Goodness-of-Fit Tests for Perturbed Dynamical Systems

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1 Goodness-of-Fit Tests for Perturbed Dynamical Systems arxiv:93.461v1 [math.st] 6 Mar 9 Yury A. Kutoyants Laboratoire de Statistique et Processus, Université du Maine 785 Le Mans, Cédex 9, France Abstract We consider the goodness of fit testing problem for stochastic differential equation with small diffusion coefficient. The basic hypothesis is always simple and it is described by the known trend coefficient. We propose several tests of the type of Cramer-von Mises, Kolmogorov-Smirnov and Chi-Square. The power functions of these tests we study for a special classes of close alternatives. We discuss the construction of the goodness of fit test based on the local time and the possibility of the construction of asymptotically distribution free tests in the case of composite basic hypothesis. MSC Classification: 6M, 6G1, 6G. Key words: Cramer-von Mises, Kolmogorov-Smirnov and Chi-Square tests, diffusion process, goodness of fit, hypotheses testing, small noise asymptotics. 1 Introduction We consider the construction of the goodness-of-fit(gof) tests for dynamical system withsmall noise, i.e., theobservations X ε = {X t, t T} arefrom the homogeneous stochastic differential equation dx t = S(X t ) dt+ε σ(x t ) dw t, X = x, t T (1) withdeterministicinitialvaluex andknowndiffusioncoefficientε σ( ) >. All statistical inference concerns the trend coefficient S( ) only. We have two hypotheses: the basic hypothesis in our consideration is always simple H : S( ) = S ( ) 1

2 and the alternative corresponds to the process (1) with a different trend coefficient S( ) S ( ). As usual in GoF testing there are two problems. The first one is to find the threshold which provides the asymptotic size α of the test and the second is to describe the behavior of the power function for some classes of alternatives. There are different ways to present such (nonparametric) alternatives and we discuss below the choices of alternatives. The problem considered corresponds to the asymptotics of the small noise, i.e., we study the properties of the tests as ε. We suppose that the trend and diffusion coefficients satisfy the Lipshits condition S (x) S (y) + σ(x) σ(y) L x y () hence the equation (1) has a unique strong solution [15] and we have the estimates { } P sup X t x t > δ Ce c ε, sup E X t x t Cε, (3) t T t T where C >,c > are some generic constants (see [5], [1]). Here the function {x t, t T} is solution of the limit (deterministic) ordinary equation dx t dt = S (x t ), x, t T. (4) Our goal is to present some goodness-of-fit tests for this stochastic model which are similar to the well-known in classical statistics Cramer-von Mises (C-vM), Kolmogorov-Smirnov (K-S) and Chi-Square (Ch-S) tests. Remind that these classical tests are distribution free, i.e., their limit distributions do not depend on the basic hypothesis and therefore the problem of the choice of the threshold is universal for all tests of such types. Moreover these tests are consistent against any fixed alternative. The GoF tests proposed below for the model (1) have the similar properties. Letusrecallthebasicpropertiesoftheclassicaltests. Supposethatweobserve n independent identically distributed random variables (X 1,...,X n ) = X n with continuous distribution function F (x) and the basic hypothesis is simple : H : F (x) = F (x), x R. Then the Cramér-von Mises Wn and Kolmogorov-Smirnov D n statistics are ] Wn [ˆFn = n (x) F (x) df (x), D n = sup n ˆFn (x) F (x) x respectively. Here ˆF n (x) = 1 n n j=1 1 {Xj <x}

3 is the empirical distribution function. Let us denote by {W (s), s 1} a Brownian bridge, i.e., a continuous Gaussian process with EW (s) =, EW (s)w (t) = t s st. Then the limit behavior of these statistics can be described with the help of this process as follows Wn = W (s) ds, Hence the corresponding C-vM and K-S tests D n = sup W (s). s 1 ψ n (X n ) = 1 {W n >c α}, φ n (X n ) = 1 {Dn>d α} with constants c α,d α defined by the equations { } { } P W (s) ds > c α = α, P sup W (s) > d α = α s 1 are of asymptotic size α. We see that these tests are distribution-free (the limit distributions do not depend on the function F ( )) and are consistent against any fixed alternative (see, for example, Durbin [3], Lehmann and Romano [14]). It is interesting to study these tests for non degenerate set of alternatives, i.e., for alternatives with limit power function is grater than α and less than 1. It can be realized on the close nonparametric alternatives of the special form making this problem asymptotically equivalent to the signal in Gaussian noise problem. Let us put F (x) = F (x)+ 1 n x h(f (y)) df (y), where the function h( ) describes the alternatives. We suppose that h(s) ds =, h(s) ds <. Then we have the following convergence (under fixed alternative, given by the function h( )): [ s Wn = h(v)dv +W (s)] ds, s D n = sup h(v)dv +W (s) s 1 3

