0 is a finite constant depending only on φ and ind ependent of θ ; Assumption 3.2. C φ = 0 and K φ 6 = 0 is a finite constant depending only on φ and independent of θ ; Assumption 3.3. C φ 6 = 0 and K φ 6 = 0 ar e finite constants depending only on φ and independent of θ ; Remark 3.1. The above Assumptio ns ar e not so str ong. In fact, f or example the α -diver gences D φ α ( θ , θ 0 ) satisfy the Assumptions 3.1 and 3.3, while for the power- d iver gences D φ λ ( θ , θ 0 ) it’ s easy to verify that C φ = φ ′ (1) = 0 . Clearly , the quantity D φ ( θ , θ 0 ) measures the discrepancy between θ and the true value of the parameter θ 0 and is an ideal candi date to construct a t est statis tics. Let ˜ θ n ( X n ) be any consistent estimator of θ 0 and such that Γ − 1 / 2 ( ˜ θ n ( X n ) − θ 0 ) d → N (0 , I ( θ 0 ) − 1 ) (3.1) where I ( θ 0 ) is the posit iv e definite and in vertible Fisher i nformation matrix at θ 0 equal to I ( θ 0 ) =  ( I k j b ( θ 0 )) k ,j =1 ,...,p 0 0 ( I k j σ ( θ 0 )) k ,j =1 ,...,q  where I k j b ( θ 0 ) = Z 1 σ 2 ( β 0 , x ) ∂ b ( α 0 , x ) ∂ α k ∂ b ( α 0 , x ) ∂ α j µ θ 0 ( dx ) I k j σ ( θ 0 ) = 2 Z 1 σ 2 ( β 0 , x ) ∂ σ ( β 0 , x ) ∂ β k ∂ σ ( β 0 , x ) ∂ β j µ θ 0 ( dx ) W e indicate with Γ the ( p + q ) × ( p + q ) matrix Γ =  1 n ∆ n I p 0 0 1 n I q  and I p is th e p × p identity m atrix. Using the app roximated l ikelihood f n ( X n , θ ) and f n ( X n , θ 0 ) , the φ -diver gence in (1.1) becomes D φ ( θ , θ 0 ) = E θ 0 φ  f n ( X n , θ ) f n ( X n , θ 0 )  (3.2) 7 T o construct a test statis tics we replace θ by the estimator ˜ θ n ( X n ) and, having only o ne singl e observation of X n , i.e. only one observed trajectory , we estimate (3.2) with D φ ( ˜ θ n ( X n ) , θ 0 ) = φ f n ( X n , ˜ θ n ( X n )) f n ( X n , θ 0 ) ! (3.3) Please notice that, con versely to the i.i.d. case, there is no integral i n the definition of (3.3). W e wi ll discuss thi s point after the presentation of the T heorem 3.1. The proposed test for testing H 0 : θ = θ 0 versus H 1 : θ 6 = θ 0 is realized as D φ ( ˜ θ n ( X n ) , θ 0 ) = 0 versus D φ ( ˜ θ n ( X n ) , θ 0 ) 6 = 0 . Theor em 3.1. Under H 0 : θ = θ 0 , Assumptions 2.1-2.3, con ver gence (3.1) , we have that i) if fu nction φ ( · ) sa tisfies Assumptio n 3.1, then D φ ( ˜ θ n ( X n ) , θ 0 ) d → C φ χ 2 p + q (3.4) ii) if f unction φ ( · ) satisfies Assumpti on 3.2, then D φ ( ˜ θ n ( X n ) , θ 0 ) d → K φ 2 Z p + q (3.5) wher e p Z p + q = χ 2 p + q . iii) if functi on φ ( · ) sat isfies Assumption 3.3, then D φ ( ˜ θ n ( X n ) , θ 0 ) d → 1 2 ( C φ χ 2 p + q + ( C φ + K φ ) Z p + q ) (3.6) Remark 3.2 . It’ s clear that for C φ = 0 f r om (3.6) we immediately reobtain the con ver gence r esult (3.5) . Remark 3.3. If we cons ider the limits as α → − 1 for φ α ( x ) of the α -diver gences, i.e. we consider the K ullback-Leibler diver gence, we have φ ( x ) = lim α →− 1 φ α ( x ) = − log ( x ) for which C φ = − 1 and K φ = 1 . In that case, (3.6) r educes to the standar d r esult for the likelihood ratio test statist ics. The con ver gence in Theorem 3 .1 may appear s omewhat strange if o ne thin ks about t he usual results on φ -dive rgences for i .i.d. o bservations. T he main differ - ence in diffusion m odels, is that our esti mate of the diver gence h as not the us ual 8 form of an expected value, i.e. it est imates the expected value with one observa- tion only . This is why , in the i .i.d case, the first term in the T aylor expansion of D φ vanishes being the expected v alue of the score functi on, wh ile