On ADF goodness-of-fit tests for perturbed dynamical systems

Transcription

1 Bernoulli 21(4), 215, DOI: 1.315/14-BEJ65 arxiv: v2 [math.st] 29 Sep 215 On ADF goodness-of-fit tests for perturbed dynamical systems YURY A. KUTOYANTS 1 Laboratoire de Statistique et Processus, Université du Maine, 7285 Le Mans, France and Laboratory of Quantitative Finance, Higher School of Economics, Moscow, Russia. kutoyants@univ-lemans.fr We consider the problem of construction of goodness-of-fit tests for diffusion processes with a small noise. The basic hypothesis is composite parametric and our goal is to obtain asymptotically distribution-free tests. We propose two solutions. The first one is based on a change of time, and the second test is obtained using a linear transformation of the natural statistics. Keywords: asymptotically distribution free test; Cramér von Mises tests; diffusion processes; goodness of fit test; perturbed dynamical systems 1. Introduction We consider the following problem. Suppose that we observe a trajectory X ε {X t, t T} of the following diffusion process: dx t S(t,X t )dt+εσ(t,x t )dw t, X x, t T, (1.1) where W t, t T is a Wiener process, σ(t,x) is known smooth function, the initial value x is deterministic and the trend coefficient S(t,x) is a unknown function. Here ε (,1) is a given parameter. We have to test the composite (parametric) hypothesis H :dx t S(ϑ,t,X t )dt+εσ(t,x t )dw t, X x, t T (1.2) against alternative H 1 :not H. Here S(ϑ,t,x) is a known smooth function of ϑ and x. The parameter ϑ Θ is unknown and the set Θ R d is open and bounded. Let us fix some value α (,1) and consider the class of tests of asymptotic (ε ) size α: K α { ψ ε :E ϑ ψε α+o(1)} for all ϑ Θ. The test ψ ε ψ ε (X ε ) is the probability to reject the hypothesis H and E ϑ stands for the mathematical expectation under hypothesis H. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 215, Vol. 21, No. 4, This reprint differs from the original in pagination and typographic detail c 215 ISI/BS

2 2 Y.A. Kutoyants Our goal is to find goodness-of-fit (GoF) tests which are asymptotically distribution free (ADF), that is, we look for a test statistics whose limit distributions under null hypothesis do not depend on the underlying model given by the functions S(ϑ, t, x), σ(t,x) and the parameter ϑ. This work is a continuation of the study Kutoyants [9], where an ADF test was proposed in the case of simple basic hypothesis. The behaviour of stochastic systems governed by such equations (called perturbed dynamical systems) is well studied, see, for example, Freidlin and Wentzell [3] and the references therein. Estimation theory (parametric and non-parametric) for such models of observations is also well developped, see, for example, Kutoyants [8] and Yoshida [17, 18]. Let usremind the well-knownbasic resultsin this problemforthe i.i.d. model. We start withthesimplehypothesis.supposethatweobserveni.i.d.r.v. s(x 1,...,X n )X n with a continuous distribution function F(x), and the basic hypothesis is H :F(x) F (x), Then the Cramér von Mises statistic is D n n [ˆF n (x) F (x)] 2 df (x), x R. ˆFn (x) 1 n n j1 1 {Xj<x}, where ˆF n (x) is the empirical distribution function. Denote by K α the class of tests of asymptotic (n ) size α (,1), that is, K α { ψ:e ψα+o(1)}. We have the convergence (under hypothesis H ) B n (x) n(ˆf n (x) F (x)) B(F (x)), where B( ) is a Brownian bridge process. Hence, it can be shown that and the Cramér von Mises test D n δ B(s) 2 ds, ψ n (X n )1 {Dn>c α} K α, P{δ>c α }α is asymptotically distribution-free (ADF). The situation changes in the case of parametric basic hypothesis: H :F(x)F(ϑ,x), ϑ Θ, where Θ(α,β). If we introduce the similar statistic ˆD n n [ˆF n (x) F(ˆϑ n,x)] 2 df(ˆϑ n,x),

3 ADF GoF tests for dynamical systems 3 where ˆϑ n is the maximumlikelihood estimator(mle), then (under regularityconditions) we have U n (x) n(ˆf n (x) F(ˆϑ n,x))b n (x) n(ˆϑ n ϑ) F(ϑ,x)+o(1). For the MLE, we can use its representation n(ˆϑn ϑ) 1 n n j1 l(ϑ,x j ) I(ϑ) +o(1), l(ϑ,x)lnf(ϑ,x). All this allows us to write the limit U( ) of the statistic U n ( ) as follows: l(ϑ,y) x l(ϑ, y) U n (x) B(F(ϑ,x)) db(f(ϑ,y)) df(ϑ,y) I(ϑ) I(ϑ) B(t) h(ϑ, v) db(v) h(ϑ,v)dv U(ϑ,t), 1 where tf(ϑ,x) and we put h(ϑ,t)i(ϑ) 1/2 l(ϑ,f ϑ (t)). If ϑ Θ R d, then we obtain a similar equation U(ϑ,t)B(t) h(ϑ, v) db(v), h(ϑ,v)dv, (1.3) where, is the scalar product in R d. This presentation of the limit process U(ϑ,t) can be found in Darling [2]. Of course, the test ˆψ n 1 { ˆDn>c is not ADF and the choice of the threshold c α} α can be a difficult problem. One way to avoid this problem is, for example, to find a transformation L W [U](t)w(t), where w( ) is the Wiener process. This transformation allows to write the equality L W [U](F(ϑ,x)) 2 df(ϑ,x) w(t) 2 dt. Hence, if we prove the convergence D n L W [U n ](x) 2 df(ˆϑ n,x), then the test ψ n 1 { Dn>c α}, with P( > c α) α is ADF. Such transformation was proposed in Khmaladze [6]. In the present work, we consider a similar problem for the model of observations (1.1) with parametric basic hypothesis (1.2). Note that several problems of GoF testing for the model of observations (1.1) with simple basic hypothesis Θ{ϑ } were studied in Dachian and Kutoyants [1], Iacus and Kutoyants [5], Kutoyants [9]. The tests considered

