On ADF Goodness-of-Fit Tests for Perturbed Dynamical Systems

Transcription

1 On ADF Goodness-of-Fit Tests for Perturbed Dynamical Systems arxiv: v1 [math.st] 3 Mar 214 Yu.A. Kutoyants Laboratoire de Statistique et Processus, Université du Maine Le Mans, FRANCE Abstract We consider the problem of the construction of the goodness-of-fit tests for diffusion processes with 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 the change of time and the second test we obtain with the help of some linear transformation of the natural statistics. MSC 2 Classification: 62M2, 62G1, 62G2. Key words: Cramér-von Mises tests, perturbed dynamical systems, diffusion processes, goodness of fit test, asymptotically distribution free test. 1 Introduction We consider the following problem. Suppose that we observe a trajectory X ε {X t, t T} of diffusion process dx t S(t,X t dt+εσ(t,x t dw t, X x, t T, (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 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 (2 against alternative H 1 : not H. 1

2 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 { ψε : 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 means the mathematical expectation under hypothesis H. Our goal is to find the goodness-of-fit (GoF tests, which are asymptotically distribution free (ADF, i.e., we seek the test statistics whose limit distributions under hypothesis do not depend on the underlying model given by the functions S(ϑ,t,x,σ(t,x and parameter ϑ. This work is a continuation of the study [9], where the ADF test was proposed in the case of simple basic hypôthesis. The behaviour of the stochastic systems governed by such equations (called perturbed dynamical systems is well studied, see, e.g., [3] and the references there in. The estimation theory (parametric and non parametric for such models of observations is well developped too, see, e.g., [8] and [15],[16]. Let us remind the well-known related basic results in the case of the i.i.d. observations model. We start with the simple hypothesis. Suppose that we observe n i.i.d. r.v. s (X 1,...,X n X n with continuous distribution function F (x. The basic hypothesis is H, F (x F (x, x R. Let us denote by K α the class of tests of asymptotic (n size α (,1, i.e.; K α { ψ : E ψ α+o(1 }. As usual ψ is the probability to reject the hypothesis H. One possibility of the costruction of the test belonging to K α is to use the Cramér-von Mises statistic ] 2dF D n n [ˆFn (x F (x (x, ˆFn (x 1 n 1I {Xj <x}. n Here ˆF n (x is the empirical distribution function. The test is of the form ψ n (X n 1I {Dn>c α}. The choice of the constant c α is based on the following well-known properties. We have the weak convergence (under hypothesis H B n (x n(ˆfn (x F (x B(F (x, j1 2

3 where B( is a Brownian bridge process. Hence, it can be shown that D n δ and the Cramér-von Mises test B(s 2 ds, ψ n (X n 1I {Dn>c α} K α, P{δ > c α } α is asymptotically distribution-free. In the case of parametric basic hypothesis: H : F (x F (ϑ,x, ϑ Θ, where Θ (α,β the situation changes. If we introduce the similar statistic ˆD n n [ˆFn (x F ] 2dF (ˆϑn,x (ˆϑn,x, where ˆϑ n isthemaximum likelihoodestimator (MLE, then(under regularity conditions we have U n (x n(ˆfn (x F (ˆϑn,x n(ˆfn (x F (ϑ,x + ( n F (ϑ,x F (ˆϑn,x B n (x n(ˆϑn ϑ F (ϑ,x+o(1. For the MLE we can use its representation (brlow l(ϑ,x lnf (ϑ,x n (ˆϑn ϑ 1 n n j1 l(ϑ,x j I(ϑ +o(1 l(ϑ,x I(ϑ db n(x+o(1. All these allow us to write the limit U ( of the statistic U n ( as follows U n (x B(F (ϑ,x B(t where t F (ϑ,x and we put l(ϑ,y I(ϑ db(f (ϑ,y h(ϑ,v db(v x h(ϑ,v dv U (ϑ,t, l(ϑ,y I(ϑ df (ϑ,y h(ϑ,t l ( ϑ,f 1 ϑ (t I(ϑ, h(ϑ,v 2 dv 1. 3