in our case it remains only th e score function whi ch, as usual, con verges to a Gaussian random var i able. For the same reason, in the second term of the T aylor expansion, in the i.i.d. case appears the expected v alu e of the second order deri vativ e which con- ver ges to the Fisher information and, in our case, we ha ve not the expected v alue, hence the con ver gence to th e square of the χ 2 emer g es. If one wants to emulate the standard results for the i.i .d. case, it is still p ossible to work on the i n variant density of the diffusion process. In th at case, th e φ - div ergence takes t he us ual form of th e i.i .d. case because t he i n variant d ensity hav e the explicit form. Indeed, let s ( x, θ ) = exp  − 2 Z x ˜ x b ( y , θ ) σ 2 ( y , θ ) d y  , m ( x, θ ) = 1 σ 2 ( x, θ ) s ( x, θ ) be the s cale and speed fun ctions of the diffusion, with ˜ x some value in the state space of the dif fusio n p rocess. Let M = R m ( x, θ ) d x , then π ( x, θ ) = m ( x, θ ) / M is the in var i ant density of the diffusion process. In this case, it is possible t o define the φ -diver gence as D φ ( ˜ θ n , θ 0 ) = Z φ π ( x, ˜ θ n ) π ( x, θ 0 ) ! π ( x, θ 0 )d x and the standard results follows. Remark 3.4. In our applicatio n, to derive and estimator , we consider further the local gaussi an appr oximation of the same transiti on d ensity (see, Y oshida, 1992) g n ( X n , θ ) = n X i =1 g n (∆ n , X t i − 1 , X t i , θ ) (3.7) wher e g ( t, x, y , θ ) = − 1 2 log(2 π t ) − log σ ( β , x ) − [ y − x − tb ( α, x )] 2 2 tσ 2 ( β , x ) The approximate maximum likelihood estimator ˆ θ n ( X n ) based on (3.7) is then defined as ˆ θ n ( X n ) = a r g sup θ g n ( X n , θ ) (3.8) Under the condition n ∆ 2 n → 0 (see Theor em 1 in Ke s sler , 1997) the estima- tor ˆ θ n ( X n ) in (3.8) satisfies (3.1) . Hence, the r esult of Theor em 3.1 applies for ˜ θ n ( X n ) = ˆ θ n ( X n ) . 9 Remark 3.5. I n Theor em 3.1 ther e i s no need to i mpose C φ = 0 and K φ = 1 as, e.g. in Morales et al. (1997). Of course, in our case the constants C φ and K φ enter in the asympt otic distribution of the test statisti cs. The con ver gence r esul t is a lso i nter esting because, contrary to the i.i.d case, the rate of con ver gence of the estimat ors of θ for th e drift and diffusion coef ficients ar e differ ent and a r e r espectively equal to √ n ∆ n and √ n . Remark 3.6. As r emarked in Uchida and Y oshida (2001), it is al ways bett er to d e- rive appr oximate ML estimators and the test statistics on differ ent approximations of the true likelihood to avoid cir cularities. 4 Numeric al analysis Although asym ptotic properties h a ve been obtain ed, what really matters i n appli- cation is the behaviour of the test stati stics under fine sample setup. W e st udy the empirical performance of the test for small samples in terms o f le vel of the t est and power u nder so me alt ernativ es. In the analy sis we consider the estimator (3.8) and the following quantities • estimated α -diver gences D α ( ˆ θ n ( X n ) , θ 0 ) = φ α f n ( X n , ˆ θ n ( X n )) f n ( X n , θ 0 ) ! with φ α ( x ) = 4(1 − x 1+ α 2 ) / (1 − α 2 ) , wi th C α = 2 α − 1 and K φ = 1 . W e consider α ∈ {− 0 . 99 , − 0 . 90 , − 0 . 75 , − 0 . 50 , − 0 . 25 , − 0 . 10 } ; • estimated power -dive rgences D λ ( ˆ θ n ( X n ) , θ 0 ) = φ λ f n ( X n , ˆ θ n ( X n )) f n ( X n , θ 0 ) ! with φ λ ( x ) = ( x λ +1 − x − λ ( x − 1)) / ( λ ( λ + 1 ) ) , with C λ = 0 , K λ = 1 . W e consid er λ ∈ { − 0 . 99 , − 1 . 20 , − 1 . 50 , − 1 . 75 , − 2 . 