4 We see that this problem is asymptotically equivalent to the following signal in Gaussian noise problem: dy s = h (s) ds+dw (s), s 1. (5) Indeed, if we use the statistics W = Y s ds, D = sup Y s s 1 then under hypothesis h( ) and alternative h( ) the distributions of these statistics coincide with the limit distributions of W n and D n under hypothesis and alternative respectively. C-vM and K-S type tests.1 Choice of the thresholds We test the basic simple hypothesis H and our goal is to study the GoF tests of asymptotic size α (,1). Let us denote the class of such tests as K α = {ψ ε : E ψ ε (X ε ) = α+o(1)}, where E is expectation under hypothesis. We use the following regularity condition. Condition R. The function S ( ) has two continuous bounded derivativess (x),s (x)andthefunctionσ(x)hasonecontinuousboundedderivative σ (x) and the both functions are positive : S (x) >,σ (x) > for x x. Remind that under this condition the equation (1) has a unique strong solution andthis solution converges uniformly on t [,T] to the deterministic solution of the equation (4) (see, e.g., [15] [1]). Moreover, the stochastic process X t is differentiable w.r.t. ε at the point ε = and its derivative x (1) t satisfies the linear equation dx (1) t = S (x t )x (1) t dt+σ(x t ) dw t, x (1) =. (6) For the proof see, e.g., [1], Lemma 3.3. To construct the C-vM and K-S type tests we use the statistics [ ( ) σ(xt ) T ( ) Xt x t δ ε = dt] S (x t ) εs (x t ) σ(x t ) dt. 4

5 and γ ε = [ ( ) ] 1/ σ(xt ) dt sup S (x t ) t T X t x t εs (x t ) respectively. Below c α and b α are solutions of the equations { } { } P wv dv > c α = α, P sup w v > b α = α, (7) v 1 where w v, v 1 is some Wiener process. Proposition 1 Let the condition R be fulfilled then the tests ψ ε = 1 {δε>c α} and φ ε = 1 {γε>b α} belong to the class K α. Proof. The stochastic process ε 1 (X t x t ) converges in probability uniformly on t [,T] to the limit x (1) t - solution of the linear equation (6), i.e., for any κ > { } P sup X t x t x (1) t ε > κ. t T This solution can be written explicitly as t { t } x (1) t = exp S (x v)dv σ(x s ) dw s. s Below we follow [1], where the similar calculus were done. Using () we can write t t S (x S v)dv = (x v ) dx t v s s S (x v ) dv dv = S (x v ) s S (x v ) dx v = xt S = (x) xt ( ) x s S (x) dx = S (x t ) lns (x) dx = ln S (x s ). (8) Hence x (1) t = S (x t ) t x s ( ) σ(x s ) t S (x s ) dw σ(x s ) s = S (x t ) W S (x s ) ds where W ( ) is some Wiener process. Further ( ) ( Xt x T ) ( t ) t εs (x t ) σ(x s ) σ(x s ) t dt W S (x s ) ds σ(x s ) d S (x s ) ds = ut W (u) du = u T 5 w v dv,

6 where we put u = t σ(x s ) S (x s ) ds = v σ(x s ) S (x s ) ds = v u T. Here w v = u 1/ T W (u T v), v 1 is a Wiener process. Hence (under hypothesis H ) δ ε w v dv. For the statistic γ ε the similar consideration leads to the relation γ ε sup w v. v 1 Therefore for the first type errors we have { } α ε (δ ε ) = P {δ ε > c α } P wv dv > c α = α, and { } α ε (γ ε ) = P {γ ε > b α } P sup w v > b α = α. v 1 The Proposition 1 is proved.. Similar tests. Note that the similar result we have if we use the statistics [ ( ) σ(xt ) T ( ) Xt x t δ ε = dt] S (X t ) εs (X t ) σ(x t ) dt and γ ε = [ ( ) ] 1/ σ(xt ) dt sup S (X t ) t T X t x t εs (X t ). because, as we mentioned above, the process X t converges uniformly on t [,T] to the deterministic solution x t and this implies the convergence δ ε w vdv, γ ε sup w v. v 1 6