4 If the parameter ϑ Θ R d, then we obtain the similar equation U (ϑ,t B(t h(ϑ,v db(v, h(ϑ,v dv, (3 where, is the scalar product in R d. Thispresentation ofthelimitprocessu(ϑ,tcanbefoundin[2]ofcourse the test ˆψ n 1I {ˆDn>c α} is not ADF and the choice of the threshold c α can be a difficult problem. One way to avoid this problem is, for example, to find transformation L W [U](t w(t, where w( is a Wiener process. This transformation provide us the equality Hence if we prove the convergence D n L W [U](F (ϑ,x 2 df (ϑ,x L W [U n ](x 2 df (ˆϑn,x, w(t 2 dt. then the test ψ n 1I { Dn>c α}, with P( > c α α is ADF. Note that such transformation was proposed by Khmaladze [6]. Our second test is based on this transformation. At the present work we consider the similar problem for the model of observations (1 with parametric basic hypothesis (2. Note that several problems of GoF testing for the model of observations (1 with simple basic hypothesis Θ {ϑ } were studied in the works [1], [5], [9]. The tests considered there are mainly based on the normalized difference ε 1 (X t x t, where x t x t (ϑ is solution of the equation (2 as ε. This statistic is in some sense similar to the normalized difference n(ˆfn (x F (x used in the GoF problems for the i.i.d. r.v. s model. We propose two GoF ADF tests. Let us first remind the related results in the case of simple hypothesis (from [9]. Suppose that the observed homogeneous diffusion process under hypothesis is dx t S (X t dt+εσ(x t dw t, X x, t T, where S (x is some known smooth function. Denote x t X t ε. The process X t x t as ε and we construct the GoF test on the basis of the statistic v ε (t ε 1 (X t x t. The limit of this statistic is Gaussian process. 4

5 This process can be transformed into Wiener process as follows. Introduce the statistic [ ( 2 2 σ(xt T ( 2 Xt x t δ ε dt] S (x t εs (x t 2 σ(x t 2 dt. The following convergence δ ε w(s 2 ds was proved andtherefore the test ˆψ ε 1I {δε>c α} with P( > c α α is ADF. Consider now the hypotheses testing problem (1-(2. The solution x t of the equation (2 as ε depends on ϑ Θ R d, i.e., 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 (. Let us denote v(t the limit of v ε (t as ε and suppose that we found the transformation L U [ˆv]( of ˆv( into the Gaussian process U (ϑ,t W (t h(ϑ,s dw (s, h(ϑ,s ds, t 1 with some vector-function h(ϑ, s satisfying h(ϑ,s h(ϑ,s ds J. Here J is d d unit matrix. The next steps are two transformations of U ( : one into the Brownian bridge L B [U](s B(s and another into the Wiener process L W [U](s w(s respectively. This allows us to construct the ADF GoF tets as folows. Let us introduce (formally the statistics δ ε (L B [L U [ˆv ε ]](t 2 dt, ε and suppose that we proved the convergences then the tests δ ε δ B(s 2 ds, ε (L W [L U [ˆv ε ](t] 2 dt, w(s 2 ds, ˆψ ε 1I {δε>d α}, P(δ > d α α, ˆΨε 1I { ε>c α}, P( > c α α, belong to the class K α and are ADF. Our goal is to realize this program. The similar result for ergodic diffusion processes was presented in the works [1] (simple basic hypothesis and [7] (parametric basic hypothesis. 5