00 , − 2 . 50 } ; • likelihood ratio statistic D log ( ˆ θ n ( X n ) , θ 0 ) = − log f n ( X n , ˆ θ n ( X n )) f n ( X n , θ 0 ) ! 10 For D α and D λ , t he threshold of the rejection region of the test are calculated using formula (3.6) as the empirical quantiles of (3.6) of 100000 simulations of the ran- dom va riable χ 2 p + q . For D log is again used formula (3.6) b ut exact quantiles of the random variable χ 2 p + q are used. Because the interest i s in testing D φ = 0 against D φ 6 = 0 , whene ver f n ( X n , ˜ θ n ( X n )) > f n ( X n , θ 0 ) we exchange t he numerator and the denominato r to a void negati ve signs in the test statis tics. Usually , this is not going to happen if φ is con vex and φ ′ (1) = 0 (see, e.g. Morales et al. , 1997). W e ev aluate the emp irical le vel of the test calculated as the number of times the test rejects the null hypothesi s under the true model, i.e. ˆ α n = 1 M M X i =1 1 { D φ >c α } where 1 A is the indicator function of set A , M = 1 0000 is the number of simu- lations and c α is t he (1 − α )% q uantile of the proper distribution. Similarly we calculate the power of the test under alternative m odels as ˆ β n = 1 M M X i =1 1 { D φ >c α } In our experiments we consider the two families of stochastic processes borrowed from finance • the V asicek (V AS) model d X t = κ ( α − X t )d t + σ X t d W t where, in finance, σ is interpreted as v o latility , α is t he long-run equili b- rium value of the proce s s and κ i s the speed of reversion. Let ( κ 0 , α 0 , σ 2 0 ) = (0 . 85837 , 0 . 089102 , 0 . 0 021854) , we con sider three d iffe rent sets of hypothe- ses for the parameters model θ = ( κ, α , σ 2 ) V AS 0 ( κ 0 , α 0 , σ 2 0 ) V AS 1 (4 · κ 0 , α 0 , 4 · σ 2 0 ) V AS 2 ( 1 4 κ 0 , α 0 , 1 4 · σ 2 0 ) The int eresting facts are that V AS 0 , V AS 1 and V AS 2 hav e all t he same s ta- tionary distributions N ( α 0 , σ 2 0 / (2 κ 0 )) , a Gaussian transition density N  α 0 + ( x 0 − α 0 ) e − κt , σ 2 0 (1 − e − 2 κt ) 2 κ 0  11 and cov ariance function giv en by Co v ( X s , X t ) = σ 2 0 2 κ 0 e − κ ( s + t )  e − 2 κ ( s ∧ t ) − 1  and both show a strong dependency of th e cov ariance as a function of κ , which makes this model interesting in comparison with the i.i.d. setting; • the Cox-Ingersoll-Ross (CIR) model d X t = κ ( α − X t )d t + σ p X t d W t Let ( κ 0 , α 0 , σ 2 0 ) = ( 0 . 89218 , 0 . 0 9 045 , 0 . 032 742) , we consider different sets of hypotheses for the parameters model θ = ( κ, α , σ 2 ) CIR 0 ( κ 0 , α 0 , σ 2 0 ) CIR 1 ( 1 2 · κ 0 , α 0 , 1 2 · σ 2 0 ) CIR 2 ( 1 4 · κ 0 , α 0 , 1 4 · σ 2 0 ) This model has a transition densit y of χ 2 -type, hence local gaussian approx- imation is less likely to hold for non negligible values of ∆ n . The parameters of the above models, hav e b een chosen according to Pritsker (1998) and Chen et al. (200 8), in particular V AS 0 corresponds to th e m odel esti- mated by A ¨ ıt-Sahalia (1996) for real interest rates data. W e study the lev el and the po wer of the three family of test statis tics for dif fer- ent values of ∆ n ∈ { 0 . 1 , 0 . 0 01 } and n ∈ { 5 0 , 100 , 500 } . For the same trajectory , hence we simulate 1000 observations and we extract only that l ast n observations. Disregarding the first part of the trajectory ensures t hat the process i s in the sta- tionary state. The results of t hese simulations are reported in the T ables 1-9. W e point out that in the T ables 2, 4, 7 and 9, in the column “model ( α , n )” the α corresponds to the true leve l of t he test u sed to calculate c α . The other α ’ s in th e first ro w of the tables correspond to the α in φ α -div ergences. Summary of the analysis for the V asicek model It turns out that α -diver gences are not very goo d in t erms of est imated l e vel of the test, b ut thei r power function beha ves as expected. It also emerges that for λ = − 0 . 