4 4 Y.A. Kutoyants there are mainly based on the normalized difference ε 1 (X t x t ), where x t x t (ϑ ) is a solution of equation (1.2) for ε. This statistic is in some sense similar to the normalized difference n(ˆf n (x) F (x)) used in the GoF problems for i.i.d. models. We propose two GoF ADF tests. Note that the construction of the first test is in some sense close to the one considered in Kutoyants [11] and based on the score function process. These tests are originated by the different processes but after our first transformation of the normalized difference ε 1 (X t x t (ˆϑ ε )) we obtain the same integrals to calculate as those in Kutoyants [11]. Let us remind the related results in the case of simple hypothesis (see Kutoyants [9]). Suppose that the observed homogeneous diffusion process under null hypothesis is dx t S (X t )dt+εσ(x t )dw t, X x, t T, where S (x) is a known smooth function. Denote x t X t ε. We have X t x t as ε and we construct a GoF test based on statistic v ε (t)ε 1 (X t x t ). The limit of this statistic is a Gaussian process. This process can be transformed into the Wiener process as follows: introduce the statistic [ ( σ(xt ) δ ε S (x t ) The following convergence: ) 2 dt] 2 δ ε ( ) 2 Xt x t εs (x t ) 2 σ(x t ) 2 dt. w(s) 2 ds was proved and therefore the test ˆψ ε 1 {δε>c α} with P( >c α )α is ADF. Consider now the hypotheses testing problem (1.1) and (1.2). The solution x t of equation (1.2) for ε depends on ϑ Θ R d, that is, x t x t (ϑ). The statistic ˆv ε (t) ε 1 (X t x t (ˆϑ ε )) (here ˆϑ ε is the MLE) is in some sense similar to U n ( ). Denote by ˆv(t) the limit of ˆv ε (t) as ε and suppose that we know the transformation L U [ˆv]( ) of ˆv( ) into the Gaussian process U(ϑ,t)W(t) h(ϑ,s)dw(s), with a vector-function h(ϑ, s) satisfying h(ϑ,s)ds, t 1 h(ϑ,s)h(ϑ,s) dsj. Here J is the d d unit matrix. The next steps are two transformations of U( ): one transformation into the Brownian bridge L B [U](s)B(s) and another one into the Wiener process L W [U](s)w(s),

5 ADF GoF tests for dynamical systems 5 respectively. This allows us to construct the ADF GoF tets as follows: let us introduce (formally) the statistics δ ε (L B [L U [ˆv ε ]](t)) 2 dt, ε and suppose that we have proved the convergences (L W [L U [ˆv ε ](t)]) 2 dt, δ ε δ B(s) 2 ds, ε w(s) 2 ds. Then the tests ˆψ ε 1 {δε>d α}, P(δ >d α )α, ˆΨε 1 { ε>c α}, P( >c α )α, belong to the class K α and are ADF. Our objective is to realize this program. A similar result for ergodic diffusion processes is contained in Kutoyants [12] (simple basic hypothesis) and Kleptsyna and Kutoyants [7] (parametric basic hypothesis). 2. Auxiliary results We have the following stochastic differential equation: dx t S(ϑ,t,X t )dt+εσ(t,x t )dw t, X x, t T, (2.1) where ϑ Θ, Θ is an open bounded subset of R d and ε is a small parameter, that is, we study this equation in the asymptotics of small noise ε. Introduce the Lipschitz condition and that of linear growth: C1. The functions S(ϑ,t,x) and σ(t,x) satisfy the relations S(ϑ,t,x) S(ϑ,t,y) + σ(t,x) σ(t,y) L x y, S(ϑ,t,x) + σ(t,x) L(1+ x ). Recall that by these conditions the stochastic differential equation (2.1) has a unique strong solution (Liptser and Shiryaev [14]), and moreover this solution X ε {X t, t T} converges uniformly, with respect to t, to the solution x T {x t, t T} of the ordinary differential equation dx t dt S(ϑ,t,x t), x, t T. (2.2) Observe that x t x t (ϑ) (for the proof see Freidlin and Wentzell [3], Kutoyants [8]). C2. The diffusion coefficient σ(t,x) 2 is bounded away from zero inf t T,x σ(t,x)2 >.

6 6 Y.A. Kutoyants Conditions C1 and C2 provide the equivalence of the measures {P (ε) ϑ,ϑ Θ} induced on the measurable space (C T,B T ) by the solutions of equation (2.1) (Liptser and Shiryaev [14]). Here C T is the space of continuous functions on [,T] with uniform metrics and B T is the Borelian σ-algebra of its subsets. The likelihood ratio function is { L(ϑ,X ε S(ϑ,t,X t ) T )exp ε 2 σ(t,x t ) 2 dx S(ϑ,t,X t ) 2 } t 2ε 2 σ(t,x t ) 2 dt, ϑ Θ, and the maximum likelihood estimator (MLE) ˆϑ ε is defined by the equation L(ˆϑ ε,x ε ) supl(ϑ,x ε ). ϑ Θ The following regularity conditions (smoothness and identifiability) provides us necessary properties of the MLE. Below x t x t (ϑ ). C3. The functions S(ϑ, t, x) and σ(t, x) have two continuous bounded derivatives w.r.t. x and the function S(ϑ,t,x) has two continuous bounded derivatives w.r.t. ϑ. For any ν > inf inf ϑ Θ ϑ ϑ >ν and the information matrix (d d) is uniformly non-degenerate: ( ) 2 S(ϑ,t,xt ) S(ϑ,t,x t ) dt> σ(t,x t ) Ṡ(ϑ,t,x t I(ϑ ) )Ṡ(ϑ,t,x t ) σ(t,x t ) 2 dt inf inf ϑ Θ λ 1 λ I(ϑ )λ>. We denote by a prime the derivatives w.r.t. x and t, and by a dot those w.r.t. ϑ, that is, for a function f f(ϑ,t,x) we write f (ϑ,t,x) f(ϑ,t,x), x f t f(ϑ,t,x) (ϑ,t,x), t f(ϑ,t,x) f(ϑ,t,x). ϑ Of course, in the case of d>1 the derivative ḟ(ϑ,t,x) is a column vector. If the conditions C2 and C3 hold, then the MLE admits the representation ε 1 (ˆϑ ε ϑ)i(ϑ) 1 Ṡ(ϑ,t,x t ) σ(t,x t ) dw t +o(1). (2.3)