6 2 Auxiliary results We have the stochastic differential equation dx t S(ϑ,t,X t dt+εσ(t,x t dw t, X x, t T, (4 where ϑ Θ, Θ is an open bounded subset of R d and ε is a small parameter, i.e., we study this equation in the asymptotics ε. Introduce the Lipschitz and linear growth conditions 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. Remind that by these conditions the stochastic differential equation (4 has a unique strong solution [12], andmoreover this solutionx 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. (5 Note that the function x t x t (ϑ (see, e.g., [3], [8]. C2 : The diffusion coefficient σ(t,x 2 is separated from zero inf t T,x σ(t,x2 >. { } The conditions C1 C2 provide the equivalence of the measures P (ε ϑ,ϑ Θ induced in the measurable space (C T,B T by the solutions of equation (4 [12]. Here C T is the space of continuous on [,T] functions with uniform metrics and B T is the Borelian σ - algebra of its subsets. The likelihood ratio function is L(ϑ,X ε exp { S(ϑ,t,X t T ε 2 σ(t,x t 2 dx t } S(ϑ,t,X t 2 2ε 2 σ(t,x t 2 dt, ϑ Θ, and the maximum likelihood estimator (MLE ˆϑ ε is defined by the equation L (ˆϑε,X ε supl(ϑ,x ε. ϑ Θ The following regularity conditions (smoothness and identifiability provides us the necessary properties of the MLE. Below x t x t (ϑ. 6

7 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 > ( 2 S(ϑ,t,xt S(ϑ,t,x t inf inf dt > ϑ Θ ϑ ϑ >ν σ(t,x t and the information matrix (d d I(ϑ is uniformly non degenerate: Ṡ(ϑ,t,x t Ṡ(ϑ,t,x t σ(t,x t 2 dt inf inf ϑ Θ λ 1 λ I(ϑ λ >. We denote by prime the derivatives w.r.t. x and t and by dot we denote the derivatives w.r.t. ϑ, i.e., for a function f f (ϑ,t,x we write f (ϑ,t,x f (ϑ,t,x, f t x f (ϑ,t,x (ϑ,t,x, f (ϑ,t,x t f (ϑ,t,x. ϑ Of course, in the case of d > 1 the derivative ḟ (ϑ,t,x is a vector-column. If the conditions C2 and C3 are fulfilled, then the MLE admits the representation ε (ˆϑε 1 ϑ I(ϑ 1 Ṡ(ϑ,t,x t σ(t,x t dw t +o(1. (6 Here x t x t (ϑ. For the proof see [8]. Note that X t X t (ε (solution of the equation (4 under condition C3 is continuosly differentiable w.r.t. ε. Denote the derivatives X (1 t X t ε, t X t ε, t T. ε x(1 The equations for X (1 t and x (1 t dx (1 t S (ϑ,t,x t X (1 t dt+ are [ ] εσ (t,x t X (1 t +σ(t,x t dw t, X (1 and dx (1 t S (ϑ,t,x t x (1 t dt+σ(t,x t dw t, x (1, (7 7

8 respectively. Hence x (1 t, t T is a Gaussian process and it can be written as x (1 t Let us denote { ψ(t exp We can write X t x t (ˆϑ ε ε { } exp S (ϑ,v,x v dv σ(s,x s dw s. (8 s } { } S (ϑ,v,x v dv, ψ ε (t exp S (ˆϑε,v,X v dv. X t x t (ϑ x t (ϑ x t (ˆϑε + ε ε X (1 t (ˆϑ ε ϑ,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(tv (t+o(1, where V (t ψ(t 1 x (1 t ψ(t 1 I(ϑ 1 Ṡ(ϑ,s,x s σ(s,x s dw s, ẋ t (ϑ. Introduce the random process U (ϑ,t ψ(s dv (s. σ(s,x s Lemma 1 We have the equality U (ϑ,t W t h(ϑ,sdw s, where the vector function h(ϑ,s ds, t T (9 h(ϑ,t I(ϑ 1/2 Ṡ(ϑ,t,x t σ(t,x t. (1 8