7 These tests can be slightly simplified, if we replace the equation (4) by the following one d ˆX t = S (X t ), ˆX = x, (9) dt i.e., we put ˆX t = x + t S (X s ) ds, t T, (we need not to solve the equation (4), just to calculate the integral, using observations) and introduce the statistic [ ] ( T ˆδ ε = σ(x t ) dt σ(x t ) X t ˆX ) t dt ε Then under hypothesis H where ε σ(x t ) ( X t ˆX ) [ t ] t dt = σ(x t ) σ(x s )dw s dt = = τ T = τt [ ( t W σ(x s ) ds)] d W (r) dr Hence we have the convergence τ o T ( t ) σ(x s ) ds W (r) dr = (τt o ) wv dv, σ(x s ) ds τt o = σ(x s ) ds, ˆδ ε = w vdv. v = r. τt o Of course, we have the distribution free limit for the corresponding statistic ˆγ ε too..3 Partially observed linear system Suppose that we observe a random process X T = {X t, t T} and we have to test the hypothesis H that this process comes from the following linear partially observed system dy t = A t Y t dt+εb t dv t, Y = y, t T, dx t = C t Y t dt+εσ t dw t, X =, t T, 7

8 where A t,b t, C t and σ t are known functions and V t and W t are independent Wiener processes. We suppose as well that these functions satisfy the usual conditions which allows us to write the equations of filtration (see Liptser and Shiryaev [15]). If the hypothesis is true then according to well known Kalman-Bucy theory(see Liptser and Shiryaev [15], Theorem 1.1) the conditional expectation M t = E (Y t X s, s t) satisfies the equations dm t = A t M t dt+ C tγ(t) [dx ε σt t C t M t dt], dγ(t) = A t γ(t) C t γ(t) +ε B dt ε σt t with initial values M = E Y = y and γ() = E (Y E Y ) =. Note that if we put Γ t = γ(t)ε, then this system can be rewritten as dm t = A t M t dt+ C tγ t [dx t C t M t dt], M = y, dγ t (t) dt σ t = A t Γ t C t Γ t σ t +B t, Γ =. Remind as well that the observed process admits the representation dx t = C t M t dt+εσ t d W t, t T, where W t is innovation Wiener process defined by this equality. This representation suggests to define the statistic ( ) t δ ε = ε σ(t) dt σ(t) [X t C s M s ds] dt. Elementary calculations yield ε σ(t) [X t = t C s M s ds] dt = [ t ] σ(t) σ s d W s dt [ ( t )] ( ) σ(t) W σs ds dt = σ(t) dt wv dv. Hence the statistic δ ε (under hypothesis) is δ ε = w v dv and the corresponding test ψ ε = 1 {δε>c α} is distribution free. 8

9 .4 Alternatives Let us consider nonparametric alternatives S( ) S ( ), which correspond to the equation dx t = S(X t ) dt+εσ(x t ) dw t, X = x, t T. We suppose that the functions S( ) and σ( ) satisfy the conditions, S(x) S(y) + σ(x) σ(y) L x y, S(x) + σ(x) L(1+ x ). The set of such functions we denote as S(L) and the limit of the stochastic process X t, t T we write as x t (S), t T. There are many ways to introduce the class of alternatives. We consider two of them. Let us define the sets F r = {S( ) S(L) : x. (S) x. r}, G r = {S( ) S(L) : S(x. ) S (x. ) r}, where is L (,T)-norm, say, S(x. ) S (x. ) = [S(x t ) S (x t )] dt. Here x t, t T is solution of the equation (4) (under hypothesis H ). Let us start with the problem H : S( ) = S ( ), H 1 : S( ) F r To show the consistency of the test ψ ε note that δ ε κε 1 X. x. with some κ >. We have { inf P S {δ ε > c α } inf P S κε 1 X. x. > } c α S F r S F r { inf P S ε 1 x. (S) x. ε 1 X. x. (S) > } c α S F r { 1 sup P S ε 1 X. x. (S) > ε 1 x. (S) x. κ 1 } c α S F r 1 ( ε 1 r κ 1 c α ) sup S F r E S ( ) Xt x t (S) dt 1 Cε. Therefore even if r = r ε such that ε 1 r ε, then the C-vM test is minimax consistent, i.e., its power function tends to 1 uniformly on 9 ε