7 ADF GoF tests for dynamical systems 7 Here, x t x t (ϑ). For the proof see, Kutoyants [9]. Note that X t X t (ε) (solution of equation (2.1)) under condition C3 is continuously differentiable w.r.t. ε. Denote the derivatives X (1) t X t ε, The equations for X (1) t and x (1) t are and t X t, ε ε x(1) t T. dx (1) t S (ϑ,t,x t )X (1) t dt+[εσ (t,x t )X (1) t +σ(t,x t )]dw t, X (1) dx (1) t S (ϑ,t,x t )x (1) t dt+σ(t,x t )dw t, x (1), (2.4) respectively. Hence x (1) t, t T is a Gaussian process and it can be written as x (1) t { } exp S (ϑ,v,x v )dv σ(s,x s )dw s. (2.5) s Denote { ψ(t)exp } S (ϑ,v,x v )dv, { ψ ε (t)exp } S (ˆϑ ε,v,x v )dv. We can write where X t x t (ˆϑ ε ) ε X t x t (ϑ) + x t(ϑ) x t (ˆϑ ε ) ε ε X (1) (ˆϑε ϑ) t,ẋ t (ϑ) +o(1) ε x (1) t I(ϑ) 1 Ṡ(ϑ,s,x s ) dw s,ẋ t (ϑ) +o(1) σ(s,x s ) ψ(t)v(t)+o(1), V(t)ψ(t) 1 x (1) t ψ(t) 1 I(ϑ) 1 Introduce the random process ψ(s) U(ϑ,t) σ(s,x s ) dv(s). Ṡ(ϑ,s,x s ) dw s,ẋ t (ϑ). σ(s,x s )

8 8 Y.A. Kutoyants Lemma 1. We have the equality U(ϑ,t)W t h(ϑ,s)dw s, where is a vector-valued function. h(ϑ,t)i(ϑ) 1/2Ṡ(ϑ,t,x t) σ(t,x t ) Proof. The solution of equation (2.4) can be written (see (2.5)) as For the vector ẋ t (ϑ), we can write ẋ t (ϑ) The solution of this equation is Introduce two stochastic processes and Then we can write U(ϑ,t) x (1) ψ(t)σ(s,x s ) t dw s. ψ(s) S (ϑ,s,x s )ẋ s (ϑ)ds+ h(ϑ,s)ds, t T, (2.6) Ṡ(ϑ,s,x s ) ẋ t (ϑ)ψ(t) ds. ψ(s) v 1 (t)ψ(t) 1 x (1) t v 2 (t)ψ(t) 1 ẋ t (ϑ) ψ(s) σ(s,x s ) dv(s) ψ(s) σ(s,x s ) dv 1(s) I(ϑ) 1 Ṡ(ϑ,s,x s ) σ(s,x s ) W(t) I(ϑ) 1/2 Ṡ(ϑ,s,x s )ds. ψ(s) 1 σ(s,x s )dw s ψ(s) 1 Ṡ(ϑ,s,x s )ds. ψ(s) dw s, σ(s,x s ) dv 2(s) Ṡ(ϑ,s,x s ) t dw s,i(ϑ) 1/2 Ṡ(ϑ,s,x s ) ds σ(s,x s ) σ(s,x s ) (2.7)

9 ADF GoF tests for dynamical systems 9 W t h(ϑ,s)dw s, Introduce the random process h(ϑ,s)ds. u(ϑ,r)t 1/2 U(ϑ,rT), r 1 and denote Ṡ(ϑ,rT,x rt I 1 (ϑ) )Ṡ(ϑ,rT,x rt) σ(rt,x rt ) 2 dr, h(ϑ,r) I 1 (ϑ) 1/2Ṡ(ϑ,rT,x rt) σ(rt,x rt ), w r T 1/2 W rt. Then we can write u(ϑ,r)w r and therefore r h(ϑ,q)dw q, h(ϑ,q)dq, r 1, (2.8) h(ϑ,q) h(ϑ,q) dqj. Note that u( ) is in some sense a universal limit which appears in the problems of goodness of fit testing for stochastic processes. For example, the same limit is obtained in the case of ergodic diffusion process and in the case of inhomogeneous Poisson process (Kutoyants [1]). The main difference with the i.i.d. case is due to the Wiener process here, while in the i.i.d. case the Brownian bridge B(t), t 1 appears (see (1.3)). Of course, we can immediately replace B(t) by a Wiener process B(t)W t W 1 t and this will increase the dimension of the vector h(ϑ, ). In the case of vector-valued parameter ϑ, this change is not essential and will slightly modify calculations of the test statistics for the first type test. At the same time if the parameter ϑ is one-dimensional, then we can easily construct the second-type goodness-of-fit test for stochastic processes and it remains unclear how to construct such tests in the i.i.d. case. The difference will be explained in Section 3.2. In the construction of a GoF test, we will use another condition. C4. The functions S(ϑ,t,x), Ṡ(ϑ,t,x) and σ(t,x) have continuous bounded derivatives w.r.t. t [,T]. 3. Main results Suppose that we observe a trajectory X ε (X t, t T) of the following diffusion process: dx t S(t,X t )dt+εσ(t,x t )dw t, X x, t T. (3.1)

10 1 Y.A. Kutoyants We have to test the basic parametric hypothesis H :S(t,x)S(ϑ,t,x), t T,ϑ Θ, that is, that the observed process (3.1) has the stochastic differential dx t S(ϑ,t,X t )dt+εσ(t,x t )dw t, X x, t T (3.2) with some ϑ Θ. Here S(ϑ,t,x) and σ(t,x) are known strictly positive smooth functions and Θ R d is an open convex set. We have to test this hypothesis in the asymptotics of a small noise (as ε ). Our goal is to construct such statistics v ε [X ε ]( ), V ε [X ε ]( ) that (under hypothesis H ) δ ε v ε [X ε ](t) 2 dt δ ε V ε [X ε ](t) 2 dt B(s) 2 ds, w(s) 2 ds, where B( ) and w( ) are the Brownian bridge and the Wiener process, respectively. Then we introduce the tests ˆψ ε 1 {δε>d α}, ˆΨε 1 { ε>c α} with the thresholds c α and d α satisfying the equations These tests will belong to the class P(δ >d α )α, P( >c α )α. (3.3) } K α { ψε :lime ϑ ψε α, ϑ Θ ε and will be ADF. We propose these tests in the Sections 3.1 and 3.2 below. We call ˆψ ε the first test and ˆΨ ε the second test First test Theconstructionofthe firstadf GoF testis basedonthe followingwellknownproperty. Suppose that we have a Gaussian process U(t), t T satisfying the equation U(t)w(t) h(s) dw(s) h(s) ds, h(s) 2 ds1.