99 , the power div ergence cannot identify as wrong model V AS 1 for small sample size n = 50 and ∆ n = 0 . 001 (T able 3, ro w 2), alt hough this is not the case for the po wer-di vergences and the likelihood ratio test (T ables 2 and 1, row 2). 12 In general power diver gences for λ in {− 0 . 99 , − 1 . 2 0 , − 1 . 50 , − 1 . 75 , − 2 . 00 } hav e alw ays ve ry small estimated level and high power under the s elected a l terna- tiv es. The α -diver gences, do not behav e very good and, t he way they are defined, only approximate the likelihood ratio for α = − 0 . 99 . The power dive rgences are, on ave rage, b etter t han the l ikelihood ratio test in terms of both empirical lev el ˆ α and power ˆ β under the selected alternatives. Summary of the analysis f o r the CIR model The same a verage considerations apply to the case of CIR model. T he diff erence is that, for small sampl e size, all test s tatistics ha ve lo w power under the alternati ve CIR 1 while CIR 2 doesn’t present particular problems. 5 Pr o ofs The following important Lemmas are useful to prove the Theorem 3.1. Lemma 5.1 (Kessler , 1997) . Under the assumptio ns 2.1-2.3, as n ∆ 2 n → 0 the following hold true Γ 1 2 ∇ θ g n ( X n , θ 0 ) p → N (0 , I ( θ 0 )) (5.1) Lemma 5.2 (Uchida and Y oshid a, 2005) . Under the assumpt ions 2.1-2.3, as n ∆ 2 n → 0 the followin g hold true Γ 1 2 ∇ θ l n ( X n , θ 0 ) = Γ 1 2 ∇ θ g n ( X n , θ 0 ) + o p (1) (5.2) Lemma 5.3 (Uchida and Y oshid a, 2005) . Under the assumpt ions 2.1-2.3, as n ∆ 2 n → 0 the followin g hold true Γ 1 2 ∇ 2 θ l n ( X n , θ 0 )Γ 1 2 p → −I ( θ 0 ) (5.3) Pr oof of Theor em 3.1. W e st art by applying del ta metho d. W e denote the gradient vector by ∇ θ = [ ∂ /∂ θ i ] , i = 1 , . . . , p + q and s imilarly the Hessian matrix by ∇ 2 θ = [ ∂ 2 /∂ θ i ∂ θ j ] , i, j = 1 , . . . , p + q . i ) W e can write t hat D φ ( ˜ θ n ( X n ) , θ 0 ) = D φ ( θ 0 , θ 0 ) + [ ∇ θ D φ ( θ 0 , θ 0 )] T ( ˜ θ n ( X n ) − θ 0 ) + o p (1) = [ ∇ θ D φ ( θ 0 , θ 0 )] T ( ˜ θ n ( X n ) − θ 0 ) + o p (1) because D φ ( θ 0 , θ 0 ) = 0 . Noting that for k = 1 , ..., p + q ∂ ∂ θ k  φ  f n ( · , θ ) f n ( · , θ 0 )  = 1 f n ( · , θ 0 ) φ ′  f n ( · , θ ) f n ( · , θ 0 )  ∂ f n ( · , θ ) ∂ θ k 13 by Assumpt ion 3.1 follows that ∇ θ D φ ( θ 0 , θ 0 ) = C φ ∇ θ l n ( X n , θ ) | θ = θ 0 = C φ ∇ θ l n ( X n , θ 0 ) and therefore D φ ( ˜ θ n ( X n ) , θ 0 ) = C φ h Γ 1 2 ∇ θ l n ( X n , θ 0 ) i T Γ − 1 2 ( ˜ θ n ( X n ) − θ 0 ) + o p (1) (5.4) From (5.4) by means of Lemm a 5.2-5.1 and Slutsky’ s Theorem im mediately fol- lows D φ ( ˜ θ n ( X n ) , θ 0 ) d → C φ χ 2 p + q ii ) Since for k , j = 1 , ..., p + q ∂ 2 ∂ θ k ∂ θ j  φ  f n ( · , θ ) f n ( · , θ 0 )  = 1 f 2 n ( · , θ 0 ) φ ′′  f n ( · , θ ) f n ( · , θ 0 )  ∂ f n ( · , θ ) ∂ θ k ∂ f n ( · , θ ) ∂ θ j + 1 f n ( · , θ 0 ) φ ′  f n ( · , θ ) f n ( · , θ 0 )  ∂ 2 f n ( · , θ ) ∂ θ k ∂ θ j follows that D φ ( ˜ θ n ( X n ) , θ 0 ) = 1 2 [Γ − 1 / 2 ( ˜ θ n ( X n ) − θ 0 )] T Γ 1 / 2 ∇ 2 θ D φ ( θ 0 , θ 0 )Γ 1 / 2 × Γ − 1 / 2 ( ˜ θ n ( X n ) − θ 0 ) + o p (1) = K φ 2 [Γ − 1 / 2 ( ˜ θ n ( X n ) − θ 0 )] T Γ 1 / 2 ∇ θ l n ( X n , θ 0 )[Γ 1 / 2 ∇ θ l n ( X n , θ 0 )] T × Γ − 1 / 2 ( ˜ θ n ( X n ) − θ 