10 S( ) F r. The same is true for the Kolmogorov-Smirnov test. It is not so in the case of the hypotheses testing problem H : S( ) = S ( ), H 1 : S( ) G r It will be more convenient to write S(x) S (x) as εh(x)σ(x) S (x) 1, where h( ) is such that S( ) G r. This corresponds to the model dx t = S (X t ) dt+ε h(x t)σ(x t ) S (X t ) dt+εσ(x t ) dw t, t T (1) with the same initial value X = x. The case h( ) corresponds to the hypothesis H. We start with the study of the limit behavior of the statistic δ ε under a fixed alternative h( ). Let us denote by x h t the solution of the equation and write dx h t dt It is easy to see that ) ( ) = S x h σ ( x h t +ε t ( ) h ( ) x h S x h t, x h = x ; t X t x t ε = X t x h t ε + xh t x t. ε X t x h t x (1) t, t T ε in probability and the direct calculations yield x h t x t ( ) t S x h = s S (x s ) t σ ( x h ds+ s) h ( ) x h s ε ε S (x h s ) ds t ( ) x h = s x s ) t S ( x h σ ( ) h ( ) x h ε s ds+ s x h s ds. S (x h s) Hence xh t xt ε Therefore converges to the function ẋ h t which is solution of the equation dẋ h t dt ẋ h t = t = S (x t ) ẋ h t + σ(x t) = S (x t ) { t exp s t S (x t ) h(x t), ẋ h =. } S (x σ(xs ) v) dv S (x s ) h(x s) ds σ(x s ) S (x s ) h(x s) ds 1

11 and Let us put X t x t εs (x t ) Z(x t) x(1) S (x t ) = = t xt x u(x) = σ(x s ) t +ẋ h t S (x s ) h(x s) ds+w σ(x) 3 h(x) dx+w S (x) x ( t ( xt x σ(y) x S (y) 3 dy [,u T] ) σ(x s ) S (x s ) ds σ(x) S (x) 3 dx and denote x(u) the function inverse to u(x) These relations yield the following representation for the limit of the test statistic Z(x t ) σ(x t ) ( T ) Z(x t ) xt σ(y) δ ε δ = u T S (x t ) dt = d u T x S (y) 3 dy ( xt ) Z(x) x σ(y) = d x u T x S (y) 3 dy ut [ u = u T h(x(z))dz +W (u)] du [ v ] = h (s)ds+w v ds, (11) where we put u = u T v and denoted h (s) = u 1/ T h(x(u Ts)) and w v = u 1/ T W (u T v). It is easy to see that w v, v 1 is standard Wiener process. Remind that if the observed process is of type signal in white Gaussian noise: dy s = h(s) ds+dw s, s 1, then the natural distance between hypothesis h(s) and h(s) is L (,1): In our case it corresponds to h (s) ds = h(s) ds ρ. xt x h(x) σ(x) S (x) 3 dx ρ. 11 ).

12 The convergence (11) provides the limit of the power function given in the next proposition. Proposition let the condition R be fulfilled, then for any function h( ) H r we have { [ v ] β(δ ε,h) = P h {δ ε > c α } = P h (s)ds+w v ds > c α }+o(1). Let us define the composite (nonparametric) alternative as H 1 : h( ) H r, where H r = { h( ) : } h(x. )σ(x. ) S (x. ) r. ε Note that x. here is from hypothesis H. As h( ) is an arbitrary function, this alternative coincides with G r. Let us show that without regularity conditions the problem of hypotheses testing is degenerate in the following sens: we have for the power of the test inf β(δ ε,h) α S G r as ε even for fixed r >. The condition S( ) S(L) already provides the control of the first derivative, that is why we have to weaken it slightly. At particularly, we suppose that S( ) S(L), but L > can be not the same for different functions S( ). Let us put h n (x) = c S (x) σ(x) cos(n(x x )) then, S n (x) S (x) = εc S (x) cos(n(x x )) and L = Kcεn with some K >. Further, using cos ϕ = 1 cos(ϕ) we obtain xt h n (x) σ(x) xt c S (x) 3 dx = x S (x) x σ(x) dx c xt cos(n(x x )) S xt (x) c S (x) dx σ(x) σ(x) dx x 1 x

13 as n. Remind that S (x) and σ(x) are continuous positive functions. The constant c = c(ε) > can be chosen from the condition c 4 xt x S (x) r dx σ(x) ε. Hence h n ( ) H r. On the other hand x h n (y) σ(y) c(ε) sin(n(x x )) 3 dy = x S (y) n S (x) + c(ε) x (y)sin(n(y x )) n S (y) dy x S uniformly in x [x,x T ] if we put, say, n = n(ε) = c(ε). Therefore lim inf β(δ ε,h) lim inf β(δ ε h( ) H r ε ε,h n ) = α. h n( ) Hence if we use the introduced above statistics then this hypotheses testing problem will be asymptotically degenerated. It can be shown that this is not the particular property of these two tests, but is true for any other tests too. 3 Chi Square Test If the alternative is defined by the inequality x. (S) x. r, then it isnatural to replace x t (S) by itsestimate X t andthis leads tothe test ψ ε = 1 { X. x. e α} similar to the introduced above C-vM type test. Remind that it was shown before that X t is the best in different senses estimator of x t (S) (see [1], Section 4.3). Therefore we can think that if the alternative is defined by the relation S(x. ) S (x. ) σ(x. ) r, then for construction of the good test statistics we have to replace S(x t ) by some estimator. Forget for instant that the Wiener process is not differentiable and write the observed process as Ẋ t = S(X t )+εσ(x t ) Ẇt 13