9 Proof. The solution of the equation (7 can be written (see (8 as x (1 t For the vector ẋ t (ϑ we can write ẋ t (ϑ The solution of this equation is ψ(tσ(s,x s dw s. ψ(s S (ϑ,s,x s ẋ s (ϑds+ ẋ t (ϑ ψ(t Introduce two stochastic processes and Then we can write U (ϑ,t v 1 (t ψ(t 1 x (1 t v 2 (t ψ(t 1 ẋ t (ϑ ψ(s dv (s σ(s,x s Ṡ(ϑ,s,x s ψ(s Ṡ(ϑ,s,x s ds. ds. ψ(s 1 σ(s,x s dw s, ψ(s 1 Ṡ(ϑ,s,x s ds. ψ(s σ(s,x s dv 1(s I(ϑ 1 Ṡ(ϑ,s,x s dw s, σ(s,x s W (t I(ϑ 1/2 Ṡ(ϑ,s,x s t dw s,i(ϑ 1/2 σ(s,x s W t h(ϑ,sdw s, h(ϑ,s ds. Let us introduce the random process and denote I 1 (ϑ u(ϑ,r T 1/2 U (ϑ,rt, r 1 Ṡ(ϑ,rT,x rt Ṡ(ϑ,rT,x rt σ(rt,x rt 2 dr, ψ(s σ(s,x s dv 2(s Ṡ(ϑ,s,x s σ(s,x s h(ϑ,r I 1 (ϑ 1/2 Ṡ(ϑ,rT,x rt σ(rt,x rt, w r T 1/2 W rt. 9 ds

10 Then we can write u(ϑ,r w r r h(ϑ,qdw q, h(ϑ,q dq, r 1, (11 and we have h(ϑ,q h(ϑ,q dq J. Note that u( is in some sense universal limit which appears in the problems of goodness of fit testing for stochastic processes. For example, the same limit we obtain in the case of ergodic diffusion process and in the case of inhomogeneous Poisson process [11]. The main difference with the i.i.d. case is due to the Wiener process here, when in i.i.d. case we obtain the Brownian bridge B(t, t 1 (see (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 parameter ϑ this change is not essential and will slightly modify the calculation of the test statistics for the second type test. The same time if the parameter ϑ is onedimensional, then we can easily construct the first-type goodness of fit test for stochastic processes and the construction of such test in i.i.d. case is not clear. The difference will be explained in the Section 3.2. In the construction of GoF test we will use one condition else. 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 diffusion process dx t S(t,X t dt+εσ(t,x t dw t, X x, t T. (12 We have to test the basic parametric hypothesis H, S(t,x S(ϑ,t,x, t T, ϑ Θ, i.e., that the observed process (12 has the stochastic differential dx t S(ϑ,t,X t dt+εσ(t,x t dw t, X x, t T (13 with some ϑ Θ. Here S(ϑ,t,x and σ(t,x are known smooth functions and Θ R d is an open convex set. We have to test this hypothesis in the asymptotics of small noise (as ε. 1

11 Our goal is to construct such statistics v ε [X ε ](, V ε [X ε ]( that (under hypothesis H δ ε ε v ε [X ε ](t 2 dt δ 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. Then introduce the tests ˆψ ε 1I {δε>d α}, ˆΨε 1I { ε>c α} with the thresholds c α and d α satisfying the equations P(δ > d α α, P( > c α α. (14 These tests will belong to the class } K α { ψε : lime ϑ ψε α, ϑ Θ ε and will be ADF. We propose these tests below in the sections 3.1 and 3.2, where we call ˆψ ε the first and ˆΨ ε the second test respectively. 3.1 First test The construction of the first ADF GoF test is based on the following well known property. Suppose that we have a Gaussian process U (t, t T satisfying the equation U (t w(t Introduce the process b(t h(s dw(s h(s du (s h(s dw(s It is easy to see that b( b(t and E [b(tb(s] s h(v 2 dv 11 h(sds, h(s dw(s h(v 2 dv h(s 2 ds 1. h(s 2 ds. s h(v 2 dv.