0 ) + o p (1) From (5.4) by means of Lemm a 5.2-5.1 and Slutsky’ s Theorem im mediately fol- lows D φ ( ˜ θ n ( X n ) , θ 0 ) d → K φ 2 Z p + q It’ s easy to verify that the density function of the r .v . Z p + q is equal to f Z p + q ( z ) = (1 / 2) p + q 2 Γ  p + q 2  √ z p + q 2 − 1 e − √ z / 2 1 2 √ z , z > 0 (5.5) iii ) By previous c o nsiderations we hav e that D φ ( ˜ θ n ( X n ) , θ 0 ) = [ ∇ θ D φ ( θ 0 , θ 0 )] T ( ˜ θ n ( X n ) − θ 0 ) + 1 2 [Γ − 1 / 2 ( ˜ θ n ( X n ) − θ 0 )] T Γ 1 / 2 ∇ 2 θ D φ ( θ 0 , θ 0 )Γ 1 / 2 × Γ − 1 / 2 ( ˜ θ n ( X n ) − θ 0 ) + o p (1) (5.6) 14 where ∇ 2 θ D φ ( θ 0 , θ 0 ) = K φ ∇ θ l n ( X n , θ 0 )[ ∇ θ l n ( X n , θ 0 )] T + C φ 1 f ( X n , θ 0 ) ∇ 2 θ f ( X n , θ 0 ) = ( K φ + C φ ) ∇ θ l n ( X n , θ 0 )[ ∇ θ l n ( X n , θ 0 )] T + C φ ∇ 2 θ l n ( X n , θ 0 ) (5.7) Plugging in (5.6) the quantity (5.7) we deri ve, applying again Lemma 5.1-5.3, the following result D φ ( ˜ θ n ( X n ) , θ 0 ) d → 1 2  C φ χ 2 p + q + ( C φ + K φ ) Z p + q  Conclusio ns It seems t hat, as in the i.i.d. case, also for di scretely observed diff u sion processes the φ -diver gences may comp ete or i mprove the performance of t he standard li ke- lihood rati o s tatistics. In particular , the p ower div ergences are in general quite good i n terms of est imated le vel and po wer of the test e ven for moderate s ample sizes (e.g. n ≥ 100 in our simu lations). The package sde for the R statistical en vironment (R De velopment Core T eam, 2008) and freely a vailable at http ://cran.R- Proj ect.org con- tains the function sdeDiv which implem ents the φ -di vergence test statistics. References [1] A ¨ ıt-Sahalia, Y . (1996) T esting continuous-tim e model s of the spot interest rate, Rev . F ina ncial Stud. , 70 (2), 385-426. [2] A ¨ ıt-Sahalia, Y . (2002) Maximum-likelihood estimati on for di scretely- sampled dif fusi ons: a closed-form approximation apporach, Econo metrica , 70 , 223-262. [3] A ¨ ıt-Sahalia, Y ., Fan, J., Peng, H. (2006) Nonparametric transi- tion based t ests for jum p di?usi ons. W orking p aper . 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(2005) AIC for er godic dif fus ion processes from discrete observ ations, pre p rint MHF 2005-12, march 2005, F a culty of Math- ematics, K yushu University , Fukuoka, Japan. [34] Y oshida, N. (1992) Estimati on for dif fus ion processes from discrete o bser- vation, J . Multivar . Anal. , 41 (2), 220–242. 18 model (n) α = 0 . 0 1 α = 0 . 05 V AS 0 (50) 0.01 0.04 V AS 1 (50) 1.00 1.00 V AS 2 (50) 1.00 1.00 V AS 0 (100) 0.01 0.04 V AS 1 (100) 1.00 1.00 V AS 2 (100) 1.00 1.00 V AS 0 (500) 0.01 0.07 V AS 1 (500) 1.00 1.00 V AS 2 (500) 1.00 1.00 model (n) α = 0 . 0 1 α = 0 . 05 V AS 0 (50) 0.01 0.04 V AS 1 (50) 1.00 1.00 V AS 2 (50) 1.00 1.00 V AS 0 (100) 0.01 0.04 V AS 1 (100) 1.00 1.00 V AS 2 (100) 1.00 1.00 V AS 0 (500) 0.00 0.02 V AS 1 (500) 1.00 1.00 V AS 2 (500) 1.00 1.00 T able 1: Numbers represent probability of rejection und er the true generating model, with c α calculated under H 0 . Therefore, the values are ˆ α under model “ 0 ” and ˆ β ot herwise. Estimates calculated on 10000 experiments. Likelihood ratio, for ∆ n = 0 . 00 1 (up) and ∆ n = 0 . 1 (bott om). 19 model ( α , n ) α = − 0 . 99 α = − 0 . 90 α = − 0 . 75 α = − 0 . 50 α = − 0 . 25 α = − 0 . 