14 Then (formally!) Ẋ t S(x t ) as ε and we can consider Ẋ t as estimator of S(x t ). Let {ϕ j ( ),j =,±1,±...} form an orthonormal base in L (,T), thenbyparseval identity (formally!) we havethefollowing equality for the corresponding (modified S (x t ) S (X t )) statistic ] [Ẋt S (X t ) dt = ε σ(x t ) y j,ε where the Fourier coefficients y j,ε we can write as follows y j,ε = ϕ j (t) T [Ẋt S (X t )] dt = εσ(x t ) ϕ j (t) εσ(x t ) [dx t S (X t )dt]. This last integral has mathematical meaning and starting from this definition of y j,ε we can introduce the statistic δ ε = 1 4m j <m [ y j,ε 1 ] Note that under H the random variables y j,ε,j =,±1,±... are independent Gaussian y j,ε = ϕ j (t) dw t N (,1). Therefore the statistic j <m[ y j,ε 1 ] has Chi-Square distribution and the equation for c α P {δε > c α} = α can be easily solved. Moreover, m we have the convergence (under hypothesis) 1 [ y j,ε 1 ] = N (,1). 4m j <m Hence we can introduce the Ch-S test as ψ ε (Xε ) = 1 {δ ε >z α}, where m = m(ε) and z α is 1 α quantile of the Gaussian N (,1) law. This leads us to the following result. Proposition 3 Let us suppose that σ(x) >,x R, then the test ψ ε (X ε ) belongs to K α. 14

15 We see that we need not even to use the convergence X t to x t and the proposition is valid, say, for ergodic diffusion processes with ε = 1 and the asymptotic T. The choice of asymptotic is important in the calculation of the power function. Suppose that the observed process has the trend coefficient S( ) S ( ) and the condition R is fulfilled, then where ε j y j,ε = ϕ j (t)[s(x t ) S (X t )] εσ(x t ) dt+ ϕ j (t) dw t = z j,ε +ζ j, ( ) zj,ε = S(Xt ) S (X t ) dt S(x. ) S (x. ) σ(x t ) σ(x. ) r. Hence for any fixed contiguous alternative S(x) = S(x) + εh(x) σ(x) we have δ ε = 1 4m As m the relation j <m j <m ζ j 4m holds. To have non degenerate limit j <m z j,ε ζ j + 1 4m j <m z j,ε = h(x.) (1+o(1)) r ε (1+o(1)) 1 4m j <m zj,ε = r u ε (1+o(1)) u 1 4m we can put m = r 4 /(4ε 4 ). Note that in this case E 4m j <m z j,ε ζ j = 1 m j <m Hence for the power function we obtain the limit z j,ε = r u ε m z j,ε.. β(δ ε,h) P{ζ > z α u}, ζ N (,1). Another wayistofixfirstm suchthatmε 1/ andthentoconsider the alternatives running to the hypothesis : r = r(ε) = ε 4m. 15

16 Of course, this test is not uniformly consistent with the same explication as above. To have uniformly consistent and minimax GoF testing we need to control the derivatives. If we suppose that the function h( ) defining the alternative is k times differentiable and the L norm of the k-derivative is bounded by some constant, then we can show that the test δ ε = [ w j y j,ε 1 ] j <m with weights w i = z (1 i k), z = m m i= m [ 1 i ] k m and special choice of m is asymptotically minimax. The proof is based on the approach developed by Ermakov [4] and Ingster and Suslina [9] for the model signal in white Gaussian noise. See as well the similar problem for Poisson processes studied by Ingster and Kutoyants [8]. 4 On local time and GoF testing The local time of the diffusion process (1) is defined as ε Λ T (x) = lim ν ν and admits the Tanaka-Meyer representation (see [18]) Λ T (x) = X T x x x 1/4 1 { Xt x ν} σ(x t) dt (1) sgn(x t x) dx t. The local time Λ T (x) recently started to play an important role in statistical inference [11],[1],[1]. Note that in ergodic case (ε 1 and T ), the local time is asymptotically normal : η T (x) = ( ) ΛT (x) T f (x) = N (,d f (x) ), Tσ(x) where f (x) is invariant density and ( ) 1{ξ>x} F (ξ) d f (x) = 4f (x) E. σ(ξ)f (ξ) 16