12 Then Let us put τ s δ h(v 2 dv, b(t B(τ, τ 1. ( 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 ( t ψ ε (s U ε (t σ(s,x s d X s x s (ˆϑ ε U (ϑ,t, t T, εψ ε (s where U (ϑ,t w(t Remind that h(ϑ,s dw(s h(ϑ,sds, h(ϑ,s I(ϑ 1/2 Ṡ(ϑ,s,x s σ(s,x s, I(ϑ Introduce (formally the statistic ( 2 ˆδ ε h(ˆϑε,s du ε (s h(ˆϑ ε,t 2 dt. If we prove that ( 2 h(ˆϑε,s du ε (s h(ˆϑ ε,t 2 dt h(ϑ,s 2 ds 1. Ṡ(ϑ,s,x s 2 σ(s,x s 2 ds. ( 2 h(ϑ,s du (ϑ,s h(ϑ,t 2 dt then the test ˆψ ε 1I {δε>c α} will be ADF. The main technical problem in the realization of this program is the definition of the stochastic integral h(ˆϑε,s du ε (s 12

13 which contains the MLE ˆϑ ε ˆϑ ε (X t, t T. We will proceed as follows: first we make the formal differentiation and integration and then we take the final expressions, which do not contain the stochastic integrals, as the starting statistics. Let us 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 Q ( ϑ,t,x t Xt Ṡ(ϑ,t,y I(ϑ σ(t,y 2 dy x Xs Ṡ x s(ϑ,s,yσ(s,y 2Ṡ(ϑ,s,yσ s(s,y dyds, I(ϑ σ(s,y 3 Ṡ(ϑ,s,X s D(ϑ,s,X s I(ϑ σ(s,xs 2 ds, 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 are fulfilled then the test ˆψ ε 1I {δε>c α}, P{δ > c α } α is ADF and belongs to K ε. Proof. We can write (formally ( ψ ε (s t U ε (t σ(s,x s dv ψ ε (s ε(s σ(s,x s d X s x s (ˆϑ ε ψ ε (sε dx s εσ(s,x s dx t s εσ(s,x s where we used the equality S(ˆϑ ε,s,x s (ˆϑ ε εσ(s,x s S (ˆϑ ε,s,x s + εσ(s,x s ( X s x s (ˆϑ ε ds D(ˆϑ ε,s,x s ds, (15 εσ(s,x s dx s (ˆϑ ε S(ˆϑ ε,s,x s (ˆϑ ε ds. 13

14 Hence (formally we obtain the following expression. h ε (ˆϑ ε,sdu ε (s Ṡ(ˆϑ ε,s,x s I(ˆϑ ε ε σ(s,x s 2 dx s Ṡ(ˆϑ ε,s,x s D(ˆϑ ε,s,x s I(ˆϑ ε εσ(s,x s 2 ds. 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 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. To avoid this difficulty in general case we replace the stochastic integral by it s robust version as follows. Introduce the function Then by Itô formula and therefore M (ϑ,t,x x Ṡ(ϑ,t,y x σ(t,y 2 dy. dm(ϑ,t,x t M t (ϑ,t,x t dt+ ε2 σ(t,x t 2 M xx 2 (ϑ,t,x t dt +M x(ϑ,t,x t dx t Ṡ(ϑ,s,X s σ(s,x s 2 dx s M(ϑ,t,X t [ ] t M s (ϑ,s,x s+ ε2 σ(s,x s 2 M xx 2 (ϑ,s,x s Xt + Ṡ(ϑ,t,y σ(t,y 2 dy x Xs x 2Ṡ(ϑ,s,yσ Xs Ṡ x s (s,y σ(s,y 3 Note that the contribution of the term ε 2 s (ϑ,s,y σ(s,y 2 dy ds ε2 2 σ(s,x s 2 M xx (ˆϑε,s,X s ds is asymptotically (ε negligeable. Therefore K ε (ϑ,t ε 1[ R ( ϑ,t,x t Q ( ϑ,t,x t] ds σ(s,x s 2 M xx(ϑ,s,x s ds. 14