11 ADF GoF tests for dynamical systems 11 Introduce the process b(t) h(s)du(s) It is easy to see that b()b(t) and E[b(t)b(s)] h(s)dw(s) h(s) dw(s) s h(v) 2 dv h(v) 2 dv s h(s) 2 ds. h(v) 2 dv. Let us put Then τ s h(v) 2 dv, b(t)b(τ), τ 1. δ ( 2 h(s)du(s)) h(t) 2 dt b(t) 2 h(t) 2 dt B(τ) 2 dτ. Suppose that the parameter ϑ is one-dimensional, ϑ Θ(a,b) and that we already proved the convergence (see Lemma 1) where ψ ε (s) U ε (t) σ(s,x s ) d ( Xs x s (ˆϑ ε ) εψ ε (s) ) U(ϑ,t), t T, U(ϑ,t)w(t) h(ϑ, s) dw(s) h(ϑ,s)ds, h(ϑ,s) 2 ds1. Recall that h(ϑ,s)i(ϑ) 1/2Ṡ(ϑ,s,x s) σ(s,x s ), I(ϑ) Ṡ(ϑ,s,x s ) 2 σ(s,x s ) 2 ds. Introduce (formally) the statistic ˆδ ε ( 2 h(ˆϑ ε,s)du ε (s)) h(ˆϑ ε,t) 2 dt.

12 12 Y.A. Kutoyants If we prove that ( 2 h(ˆϑ ε,s)du ε (s)) h(ˆϑ ε,t) 2 dt ( 2 h(ϑ,s)du(ϑ,s)) h(ϑ,t) 2 dt then the test ˆψ ε 1 {δε>c α} will be ADF. The main technical problem in carrying out this program is to define the stochastic integral h(ˆϑ ε,s)du ε (s) containingthe MLE ˆϑ ε ˆϑ ε (X t, t T). We will proceedas follows:first, we formally differentiate and integrate and then we take the final expressions, which do not contain stochastic integrals, as starting statistics. Introduce the statistics D(ϑ,s,X s )S(ϑ,s,x s (ϑ))+s (ϑ,s,x s )(X s x s (ϑ)), R(ϑ,t,X t ) Xt Ṡ(ϑ,t,y) I(ϑ)σ(t,y) 2 dy x Xs Q(ϑ,t,X t ) x Ṡ s (ϑ,s,y)σ(s,y) 2Ṡ(ϑ,s,y)σ s (s,y) I(ϑ)σ(s,y) 3 Ṡ(ϑ,s,X s )D(ϑ,s,X s ) ds, I(ϑ)σ(s,Xs ) 2 K ε (ϑ,t)ε 1 [R(ϑ,t,X t ) Q(ϑ,t,X t )], δ ε K ε (ˆϑ ε,t) 2 h ε (ˆϑ ε,t) 2 dt. The first test is given in the following theorem. Theorem 1. Suppose that the conditions C1 C4 hold. Then the test dyds, ˆψ ε 1 {δε>c α}, P{δ>c α }α is ADF and belongs to K ε. Proof. We can write (formally) ψ ε (s) U ε (t) σ(s,x s ) dv ε(s)

13 ADF GoF tests for dynamical systems 13 ψ ε (s) σ(s,x s ) d dx s εσ(s,x s ) dx s εσ(s,x s ) ( ) Xs x s (ˆϑ ε ) ψ ε (s)ε [ S(ˆϑε,s,x s (ˆϑ ε )) εσ(s,x s ) where we have used the equality D(ˆϑ ε,s,x s ) εσ(s,x s ) ds, dx s (ˆϑ ε )S(ˆϑ ε,s,x s (ˆϑ ε ))ds. Hence (formally), we obtain the following expression. Ṡ(ˆϑ ε,s,x s ) h ε (ˆϑ ε,s)du ε (s) dx s I(ˆϑ ε )εσ(s,x s ) 2 ] + S (ˆϑ ε,s,x s )(X s x s (ˆϑ ε )) ds εσ(s,x s ) Ṡ(ˆϑ ε,s,x s )D(ˆϑ ε,s,x s ) ds. I(ˆϑ ε )εσ(s,x s ) 2 (3.4) The estimator ˆϑ ε ˆϑ ε (X t, t T) and therefore the stochastic integral is not well defined because the integrand Ṡ(ˆϑ ε,s,x s ) is not a non-anticipative random function. Note that in the linear case S(ϑ,t,x)ϑQ(s,x) we have no such problem (see example below). This difficulty can be avoided in general case by at least two ways: The first one is to replace the stochastic integral by it s robust version as we show below. The second possibility is to use a consistent estimator ϑ νε of the parameter ϑ constructed after the observations X νε (X t, t ν ε ), where ν ε but sufficiently slowly. With this estimator, we can calculate the integral Ṡ( ϑ νε,s,x s ) ν ε σ(s,x s ) 2 dx s without any problem, and all limits will be the same. Such construction is discussed for a different problem in Kutoyants and Zhou [13]. Introduce the function x Ṡ(ϑ,t,y) M(ϑ,t,x) x σ(t,y) 2 dy. Then by the Itô formula dm(ϑ,t,x t )M t(ϑ,t,x t )dt+ ε2 σ(t,x t ) 2 +M x (ϑ,t,x t)dx t 2 M xx(ϑ,t,x t )dt