17 Here ξ is random variable with density f (x) and F (x) is its distribution function (see [1], Proposition 1.5). Moreover this normed difference convergence weakly to the limit Gaussian process in the space of continuous on R functions vanishing in infinity [1], Theorem This property can be used in the construction of the GoF tests as follows. Let us introduce the C-vM and K-S type statistics δ T = η T (x) dx, γ T = sup η T (x) x and the corresponding tests ψ T = 1 {δt >c α} and φ T = 1 {γt >d α}. Then using this weak convergence we can define the constants c α,d α (see [1], section 5.4 and Gassem [6]). We can consider the similar problem in the asymptotics of small noise, i.e., to use the limit behavior of the local time in the construction of the GoF tests. Let us introduce the space L (x,x T ) of square integrable functions on [x,x T ], where x t is solution of the ordinary differential equation dx t dt = S (x t ), x, where S (x) >. According to (1) we have Λ T (x) lim ε ε = lim 1 = lim ν ν 1 = lim ν ν ν 1 lim ε ν x+ν x ν 1 { Xt x ν}σ(x t ) dt 1 { xt x ν}σ(x t ) 1 dt = lim ν ν xt σ(y) σ(x) dy = S (y) S (x), for x [x,x T ]. x 1 { y x ν}σ(y) dy S (y) t We say that the random process η ε (x),x x x T converges weakly in L (x,x T ) to the random process η(x),x x x T if for any function h( ) L (x,x T ) we have xt xt h(x) η ε (x) dx = h(x) η(x) dx. x Proposition 4 Let the condition R be fulfilled, then for the local time Λ T ( ) we have the weak convergence in L (x,x T ) : η ε (x) = 1 x ( 1 ε x S (y) Λ ) ( ) T (y) x σ(y) ε σ(y) dy = η(x) = W x S (y) 3 dy, (13) where x [x,x T ] and W ( ) is a Wiener process. 17 x

18 Proof. Note that the local time allows us to write the equality (see [18]) h(x t ) dt = Suppose that h( ) C 1. We know that h(x) Λ T (x) ε σ(x) dx. and h(x t ) dt h(x t ) h(x t ) ε h(x t ) dt = dt = = xt x h (x t ) x (1) t dt h(x t ) xt S (x t ) dx t = x h(x) S (x) dx ( ) xt h σ(y) (x t ) S (x t )W x S (y) 3dy dt ( ) x h σ(y) (x) W x S (y) 3dy dx Hence if we denote ( η ε (x) = ε 1 ΛT (x) ε σ(x) 1 ) {x x x T}, x R, S (x) and then we can write g(x) = x σ(y) x S (y) 3dy, x [x,x T ] h(x) η ε (x) dx xt The same time integrating by parts we have where we put Hence h(x) η ε (x) dx = ψ ε (x) = x h (x) ψ ε (x) dx x 18 x h (x) W (g(x)) dx. h (x) ψ ε (x) dx, η ε (y) dy. xt x h (x) W (g(x)) dx.

19 We see that the values of h( ) outside of the interval [x,x T ] have no contribution in the limit. Therefore we have as well the convergence xt x h (x) ψ ε (x) dx xt x h (x) W (g(x)) dx. Remindthatthisistrueforanyfunctionh( ) C 1. Thereforetheproposition is proved. The convergence (13) suggests the construction of the following test. Let ( ) xt σ(x) xt ( δ ε = ε S (x) 3dx σ(x) x ( 1 S (x) 3 S (y) Λ ) T (y) dy) ε σ(x) dx. x Then it can be shown that x δ ε = x w v dv and the corresponding test ψ ε = 1 {δε>c α} is asymptotically distribution free. Remark. We see that despite the ergodic case, the local time random function has no limit (as process) and for small values of ε its behavior is close to white noise process. Remark. We supposed above that S (x) > for all x [x,x T ]. In the case S (x ) = the deterministic solution x t x and we have the following basic hypothesis H : dx t = εσ(x t ) dw t, X, t T. The test can be based on the statistic ( ) Xt x δ ε = dt Tεσ(x ) and it is easy to see that (under hypothesis H ) δ ε w s ds. If for some x > x we have S (x) >,x [x,x ) and S (x ) =, then by Lipschitz condition t = xt x dx S (x) S (x ) 1 L xt x dx x x = 1 L ln x x x x t and we see that the equality x t = x is impossible (well known property). 19