15 is asymptotically equivalent to K ε (ϑ,t h ε (ϑ,s du ε (s. The difference is in the omitted term of order O(ε. We have to verify the convergence the integrals δ ε K ε (ˆϑ ε,t 2 Ṡ(ˆϑ ε,t,x t 2 I(ˆϑ ε σ(t,x t 2 dt K(ϑ,t 2 Ṡ(ϑ,t,x t 2 I(ϑ σ(t,x t 2 dt. sup t T sup t T The regularity conditions C1 C3 provide us the uniform convergences X t x t (ϑ, I(ˆϑ ε I(ϑ, h ε (ˆϑ ε,t h(ˆϑ ε,t sup Ṡ(ˆϑ ε,t,x t t T I(ˆϑ ε σ(t,x t Introduce two processes Y ε (ˆϑε,t,X t Z (ˆϑε,t,X t Then we can write We have Ṡ Ṡ(ϑ,t,x t I(ϑσ(t,xt. [ ] (ˆϑε,s,X s S(ϑ,s,X s D(ˆϑ ε,s,x s R (ˆϑε,t,X t K ε (t ε 1 [ Y ε (ˆϑε,t,X t +Z σ(s,x s 2 ds, Ṡ (ˆϑε,s,X s S(ϑ,s,X s σ(s,x s 2 ds. (ˆϑε,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 (ˆϑε ϑ Ṡ ( ϑ,s,xs [ ][ ] + S (ˆϑ ε,s, X s S (ˆϑ ε,s,x s X s x s (ˆϑ ε (ˆϑε ϑ Ṡ ( ϑ,s,xs +O ( ε 2. 15

16 Therefore 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 Further, where ε 1 (Z(ˆϑ ε,t,x t Z(ϑ,t,X t (ˆϑ ε ϑ ε Ż ( ϑ,t,x t +o(1, Ż ( ϑ,t,x t Xt S(ϑ,t,y σ(t,y 2 dy x Xs sup t T x S(ϑ,s,X s S(ϑ,s,X s σ(s,x s 2 ds S s (ϑ,s,yσ(s,y 2 S(ϑ,s,yσ s (s,y σ(s,y 2 dyds. We have uniforme w.r.t. t convergence of X t to x t, hence Ż( ϑ,t,x t Ż( ϑ,t,x t. Note that for any continuosly differentiable w.r.t. s function g(s,x we have the relation xt xs g(t,y dy g(s,x s S(ϑ,s,x s ds g s (s,ydyds x x because and g(t,x s dx s g(s,x s S(ϑ,s,x s ds 1I {v: xv>x s} g(s,x s dx s g(v,x s dv dx s v Hence the function Ż(ϑ,t,xt for all t [,T]. s g(s,x s dx s g(v,x s dv dx s v xv x g v (v,ydydv. 16

17 We have by 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 + ε 2 Ṡ(ϑ,s,X s S(ϑ,s,X s εσ(s,x s 2 ds σ(s,x s 2 M xx (ϑ,s,x sds Ṡ(ϑ,s,X s σ(s,x s dw s +O(ε. Ṡ(ϑ,s,X s S(ϑ,s,X s εσ(s,x s 2 ds Therefore we obtain the convergence K ε (t K(ϑ,t, and it can be shown that this convergence is uniform w.r.t. t. This proves the convergence δ ε δ. Therefore the Theorem 1 is proved. Let us study the behaviour of the power function under alternative. Suppose that the observed diffusion process (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 the equation (5. We introduce slightly more strong condition of separability of hypothesis and alternative. Let us suppose that the function S(t,x satisfies the conditions C1,C2 and denote by y t, t T the solution of the limit (ε ordinary differential equation dy t dt S(t,y t, y x. Then we have ( ε (X 1 t x t (ˆϑ ε ε 1 (X t y t +ε 1 y t x t (ˆϑ ε y (1 t +ε 1 (y t x t (ϑ ε (ˆϑε 1 ϑ ẋ t (ϑ +o(1, where y (1 t is solution of the equation dy (1 t S (t,y t y (1 t dt+σ(t,y t dw t, y (1 17