14 14 Y.A. Kutoyants and therefore Ṡ(ϑ,s,X s ) σ(s,x s ) 2 dx s M(ϑ,t,X t ) Xt Ṡ(ϑ,t,y) x σ(t,y) 2 dy + Xs x [M s (ϑ,s,x s)+ ε2 σ(s,x s ) 2 Xs x 2Ṡ(ϑ,s,y)σ s (s,y) σ(s,y) 3 Note that the contribution of the term ε 2 2 Ṡ s(ϑ,s,y) σ(s,y) 2 dy ds ε2 2 is asymptotically (ε ) negligible. Therefore, is asymptotically equivalent to σ(s,x s ) 2 M xx(ˆϑ ε,s,x s )ds K ε (ϑ,t)ε 1 [R(ϑ,t,X t ) Q(ϑ,t,X t )] K ε (ϑ,t) h ε (ϑ,s)du ε (s). ] M xx (ϑ,s,x s) ds σ(s,x s ) 2 M xx (ϑ,s,x s)ds. The difference is in the dropped term of order O(ε). We have to verify the convergence of the integrals K ε (ˆϑ ε,t) 2 Ṡ(ˆϑ ε,t,x t ) 2 K(ϑ,t) 2 Ṡ(ϑ,t,x t ) 2 δ ε dt I(ˆϑ ε )σ(t,x t ) 2 I(ϑ)σ(t,x t ) 2 dt. Regularity conditions C1 C3 give the uniform convergences sup X t x t (ϑ), I(ˆϑ ε ) I(ϑ), t T sup h ε (ˆϑ ε,t) h(ˆϑ ε,t) sup t T t T Ṡ(ˆϑ ε,t,x t ) I(ˆϑ ε )σ(t,x t ) Ṡ(ϑ,t,x t) I(ϑ)σ(t,xt ). Introduce two processes Y ε (ˆϑ ε,t,x t Ṡ(ˆϑ ε,s,x s )[S(ϑ,s,X s ) D(ˆϑ ε,s,x s )] ) σ(s,x s ) 2 ds, Z(ˆϑ ε,t,x t )R(ˆϑ ε,t,x t ) Ṡ(ˆϑ ε,s,x s )S(ϑ,s,X s ) σ(s,x s ) 2 ds.

15 ADF GoF tests for dynamical systems 15 Then K ε (t)ε 1 [Y ε (ˆϑ ε,t,x t )+Z(ˆϑ ε,t,x t )]. We have S(ϑ,s,X s ) D(ˆϑ ε,s,x s ) S(ϑ,s,X s ) S(ˆϑ ε,s,x s )+S(ˆϑ ε,s,x s ) S(ˆϑ ε,s,x s (ˆϑ ε )) S (ˆϑ ε,s,x s )[X s x s (ˆϑ ε )] (ˆϑ ε ϑ)ṡ( ϑ,s,x s ) +[S (ˆϑ ε,s, X s ) S (ˆϑ ε,s,x s )][X s x s (ˆϑ ε )] (ˆϑ ε ϑ)ṡ( ϑ,s,x s )+O(ε 2 ). Therefore Further, where ε 1 Y ε (ˆϑ ε,t,x t ) (ˆϑ ε ϑ) ε Ṡ(ˆϑ ε,s,x s ) 2 σ(s,x s ) 2 ds+o(1). ε 1 (Z(ˆϑ ε,t,x t ) Z(ϑ,t,X t )) (ˆϑ ε ϑ) Ż(ϑ,t,X t )+o(1), ε Xt Ż(ϑ,t,X t ) x S(ϑ,t,y) σ(t,y) 2 dy Xs x S(ϑ,s,X s )S(ϑ,s,X s ) σ(s,x s ) 2 ds S s (ϑ,s,y)σ(s,y) 2 S(ϑ,s,y)σ s(s,y) σ(s,y) 2 dyds. We have uniform convergence of X t to x t w.r.t. t. Hence, sup Ż(ϑ,t,Xt ) Ż(ϑ,t,xt ). t T Note that for any continuously differentiable function g(s, x) w.r.t. s we have the relation xt x g(t,y)dy g(s,x s )S(ϑ,s,x s )ds xs x g s(s,y)dyds since g(s,x s )S(ϑ,s,x s )ds g(s,x s )dx s

16 16 Y.A. Kutoyants and g(t,x s )dx s g(s,x s )dx s Hence, Ż(ϑ,t,x t ) for all t [,T]. By the Itô formula, Z(ϑ,t,X t ) ε R(ϑ,t,Xt ) ε Ṡ(ϑ,s,X s ) εσ(s,x s ) 2 dx s Therefore, we obtain the convergence g(v,x s ) v s 1 {v:xv>x s} xv dvdx s x g v (v,y)dydv. Ṡ(ϑ,s,X s )S(ϑ,s,X s ) εσ(s,x s ) 2 ds + ε σ(s,x s ) 2 M xx 2 (ϑ,s,x s)ds Ṡ(ϑ,s,X s ) dw s +O(ε). σ(s,x s ) K ε (t) K(ϑ,t). g(v,x s ) dvdx s v Ṡ(ϑ,s,X s )S(ϑ,s,X s ) εσ(s,x s ) 2 Thisconvergencecanbeshowntobeuniformw.r.t. t.thisprovestheconvergenceδ ε δ. Therefore the Theorem 1 is proved. Let us study the behaviour of the power function under the alternative. Suppose that the observed diffusion process(1.1) has the trend coefficient S(t, x) which does not belong to the parametric family. This family we described as follows: F {S( ):S(ϑ,t,x t (ϑ)), t T,ϑ Θ}. Here x t (ϑ), t T is the solution of equation (2.2). We introduce a slightly more strong condition of separability of the basic hypothesis and the alternative. Suppose that the function S(t, x) satisfies conditions C1, C2 and denote by y t, t T the solution of the ordinary differential equation obtained for (ε) dy t dt S(t,y t), y x. ds

17 ADF GoF tests for dynamical systems 17 Then where y (1) t ε 1 (X t x t (ˆϑ ε ))ε 1 (X t y t )+ε 1 (y t x t (ˆϑ ε )) is a solution of the equation y (1) t +ε 1 (y t x t (ϑ )) ε 1 (ˆϑ ε ϑ )ẋ t (ϑ )+o(1), dy (1) t S (t,y t )y (1) t dt+σ(t,y t )dw t, y (1) and ϑ is defined by the relation ( ) 2 S(ϑ,t,yt ) S(t,y t ) T ( ) 2 S(ϑ,t,y t ) S(t,y t ) inf dt dt. (3.5) ϑ Θ σ(t,y t ) σ(t,y t ) Suppose that this equation has a unique solution ϑ. Note that ε 1 (ˆϑ ε ϑ ) is tight (see Kutoyants [8] for details). Moreover, we also suppose that the basic hypothesis and the alternative are separated in the following sense: First, formally, we write h ε (ˆϑ ε,s)du ε (s) inf ϑ Θ ( ) 2 S(ϑ,t,yt ) S(t,y t ) dt>. σ(t,y t ) Ṡ(ˆϑ ε,s,x s ) t Ṡ(ˆϑ ε,s,x s )[S(s,X s ) D(ˆϑ ε,s,x s )] dw s ds I(ˆϑ ε )σ(s,x s ) I(ˆϑ ε )εσ(s,x s ) 2 Further Ṡ(ϑ,s,y s ) t I(ϑ )σ(s,y s ) dw Ṡ(ϑ,s,X s )[S(s,X s ) S(ϑ,s,X s )] s ds. I(ϑ )εσ(s,x s ) 2 S(s,X s ) D(ˆϑ ε,s,x s ) S(s,X s ) S(ˆϑ ε,s,x s (ϑ)) S (ˆϑ ε,s,x s )(X s x s (ˆϑ ε )) S(s,X s ) S(ˆϑ ε,s,x s )+O(ε 2 ) S(s,X s ) S(ϑ,s,X s )+S(ϑ,s,X s ) S(ˆϑ ε,s,x s )+O(ε 2 ) S(s,X s ) S(ϑ,s,X s )+(ˆϑ ε ϑ )Ṡ(ϑ,s,X s )+O(ε 2 ). Therefore, Ṡ(ϑ,s,y s ) t h ε (ˆϑ ε,s)du ε (s) I(ϑ )σ(s,y s ) dw (ˆϑ ε ϑ s )Ṡ(ϑ,s,y s ) 2 ε ds I(ϑ )σ(s,y s ) 2