20 5 On composite basic hypothesis Suppose that under hypothesis H the observed diffusion process is solution of the stochastic differential equation dx t = S(ϑ,X t ) dt+εσ(x t ) dw t, X = x, t T, where the trend coefficient S(ϑ,x) is a known function which depends on unknown parameter ϑ Θ = (β,γ). Then the limit solution x t depends on the true value ϑ, i.e., x t = x t (ϑ) and the natural modification of the test statistic can be based on the normalized difference Y t (ˆϑ ε ) = X t x t (ˆϑ ε ), ε where as ˆϑ ε we cantake, say, the maximum likelihood estimator. We suppose that the functions S(ϑ, x) and σ(x) are positive and sufficiently smooth to calculate the derivatives below and to provide the usual properties of estimators. Remind that (under regularity conditions) this estimator is consistent and asymptotically normal (see [1], Theorem.). Moreover the MLE admits the representation (see [1], Theorem 3.1) ˆϑ ε = ϑ +εi T (ϑ ) 1 Ṡ(ϑ,x t (ϑ )) σ(x t (ϑ )) dw t +o(ε), where ϑ is the true value of ϑ and I T (ϑ ) is the Fisher information: I T (ϑ) = ) (Ṡ(ϑ,xt (ϑ)) dt. σ(x t (ϑ)) It can be shown (see Rabhi [17]), that in regular (smooth) case the limit distribution of δ ε = Y t (ˆϑ ε ) dt coincides with the distribution of the following integral ( x (1) t (ϑ ) I T (ϑ ) 1 ẋ t (ϑ ) ) Ṡ(ϑ,x s (ϑ )) dw s dt, σ(x s (ϑ )) where ẋ t (ϑ) = ϑ x t(ϑ) and x (1) t (ϑ) is solution of the linear equation dx (1) t = S (ϑ,x t )x (1) t dt+σ(x t ) dw t, x (1) =, t T,

21 The test is no more distribution free, but in some cases it can be done asymptotically distribution free (ADF) if the second limit T is taken [17]. Another possibility to have an ADF test is to use the estimator process ˆϑ t,ε, t T, where ˆϑ t,ε is an estimator constructed by the observations X t = {X s, s t}. Of course, we have to suppose that this estimator is consistent andasymptoticallynormalforallvaluesoft (,T]. Forexample, in the linear case dx t = ϑh(x t )dt+εσ(x t ) dw t, X = x, t T we can take the MLE process ( ) t h(x s ) ˆϑ 1 t t,ε = σ(x s ) ds h(x s ) σ(x s ) dx s, < t T. Sometimes an estimator process can have recurrent structure (see Levanoy et al. [16]). If ˆϑ t,ε is the MLE, then it can be shown (see the similar calculus above (8)) the limit of δ ε is (below x s = x s (ϑ )) S(ϑ,x t ) ( t I t (ϑ ) 1 t σ(x s ) S(ϑ,x s ) dw s Ṡ(ϑ,x s ) S(ϑ,x s ) ds S(ϑ,x t ) Z(t,ϑ ) dt. The process Z(t,ϑ ) has stochastic differential t ) Ṡ(ϑ,x s ) dw s dt σ(x s ) dz(t,ϑ ) = A(t,ϑ )dt+b(t,ϑ )dw t, Z(,ϑ ) =. with the corresponding random function A(t,ϑ ) and deterministic B(t,ϑ ). Hence t ) B(t,ϑ ) (Z(t,ϑ 1 ) A(s,ϑ )ds = W t and δ = ( Z(t,ϑ ) t A(s,ϑ ) )ds dt = T S(ϑ,x t )B(t,ϑ ) w v dv 1

22 Using empirical versions of these functions it is possible to construct an ADF test based on the following statistics δ ε = ε µ ( Yt (ˆϑ t,ε ) t A ε(s, ˆϑ ) ε )ds T S(ˆϑ ε,x t )B ε (t, ˆϑ dt = ε ) w v dv. What is empirical version of stochastic integral we explain below. The integral is started at t = ε µ, where µ (,1) because for the values t [,ε] the estimator ˆϑ t,ε is not asymptotically normal. One else ADF test can be constructed by compensating the additional random part by the following way. First we rewrite the stochastic integral (Itô formula) H ε (ϑ) = = XT x Ṡ(ϑ,X s ) σ(x s ) [dx s S(ϑ,X s )ds] = Ṡ(ϑ,y) dy ε σ(y) Ṡ(ϑ,X s ) S(ϑ,X s ) σ(x s ) ds. Ṡ(ϑ,X s ) σ(x s ) dw s Ṡ (ϑ,x s )σ(x s ) Ṡ(ϑ,X s)σ (X x ) dt σ(x s ) The last expression for H ε (ϑ) does not contain stochastic integral andwe can put theestimator, i.e., therandomvariableh ε (ˆϑε ) is well defined (empirical version). Then introduce the stochastic process Y ε (t, ˆϑ ε ) = X t x t (ˆϑ ε ) ε +I T (ˆϑ ε ) 1 ẋ t (ˆϑ ε )H ε (ˆϑ ε ). Note that it can be easily shown that ẋ t (ˆϑ ε ) ẋ t (ϑ ), I T (ˆϑ ε ) I T (ϑ ), H ε (ˆϑ ε ) H (ϑ ), where sup t T H (ϑ ) = Ṡ(ϑ,x s (ϑ )) dw s. σ(x s (ϑ )) Hence the stochastic process Y ε (t, ˆϑ ε ) converges uniformly on t [,T] to the Gaussian process x (1) t (ϑ ) and we can use the statistic δ ε = [ ] σ(x t ) S(ˆϑ ε,x t ) dt Y ε (t, ˆϑ ε ) σ(x t ) dt = S(ˆϑ ε,x t ) 4 w v dv.