10 V AS 0 (0.01, 50) 0.01 0.10 0.39 0.62 0.73 0 .77 V AS 1 (0.01, 50) 1.00 1.00 1.00 1.00 1.00 1 .00 V AS 2 (0.01, 50) 1.00 1.00 1.00 1.00 1.00 1 .00 V AS 0 (0.05, 50) 0.04 0.12 0.39 0.62 0.73 0 .77 V AS 1 (0.05, 50) 1.00 1.00 1.00 1.00 1.00 1 .00 V AS 2 (0.05, 50) 1.00 1.00 1.00 1.00 1.00 1 .00 V AS 0 (0.01, 100) 0.01 0.10 0.39 0.63 0.74 0.78 V AS 1 (0.01, 100) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.01, 100) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.05, 100) 0.04 0.11 0.40 0.63 0.74 0.78 V AS 1 (0.05, 100) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.05, 100) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.01, 500) 0.02 0.18 0.61 0.83 0.90 0.92 V AS 1 (0.01, 500) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.01, 500) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.05, 500) 0.07 0.20 0.61 0.83 0.90 0.92 V AS 1 (0.05, 500) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.05, 500) 1.00 1.00 1.00 1.00 1.00 1.00 T able 2: Numb ers represent probabilit y of rejectio n under the true generating model, with c α calculated under H 0 . Therefore, the v alu es are ˆ α un der model “ 0 ” and ˆ β ot herwise. Est imates calculated on 10000 experiments. α -diver gences, for ∆ n = 0 . 001 . 20 model ( α , n ) λ = − 0 . 99 λ = − 1 . 20 λ = − 1 . 50 λ = − 1 . 75 λ = − 2 . 00 λ = − 2 . 50 V AS 0 (0.01, 50) 0.00 0.00 0.00 0.01 0.02 0.04 V AS 1 (0.01, 50) 0.00 0.99 1.00 1.00 1.00 1.00 V AS 2 (0.01, 50) 0.40 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.05, 50) 0.00 0.00 0.00 0.01 0.03 0.06 V AS 1 (0.05, 50) 0.67 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.05, 50) 0.99 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.01, 100) 0.00 0.00 0.00 0.01 0.02 0.04 V AS 1 (0.01, 100) 0 .23 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.01, 100) 0 .88 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.05, 100) 0 .00 0.00 0.00 0.01 0.03 0.06 V AS 1 (0.05, 100) 1 .00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.05, 100) 1 .00 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.01, 500) 0 .00 0.00 0.00 0.01 0.03 0.08 V AS 1 (0.01, 500) 1 .00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.01, 500) 1 .00 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.05, 500) 0 .00 0.00 0.01 0.03 0.06 0.12 V AS 1 (0.05, 500) 1 .00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.05, 500) 1 .00 1.00 1.00 1.00 1.00 1.00 T able 3: Numb ers represent probabilit y of rejectio n under the true generating model, with c α calculated under H 0 . Therefore, the values are ˆ α under model “ 0 ” and ˆ β o therwise. Estimates calculated on 10000 experiments. Power -diver gences for ∆ n = 0 . 00 1 21 model ( α , n ) α = − 0 . 99 α = − 0 . 90 α = − 0 . 75 α = − 0 . 50 α = − 0 . 25 α = − 0 . 10 V AS 0 (0.01, 50) 0.01 0.15 0.55 0.78 0.86 0 .88 V AS 1 (0.01, 50) 1.00 1.00 1.00 1.00 1.00 1 .00 V AS 2 (0.01, 50) 1.00 1.00 1.00 1.00 1.00 1 .00 V AS 0 (0.05, 50) 0.05 0.17 0.55 0.78 0.86 0 .88 V AS 1 (0.05, 50) 1.00 1.00 1.00 1.00 1.00 1 .00 V AS 2 (0.05, 50) 1.00 1.00 1.00 1.00 1.00 1 .00 V AS 0 (0.01, 100) 0.01 0.13 0.48 0.71 0.80 0.83 V AS 1 (0.01, 100) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.01, 100) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.05, 100) 0.04 0.15 0.48 0.71 0.80 0.83 V AS 1 (0.05, 100) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.05, 100) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.01, 500) 0.00 0.06 0.25 0.54 0.69 0.74 V AS 1 (0.01, 500) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.01, 500) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.05, 500) 0.02 0.07 0.25 0.54 0.69 0.74 V AS 1 (0.05, 500) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.05, 500) 1.00 1.00 1.00 1.00 1.00 1.00 T able 4: Numb ers represent probabilit y of rejectio n under the true generating model, with c α calculated under H 0 . Therefore, the values are ˆ α under model “ 0 ” and ˆ β otherwise. Estimates calculated on 100 00 experiments. α -diver gences, for ∆ n = 0 . 