18 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. ϑ Θ σ(t,y t σ(t,y t (16 Supposethatthisequationhasaunique solutionϑ. Notethatε (ˆϑε 1 ϑ is tight (see [8] for details. Moreover, we suppose as well that the hypothesis and alternative are separated in the following sense ( 2 S(ϑ,t,yt S(t,y t inf dt >. ϑ Θ σ(t,y t First we write formally Ṡ(ˆϑ ε,s,x s h ε (ˆϑ ε,sdu ε (s dw s I(ˆϑ ε σ(s,x s [ ] Ṡ(ˆϑ ε,s,x s S(s,X s D(ˆϑ ε,s,x s ds I(ˆϑ ε εσ(s,x s 2 Further Ṡ(ϑ,s,y s t I(ϑ σ(s,y s dw s Ṡ(ϑ,s,X s [S(s,X s S(ϑ,s,X s ] ds. I(ϑ εσ(s,x s 2 S(s,X s D(ˆϑ ε,s,x s S(s,X s S(ˆϑε,s,x s (ϑ S (ˆϑε,s,X s (X s x s (ˆϑε S(s,X s S (ˆϑε,s,X s +O ( ε 2 S(s,X s S(ϑ,s,X s +S(ϑ,s,X s S (ˆϑε,s,X s +O ( ε 2 Therefore, S(s,X s S(ϑ,s,X s +(ˆϑε ϑ Ṡ(ϑ,s,X s +O ( ε 2. h ε (ˆϑ ε,sdu ε (s ε 1 Ṡ(ϑ,s,y s t 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 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 18

19 with obvious notations. For the statistic δ ε we have the relations δε ε 1 I 3 ( h( I 1 ( h( I 2 ( h( +O(ε, (17 where h( h(ϑ,s and is L 2 (,T norm. 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 ( T 2 Ṡ(ϑ,s,y s [S(s,y s S(ϑ,s,y s ] Ṡ(ϑ,t 2 I(ϑ σ(s,y s 2 ds 2 dt >. σ(t,y t This condition provides the consisteny of the test. Theorem 2 Let the conditions C1 C5 be fulfilled, then the test ˆψ ε is consistent. Proof. The proof follows from the convergence δ ε under alternative (see (17. 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 the condition C5 is not fulfilled then we have the relations Ṡ(ϑ,s,y s [S(s,y s S(ϑ,s,y s ] σ(s,y s 2 ds, for all t [,T]. This equality is possible if 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], i.e., 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 corresponding ϑ then the condition C5 is not fulfilled 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 such alternative. 19

20 3.2 Second test The second test is based on the following known transformation. Suppose that we are given a Gaussian process U(t, t 1 and d d matrix N(t defined by the relations U (t W t h(sdw s, N(t t h(s ds, (18 h(sh(s ds, N( J, (19 where J is d d unit matrix and h(t is some continuous vector function. Lemma 2 Suppose that the matrix N(t is non degenerate for all t [,1, then U (t+ s h(s N(s 1 h(vdu (v ds w(t, t 1, (2 where w( is some Wiener process. Proof. This formula was obtained by Khmaladze [6]. The proof in [6] is based un two results: one is of Hitsuda [4] and the other of Shepp [14]. There are many publications devoted to this transformation (see, for example, the paper [13] and references therein. Another direct proof is given in [7]. Note that from the representation (18-(19 it follows that h(sdu (s. (21 Suppose that the parameter 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 d d matrix. Introduce the following stochastic processes h ε (ϑ,t Ṡ(ϑ,t,X t σ(t,x t, N(ϑ,t N ε (ϑ,t t Ṡ(ϑ,s,X s Ṡ(ϑ,s,X s ds, σ(s,x s 2 t Ṡ(ϑ,s,x s Ṡ(ϑ,s,x s ds, σ(s,x s 2 and put ε 1 W T 2 ε (t 2 dt. 2