18 18 Y.A. Kutoyants ε 1 Ṡ(ϑ,s,y s )[S(s,y s ) S(ϑ,s,y s )] I(ϑ )σ(s,y s ) 2 ds+o(ε 2 ) I 1 (t) I 2 (t) ε 1 I 3 (t)+o(ε 2 ) with an obvious notation. For the statistic δ ε we have the relations δε ε 1 I 3 ( )h( ) I 1 ( )h( ) I 2 ( )h( ) +O(ε), (3.6) where h( )h(ϑ,s) and is the L 2 (,T) norm. Recall that the quantities I 1 ( )h( ) and I 2 ( )h( ) are bounded in probability. Introduce the condition C5. The functions S(ϑ,t,x),S(t,x) and σ(t,x) are such that I 3 ( )h( ) 2 ( Ṡ(ϑ,s,y s )[S(s,y s ) S(ϑ,s,y s )] I(ϑ )σ(s,y s ) 2 This condition provides consistency of the test. )2Ṡ(ϑ,t) 2 ds σ(t,y t ) 2 dt>. Theorem 2. Let conditions C1 C5 hold. Then the test ˆψ ε is consistent. Proof. The proof follows from the convergence δ ε under alternative (see (3.6)). Note that if ϑ is an interior point of Θ, then Ṡ(ϑ,s,y s )[S(s,y s ) S(ϑ,s,y s )] σ(s,y s ) 2 ds. If condition C5 does not hold, then This equality is possible if Ṡ(ϑ,s,y s )[S(s,y s ) S(ϑ,s,y s )] σ(s,y s ) 2 ds, for all t [,T]. Ṡ(ϑ,s,y s )[S(s,y s ) S(ϑ,s,y s )], for all s [,T]. An example of such invisible alternative can be constructed as follows: Suppose that the function S(ϑ,s,x) does not depend on ϑ for s [,T/2], that is, S(ϑ,s,x)S (s,x) for all ϑ Θ. Then Ṡ(ϑ,s,y s ) for s [,T/2]. Therefore if S(s,y s )S(ϑ,s,y s ) for s [T/2,T] and a corresponding ϑ then condition C5 does not hold, but we can have S(s,y s ) S (s,y s ) for s [,T/2]. This implies that the test ˆψ ε is not consistent for this alternative.

19 ADF GoF tests for dynamical systems Second test The second test is based on the following well-known transformation. Suppose that we have a Gaussian process U(t), t 1 and d d matrix N(t) defined by the relations U(t)W t h(s)dw s, h(s) ds, (3.7) N(t) t h(s)h(s) ds, N()J, (3.8) where J is the d d unit matrix and h(t) is a continuous vector-valued function. Lemma 2. Suppose that the matrix N(t) is non-degenerate for all t [,1). Then U(t)+ where w( ) is a Wiener process. s h(s) N(s) 1 h(v)du(v)dsw(t), t 1, (3.9) Proof. This formula was obtained by Khmaladze [6]. The proof there is based on two results: a result of Hitsuda [4] and another one of Shepp [16]. Observe that there are many publications dealing with this transformation (see, e.g., the paper Maglaperidze et al. [15] and the references therein). Another direct proof is given in Kleptsyna and Kutoyants [7]. Note that representation (3.7) and (3.8) implies that h(s)du(s). (3.1) Suppose that ϑ Θ. Here Θ is an open bounded subset of R d. Now h(ϑ,s), R(ϑ,t,X t ) and Q(ϑ,t,X t ) are d-vectors and the Fisher information I(ϑ) is a d d matrix. Introduce the following stochastic processes: and put h ε (ϑ,t) Ṡ(ϑ,t,X t) σ(t,x t ), Ṡ(ϑ,s,x s N(ϑ,t) )Ṡ(ϑ,s,x s) t σ(s,x s ) 2 ds, Ṡ(ϑ,s,X s N ε (ϑ,t) )Ṡ(ϑ,s,X s) t σ(s,x s ) 2 ds, ε 1 T 2 W ε (t) 2 dt.

20 2 Y.A. Kutoyants Here dx t s W ε (t) εσ(s,x s ) D(ˆϑ ε,s,x s ) ds εσ(s,x s ) +ε 1 h ε (ˆϑ ε,s) Nε (ˆϑ ε,s) 1 + [R(ˆϑ ε,s,x s ) Q(ˆϑ ε,s,x s )]ds. (3.11) We use the following convention for the matrix N: We have the following result. N 1 + { N 1, if N is non-degenerate,, if N is degenerate. Theorem 3. Suppose that conditions C2 C4 hold and the matrix N(ϑ,t) is uniformly in ϑ Θ non-degenerate for all t [,1). Then the test ˆΨ ε 1 { ε>c α}, ( P w(s) 2 ds>c α )α is ADF and belongs to K α. Proof. We have to show that under hypothesis H the convergence ε w(s) 2 ds (3.12) holds. TheconstructionoftheADF GoFtestisbasedonLemmas1and2.Wehavethesimilar to (2.6) presentation (3.7) with h(ϑ,t) defined in (2.7). Let us denote U ε ( ),h ε (ˆϑ ε, ), and N ε ( ) the empirical versions of U( ),h(ϑ, ) and N(ϑ,t)I(ϑ) 1 t Ṡ(ϑ,s,x s )Ṡ(ϑ,s,x s) σ(s,x s ) 2 ds, N(ϑ,)J, respectively: ψ ε (s) U ε (t) σ(s,x s ) dv ε(s), V ε (t) X t x t (ˆϑ ε ), ψ ε (t)ε h ε (ˆϑ ε,t)i ε (ˆϑ ε ) 1/2Ṡ(ˆϑ ε,t,x t ), σ(t,x t )