1 22 model ( α , n ) λ = − 0 . 99 λ = − 1 . 20 λ = − 1 . 50 λ = − 1 . 75 λ = − 2 . 00 λ = − 2 . 50 V AS 0 (0.01, 50) 0.00 0.00 0.00 0.01 0.02 0.05 V AS 1 (0.01, 50) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.01, 50) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.05, 50) 0.00 0.00 0.00 0.02 0.03 0.09 V AS 1 (0.05, 50) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.05, 50) 1.00 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.01, 100) 0 .00 0.00 0.00 0.00 0.01 0.05 V AS 1 (0.01, 100) 1 .00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.01, 100) 1 .00 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.05, 100) 0 .00 0.00 0.00 0.01 0.03 0.08 V AS 1 (0.05, 100) 1 .00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.05, 100) 1 .00 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.01, 500) 0 .00 0.00 0.00 0.00 0.01 0.02 V AS 1 (0.01, 500) 1 .00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.01, 500) 1 .00 1.00 1.00 1.00 1.00 1.00 V AS 0 (0.05, 500) 0 .00 0.00 0.00 0.01 0.01 0.04 V AS 1 (0.05, 500) 1 .00 1.00 1.00 1.00 1.00 1.00 V AS 2 (0.05, 500) 1 .00 1.00 1.00 1.00 1.00 1.00 T able 5: Numb ers represent probabilit y of rejectio n under the true generating model, with c α calculated under H 0 . Therefore, the values are ˆ α under model “ 0 ” and ˆ β o therwise. Estimates calculated on 10000 experiments. Power -diver gences for ∆ n = 0 . 1 23 model (n) α = 0 . 01 α = 0 . 05 CIR 0 (50) 0.02 0.11 CIR 1 (50) 0.59 0.84 CIR 2 (50) 1.00 1.00 CIR 0 (100) 0.03 0.11 CIR 1 (100) 0.96 0.99 CIR 2 (100) 1.00 1.00 CIR 0 (500) 0.02 0.09 CIR 1 (500) 1.00 1.00 CIR 2 (500) 1.00 1.00 model (n) α = 0 . 01 α = 0 . 05 CIR 0 (50) 0.01 0.04 CIR 1 (50) 0.78 0.93 CIR 2 (50) 1.00 1.00 CIR 0 (100) 0.01 0.04 CIR 1 (100) 0.99 1.00 CIR 2 (100) 1.00 1.00 CIR 0 (500) 0.00 0.02 CIR 1 (500) 1.00 1.00 CIR 2 (500) 1.00 1.00 T able 6: Nu mbers represent probabili ty of rejection under the true model, with rejection region calculated under H 0 . Likelihood ratio, fo r ∆ n = 0 . 001 (up) and ∆ n = 0 . 1 (bott om). 24 model ( α , n ) α = − 0 . 99 α = − 0 . 90 α = − 0 . 75 α = − 0 . 50 α = − 0 . 25 α = − 0 . 10 CIR 0 (0.01, 50) 0.03 0.28 0.69 0.83 0.89 0.90 CIR 1 (0.01, 50) 0.63 0.95 1.00 1.00 1.00 1.00 CIR 2 (0.01, 50) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.05, 50) 0.12 0.31 0.69 0.83 0.89 0.90 CIR 1 (0.05, 50) 0.86 0.96 1.00 1.00 1.00 1.00 CIR 2 (0.05, 50) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.01, 100) 0.03 0.28 0.69 0.85 0.89 0.91 CIR 1 (0.01, 100) 0.97 1.00 1.00 1.00 1.00 1.00 CIR 2 (0.01, 100) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.05, 100) 0.12 0.31 0.69 0.85 0.89 0.91 CIR 1 (0.05, 100) 0.99 1.00 1.00 1.00 1.00 1.00 CIR 2 (0.05, 100) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.01, 500) 0.03 0.22 0.59 0.79 0.86 0.88 CIR 1 (0.01, 500) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 2 (0.01, 500) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.05, 500) 0.09 0.24 0.59 0.79 0.86 0.89 CIR 1 (0.05, 500) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 2 (0.05, 500) 1.00 1.00 1.00 1.00 1.00 1.00 T able 7: Nu mbers represent probabi lity of rejection under the true model, with rejection region calculated under H 0 . α - div ergences, for ∆ n = 0 . 00 1 25 model ( α , n ) λ = − 0 . 99 λ = − 1 . 20 λ = − 1 . 