23 We see that the test ˆψ ε = 1 {δε>c α} based on this statistics is ADF. This test has to be consistent against any fixed alternative. Indeed, let the observed process be dx t = S(X t ) dt+εσ(x t ) dw t, X = x, t T and the limit solution x t (S) satisfies g = inf ϑ Θ x.(s) x. (ϑ) >. The MLE ˆϑ ε in this misspecified situation converges to the value ϑ which minimizes the Kullback-Leibner distance (see [1], Section.6) Hence and ϑ = arg inf ϑ Θ ( ) S(ϑ,xt (S)) S(x t (S)) dt. σ(x t (S)) ( Y ε t, ˆϑ ) ε = x (1) t (S)+ x t(s) x t (ϑ ) +I T (ϑ ) 1 ẋ t (ϑ )H (ϑ )+o(1) ε Therefore the test is consistent. References δ ε. [1] Bosq, D. and Davydov, Y. (1998) Local time and density estimation in continuous time. Math. Methods Statist., 8, 1, 45. [] Dachian, S. and Kutoyants, Yu.A. (7) On the goodness-of-fit tests for some continuous time processes, in Statistical Models and Methods for Biomedical and Technical Systems, F.Vonta et al. (Eds), Birkhäuser, Boston, [3] Durbin, (1973) Distribution Theory for Test Based on the Sample D.F. SIAM, Philadelphia. [4] Ermakov, M.S. (199) Minimax detection of a signal in a Gaussian white noise. Theory Probab. Appl., 35, [5] Freidlin, M.I., Wentsell, A.D. (1984) Random Perturbations of Dynamical Systems, Springer, N. Y. 3

24 [6] Gassem, A. (8) Goodness-of-Fit test for switching diffusion, prepublication 8-7, Université du Maine ( anis.pdf). [7] Iacus, S. and Kutoyants, Yu.A. (1) Semiparametric hypotheses testing for dynamical systems with small noise, Math. Methods Statist., 1, 1, [8] Ingster, Yu. I. and Kutoyants Yu. A., (7) Nonparametric hypothesis testing for intensity of Poisson process, Mathem. Methods Statist, 16, [9] Ingster, Yu.I. and Suslina, I.A. (3) Nonparametric Goodness-of-Fit Testing Under Gaussian Models, Springer, N.Y. [1] Kutoyants, Yu.A.(1994) Identification of Dynamical Systems with Small Noise, Kluwer, Dordrecht. [11] Kutoyants, Yu.A. (1997) Some problems of nonparametric estimation by the observations of ergodic diffusion processes. Statist. Probab. Lett., 3, [1] Kutoyants, Yu.A. (4) Statistical Inference for Ergodic Diffusion Processes, Springer, London. [13] Kutoyants, Yu.A. (8) On minimax goodness of fit testing for dynamical systems with small noise, in preparation. [14] Lehmann, E.L. and Romano, J.P. (5) Testing Statistical Hypotheses. (3rd ed.) Springer, N.Y. [15] Liptser, R.S. and Shiryayev, A.N.(1) Statistics of Random Processes. I, (nd ed.) Springer, N.Y. [16] Levanoy, D., Shwartz, A., Zeitouni, O. (1994) Recursive identification in continuous-time stochastic processes. Stochastic Process. Appl., 49, [17] Rabhi, A. (8) On the goodness-of-fit testing of composite hypothesis for dynamical systems with small noise, prepublication 8-6, Université du Maine, ( [18] Revuz, D. and Yor, M. (1991) Continuous Martingales and Brownian Motion. Springer, N.Y. 4