21 ADF GoF tests for dynamical systems 21 Ṡ(ˆϑ ε,s,x s )Ṡ(ˆϑ ε,s,x s ) I ε (ˆϑ ε ) σ(s,x s ) 2 ds, N ε (ˆϑ ε,t)i ε (ˆϑ ε ) 1 Ṡ(ˆϑ ε,s,x s )Ṡ(ˆϑ ε,s,x s ) σ(s,x s ) 2 ds. t Recall that there is a problem of definition of the integral for U ε ( ) because the integrand depends on the future. As convergence is uniform w.r.t. t [,T ν]: h ε (ˆϑ ε,t) h(ϑ,t), I ε (ˆϑ ε ) I(ϑ), N ε (ˆϑ ε,t) N(ϑ,t). The required limits can be obtained. Introduce (formally) the statistic Observe that W ε (t)u ε(t)+ h ε (ˆϑ ε,s) N ε (ˆϑ ε,s) 1 + s h(ϑ,s) N(ϑ,s) 1 h(ϑ,v) Ṡ(ϑ,s,x s) ( Ṡ(ϑ,r,x r )Ṡ(ϑ,r,x r) σ(s,x s ) σ(r,x r ) 2 s h ε (ˆϑ ε,v)du ε (v)ds. (3.13) ) 1Ṡ(ϑ,v,x v ) dr σ(v,x v ). Therefore this term does not depend on the information matrix I(ϑ) and we can replace the statistics h ε (ˆϑ ε,s) and N ε (ˆϑ ε,s) in (3.13) by h ε (ˆϑ ε,s) and N ε (ˆϑ ε,s). For the process U ε ( ), we have equality (3.4) (formally) dx t s U ε (t) εσ(s,x s ) D(ˆϑ ε,s,x s ) ds. εσ(s,x s ) Hence, we obtain the vector-valued integral Ṡ(ˆϑ ε,s,x s ) t h ε (ˆϑ ε,s)du ε (s) εσ(s,x s ) 2 dx Ṡ(ˆϑ ε,s,x s )D(ˆϑ ε,s,x s ) s εσ(s,x s ) 2 ds. Introduce the vector-function x Ṡ(ϑ,t,y) M(ϑ,t,x) x σ(t,y) 2 dy. Then by the Itô formula Ṡ(ϑ,s,X s ) Xt σ(s,x s ) 2 dx Ṡ(ϑ,t,y) s x σ(t,y) 2 dy + Xs x Xs x Ṡ s(ϑ,s,y) σ(s,y) 2 ds 2Ṡ(ϑ,s,y)σ s(s,y) σ(s,y) 3 ds+o(ε 2 ).

22 22 Y.A. Kutoyants Put K ε (t) h ε (ˆϑ ε,s)du ε (s)ε 1 [R(ˆϑ ε,t,x t ) Q(ˆϑ ε,t,x t )]. Note that we have dropped the term of order O(ε 2 ). Then formal expression (3.13) for Wε (t) can be replaced by (3.11) dx t s W ε (t) εσ(s,x s ) D(ˆϑ ε,s,x s ) ds εσ(s,x s ) +ε 1 For the first two terms of W ε (t) we have U ε (t) W t + dx s εσ(s,x s ) ˆϑε ϑ W t ε + h ε (ˆϑ ε,s) Nε (ˆϑ ε,s) 1 + [R(ˆϑ ε,s,x s ) Q(ˆϑ ε,s,x s )]ds. D(ˆϑ ε,s,x s ) ds εσ(s,x s ) S(ϑ,s,X s ) S(ˆϑ ε,s,x s (ˆϑ ε )) S (ˆϑ ε,s,x s )(X s x s (ˆϑ ε )) ds εσ(s,x s ) Ṡ( ϑ,s,x s ), ds σ(s,x s ) [S (ˆϑ ε,s, X s ) S (ˆϑ ε,s,x s )](X s x s (ˆϑ ε )) ds εσ(s,x s ) W t I(ϑ) 1 Ṡ(ϑ,s,x s ) t Ṡ(ϑ,s,x s ) dw s, σ(s,x s ) σ(s,x s ) U(ϑ,t)+o(1). Here ϑ ϑ ˆϑ ε and X s X s x s (ˆϑ ε ) X s ds +o(1) x s (ˆϑ ε ) x s (ϑ) + x s (ϑ) X s. This convergence is uniform w.r.t. s [,T]. Hence, sup U ε (t) U(ϑ,t). t T Further, similar arguments give the uniform convergence w.r.t. t [, T] h ε (ˆϑ ε,t) Ṡ(ˆϑ ε,t,x t ) σ(t,x t ) h(ϑ,t), Nε (ˆϑ ε,t) N(ϑ,t).

23 ADF GoF tests for dynamical systems 23 We have to show that K ε (t) K(ϑ,t), where Denote Then We have Therefore Further, K(ϑ,t) h(ϑ,s)dw s h(ϑ,s)dw s h(ϑ,s) h(ϑ,s) ds. Y ε (ˆϑ ε,t,x t Ṡ(ˆϑ ε,s,x s )[S(ϑ,s,X s ) D(ˆϑ ε,s,x s )] ) σ(s,x s ) 2 ds, Z(ˆϑ ε,t,x t )R(ˆϑ ε,t,x t ) Ṡ(ˆϑ ε,s,x s )S(ϑ,s,X s ) σ(s,x s ) 2 K ε (t)ε 1 [Y ε (ˆϑ ε,t,x t )+Z(ˆϑ ε,t,x t )]. S(ϑ,s,X s ) D(ˆϑ ε,s,x s ) S(ϑ,s,X s ) S(ˆϑ ε,s,x s )+S(ˆϑ ε,s,x s ) S(ˆϑ ε,s,x s (ˆϑ ε )) S (ˆϑ ε,s,x s )[X s x s (ˆϑ ε )] (ˆϑ ε ϑ),ṡ( ϑ,s,x s ) +[S (ˆϑ ε,s, X s ) S (ˆϑ ε,s,x s )][X s x s (ˆϑ ε )] (ˆϑ ε ϑ),ṡ( ϑ,s,x s ) +O(ε 2 ). ε 1 Y ε (ˆϑ ε,t,x t ) (ˆϑ ε ϑ) ε ds. Ṡ(ˆϑ ε,s,x s )Ṡ(ˆϑ ε,s,x s ) σ(s,x s ) 2 ds. where ε 1 (Z(ˆϑ ε,t,x t ) Z(ϑ,t,X t )) ˆϑ ε ϑ Ż(ϑ,t,X t )+o(1), ε Xt S(ϑ,t,y) Ż(ϑ,t,X t ) x σ(t,y) 2 dy Xs x S(ϑ,s,X s )S(ϑ,s,X s ) σ(s,x s ) 2 ds S s (ϑ,s,y)σ(s,y) 2 S(ϑ,s,y)σ s(s,y) σ(s,y) 2 dyds.