50 λ = − 1 . 75 λ = − 2 . 00 λ = − 2 . 50 CIR 0 (0.01, 50) 0.00 0.00 0.00 0.02 0.05 0.13 CIR 1 (0.01, 50) 0.00 0.01 0.28 0.55 0.71 0.87 CIR 2 (0.01, 50) 0.00 0.81 1.00 1.00 1.00 1.00 CIR 0 (0.05, 50) 0.00 0.00 0.01 0.04 0.09 0.19 CIR 1 (0.05, 50) 0.00 0.09 0.48 0.70 0.81 0.92 CIR 2 (0.05, 50) 0.00 0.97 1.00 1.00 1.00 1.00 CIR 0 (0.01, 100) 0 .00 0.00 0.00 0.02 0.05 0.14 CIR 1 (0.01, 100) 0 .00 0.21 0.83 0.95 0.98 0.99 CIR 2 (0.01, 100) 0 .00 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.05, 100) 0 .00 0.00 0.01 0.05 0.09 0.20 CIR 1 (0.05, 100) 0 .00 0.53 0.93 0.98 0.99 1.00 CIR 2 (0.05, 100) 0 .24 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.01, 500) 0 .00 0.00 0.00 0.02 0.04 0.10 CIR 1 (0.01, 500) 0 .00 1.00 1.00 1.00 1.00 1.00 CIR 2 (0.01, 500) 1 .00 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.05, 500) 0 .00 0.00 0.01 0.04 0.07 0.15 CIR 1 (0.05, 500) 0 .95 1.00 1.00 1.00 1.00 1.00 CIR 2 (0.05, 500) 1 .00 1.00 1.00 1.00 1.00 1.00 T able 8: Numbers repre s ent probability o f rejection under t he true model, with rejection region calculated under H 0 . Power - div ergences for ∆ n = 0 . 00 1 26 model ( α , n ) α = − 0 . 99 α = − 0 . 90 α = − 0 . 75 α = − 0 . 50 α = − 0 . 25 α = − 0 . 10 CIR 0 (0.01, 50) 0.01 0.14 0.54 0.77 0.85 0.87 CIR 1 (0.01, 50) 0.80 0.98 1.00 1.00 1.00 1.00 CIR 2 (0.01, 50) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.05, 50) 0.05 0.16 0.54 0.77 0.85 0.87 CIR 1 (0.05, 50) 0.94 0.98 1.00 1.00 1.00 1.00 CIR 2 (0.05, 50) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.01, 100) 0.01 0.13 0.49 0.71 0.79 0.82 CIR 1 (0.01, 100) 0.99 1.00 1.00 1.00 1.00 1.00 CIR 2 (0.01, 100) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.05, 100) 0.04 0.15 0.49 0.71 0.79 0.82 CIR 1 (0.05, 100) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 2 (0.05, 100) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.01, 500) 0.00 0.06 0.28 0.54 0.69 0.74 CIR 1 (0.01, 500) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 2 (0.01, 500) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.05, 500) 0.02 0.08 0.28 0.54 0.69 0.74 CIR 1 (0.05, 500) 1.00 1.00 1.00 1.00 1.00 1.00 CIR 2 (0.05, 500) 1.00 1.00 1.00 1.00 1.00 1.00 T able 9: Nu mbers represent probabi lity of rejection under the true model, with rejection region calculated under H 0 . α - div ergences, for ∆ n = 0 . 1 27 model ( α , n ) λ = − 0 . 99 λ = − 1 . 20 λ = − 1 . 50 λ = − 1 . 75 λ = − 2 . 00 λ = − 2 . 50 CIR 0 (0.01, 50) 0.00 0.00 0.00 0.01 0.02 0.06 CIR 1 (0.01, 50) 0.00 0.06 0.52 0.75 0.86 0.94 CIR 2 (0.01, 50) 0.00 0.99 1.00 1.00 1.00 1.00 CIR 0 (0.05, 50) 0.00 0.00 0.00 0.02 0.04 0.09 CIR 1 (0.05, 50) 0.00 0.23 0.70 0.85 0.92 0.96 CIR 2 (0.05, 50) 0.06 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.01, 100) 0 .00 0.00 0.00 0.01 0.02 0.05 CIR 1 (0.01, 100) 0 .00 0.56 0.96 0.99 1.00 1.00 CIR 2 (0.01, 100) 0 .00 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.05, 100) 0 .00 0.00 0.00 0.02 0.03 0.08 CIR 1 (0.05, 100) 0 .00 0.83 0.99 1.00 1.00 1.00 CIR 2 (0.05, 100) 0 .97 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.01, 500) 0 .00 0.00 0.00 0.00 0.01 0.02 CIR 1 (0.01, 500) 0 .00 1.00 1.00 1.00 1.00 1.00 CIR 2 (0.01, 500) 1 .00 1.00 1.00 1.00 1.00 1.00 CIR 0 (0.05, 500) 0 .00 0.00 0.00 0.01 0.02 0.04 CIR 1 (0.05, 500) 1 .00 1.00 1.00 1.00 1.00 1.00 CIR 2 (0.05, 500) 1 .00 1.00 1.00 1.00 1.00 1.00 T able 10 : Numbers represent probabil ity of rejection under the true m odel, wit h rejectio n region calculated under H 0 . Po wer-div ergences f o r ∆ n = 0 . 1 28