24 24 Y.A. Kutoyants Here S( ) is the matrix of second derivatives w.r.t. ϑ. We have uniform convergence of X t to x t w.r.t. t, hence sup Ż(ϑ,t,Xt ) Ż(ϑ,t,xt ). t T Observe that for any continuously differentiable function g(s, x) w.r.t. s we have since and xt x g(t,y)dy g(t,x s )dx s s g(s,x s )S(ϑ,s,x s )ds g(s,x s )S(ϑ,s,x s )ds g(v,x s ) v g(s,x s )dx s dvdx s g(v,x s ) 1 {v:xv>x s} dvdx s v xs x g s(s,y)dyds g(s,x s )dx s xv x g v (v,y)dydv. Hence, Ż(ϑ,t,xt ) for all t [,T]. By the Itô formula Z(ϑ,t,X t ) ε R(ϑ,t,Xt ) ε Ṡ(ϑ,s,X s ) εσ(s,x s ) 2 dx s Therefore, we obtain the convergence Ṡ(ϑ,s,X s )S(ϑ,s,X s ) εσ(s,x s ) 2 + ε σ(s,x s ) 2 M xx 2 (ϑ,s,x s)ds Ṡ(ϑ,s,X s ) dw s +O(ε). σ(s,x s ) K ε (t) ε 1 (R(ˆϑ ε,t,x t ) Q(ˆϑ ε,t,x t )) ds Ṡ(ϑ,s,X s )S(ϑ,s,X s ) εσ(s,x s ) 2 ε 1 (Y(ˆϑ ε,t,x t )+Z(ˆϑ ε,t,x t )) K(ϑ,t). Further, the matrix N ε (ˆϑ ε,s) converges uniformly in s [,T] to the matrix N(ϑ,s). Therefore, for ν > we have uniform on s [,T ν] convergence of N ε (ˆϑ ε,s) 1 + to ds

25 ADF GoF tests for dynamical systems 25 N(ϑ,s) 1. Introduce the random function y ε (s)ε 1 hε (ˆϑ ε,s) Nε (ˆϑ ε,s) 1 + [R(ˆϑ ε,s,x s ) Q(ˆϑ ε,s,x s )]. It is shown that we have convergence where sup y ε (s) y(ϑ,s), s T ν y(ϑ,s) h(ϑ,s) N(ϑ,s) 1 K(ϑ,s). Hence we also have convergence for all t [,1) W ε (t) U(ϑ,t)+ h(ϑ,s) N(ϑ,s) 1 K(ϑ,s)dsw(t). A similar argument can show that for any t 1 < <t k T we have convergence of the vectors (W ε (t 1 ),...,W ε (t k )) (w(t 1 ),...,w(t k )). Further, a direct but cumbersome calculation allows us to write the estimate E ϑ W ε (t 1 ) W ε (t 2 ) 2 C t 2 t 1, t 1,t 2 [,T ν]. These two conditions provide weak convergence of the integrals ν W ε (t) 2 dt ν w(t) 2 dt for any ν >. It can be shown that for any η> there exist ν > such that T ν E ϑ W ε (t) 2 dt η. The proof is close to that given in Maglaperidze et al. [15] for a similar integral. 4. Examples Example 1. We consider the simplest case which allows us to have an ADF GoF test for each ε, that is, no need to study statistics as ε. Observe that a similar situation is discussed in Khmaladze [6] but for a different problem. Suppose that the observed diffusion process (under hypothesis) is dx t ϑdt+εdw t, X, t 1. (4.1)

26 26 Y.A. Kutoyants Then h(ϑ,t)1, I(ϑ)1, N(ϑ,t)1 t, ˆϑ ε X 1, ε 1 (ˆϑ ε ϑ)w 1 N(,1). Further Therefore, x t (ϑ) ϑt, x (1) t (ϑ)w t, U(ϑ,t)W t W 1 t, V ε (t) U ε (t)ε 1 (X t X 1 t)w t W 1 tb(t). W ε (t)ε 1 (X t X 1 t)+ε 1 (1 s) 1 [X s X 1 s]ds and under the basic hypothesis we have Therefore, W ε (t)b(t)+ ε W ε (t) 2 dt B(s) 1 s dsw(t). w(t) 2 dt and the test ˆΨ ε 1 { ε>c α} K α satisfies the equality { E ϑˆψε P w(t) 2 dt>c α }α. Example 2. Consider the linear case dx t ϑ,h(t,x t ) dt+εσ(t,x t )dw t, X x, t T, where ϑ Θ R d and assume that the functions H(t,x) and σ(t,x) satisfy regularity conditions. In this case, we can take h ε (ϑ,t) h ε (t), that is, this vector-valued function does not depend on ϑ. Hence, the stochastic integral is well defined and the test has a simplified form. We have and so on. h ε (t) H(t,X t) σ(t,x t ), H(t,x t (ϑ))h(t,x t (ϑ)) Nε (ϑ,s) s σ(t,x t (ϑ)) 2 ds, dx t du ε (t) εσϕ(t,x t ) [ ˆϑ ε,h(t,x t (ˆϑ ε )) + ˆϑ ε,h x(t,x t ) (X t x t (ˆϑ ε ))]dt, εσ(t,x t ) W ε (t)u ε (t)+ s H(s,X s ) Nε (ˆϑ ε,s) 1 H(v,X v )du ε (v)ds

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