21 Here dx t s W ε (t εσ(s,x s D(ˆϑ ε,s,x s εσ(s,x s ds 1 [ ] +ε 1 h ε (ˆϑ ε,s Nε (ˆϑε,s R (ˆϑε,s,X s Q (ˆϑε,s,X s ds. We use the following convention for the matrix N We have the following result. N 1 + { N 1, if N is nondegenerate,, if N is degenerate. + (22 Theorem 3 Suppose that the conditions C2 C4 are fulfilled and the matrix N(ϑ,t is uniformly in ϑ Θ non degenerated for all t [,1. Then the test ˆΨ ε 1I { ε>c α}, is ADF and belongs to K α. ( P w(s 2 ds > c α α Proof. We have to show that under hypothesis H the convergence ε w(s 2 ds (23 holds. The construction of the ADF GoF test is based on the lemmae 1 and 2. We have the similar to (2 presentation (9 with h(ϑ,t defined in (1. Let us denote U ε (,h ε (ˆϑ ε,, and N ε ( the empirical versions of U (,h(ϑ, and respectively: N(ϑ,t I(ϑ 1 U ε (t t Ṡ(ϑ,s,x s Ṡ(ϑ,s,x s σ(s,x s 2 ds, N(ϑ, J, ψ ε (s σ(s,x s dv ε(s, V ε (t X t x t (ˆϑ ε, ψ ε (tε h ε (ˆϑ ε,t I ε (ˆϑ ε 1/2Ṡ(ˆϑ ε,t,x t σ(t,x t, I ε (ˆϑ ε N ε (ˆϑ ε,t I ε (ˆϑ ε 1 Ṡ(ˆϑ ε,s,x s Ṡ(ˆϑ ε,s,x s ds, σ(s,x s 2 t Ṡ(ˆϑ ε,s,x s Ṡ(ˆϑ ε,s,x s ds. σ(s,x s 2 21

22 Remind that there is a problem of the definition of the integral for U ε ( because the integrand depends on the future. As we have the uniform w.r.t. t [,T ν] convergence h ε (ˆϑ ε,t h(ϑ,t, I ε (ˆϑ ε I(ϑ, N ε (ˆϑ ε,t N(ϑ,t this allow us to prove the validity of the necessary limits. Introduce (formally the statistic W ε (t U ε (t+ Note that h(ϑ,s N(ϑ,s 1 h(ϑ,v h ε (ˆϑ ε,s N ε (ˆϑε,s 1 s + h ε (ˆϑ ε,vdu ε (v ds. (24 Ṡ(ϑ,s,x s σ(s,x s ( s Ṡ(ϑ,r,x r Ṡ(ϑ,r,x r 1 Ṡ(ϑ,v,x v σ(r,x r 2 dr σ(v,x v. Therefore this term does not depend on information matrix I(ϑ and we can replace in (24 the statistics h ε (ˆϑ ε,s and N ε (ˆϑε,s by h ε (ˆϑ ε,s and N ε (ˆϑε,s. For the process U ε ( we have the equality (15 (formally U ε (t Hence we obtain the vector-integral h ε (ˆϑ ε,sdu ε (s Introduce the vector-function Then by Itô formula Ṡ(ϑ,s,X s σ(s,x s 2 dx s dx t s εσ(s,x s D(ˆϑ ε,s,x s εσ(s,x s ds, Ṡ(ˆϑ ε,s,x s t εσ(s,x s 2 dx s M(ϑ,t,x Xt + x x Xs Ṡ(ϑ,t,y x σ(t,y 2 dy. Ṡ(ϑ,t,y σ(t,y 2 dy x Ṡ(ˆϑ ε,s,x s D(ˆϑ ε,s,x s εσ(s,x s 2 ds. Xs Ṡ x s (ϑ,s,y σ(s,y 2 ds 2Ṡ(ϑ,s,yσ s(s,y σ(s,y 3 ds+o ( ε 2. 22

23 Let us put K ε (t ] h ε (ˆϑ ε,sdu ε (s ε [R (ˆϑε 1,t,X t Q (ˆϑε,t,X t. Note that we omitted the term of order O(ε 2. All these allow us to replace the formal expression (24 for Wε (t by (22 dx t s W ε (t εσ(s,x s D(ˆϑ ε,s,x s εσ(s,x s ds 1 [ ] +ε 1 h ε (ˆϑ ε,s Nε (ˆϑε,s R (ˆϑε,s,X s Q (ˆϑε,s,X s ds. 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 + W t I(ϑ 1 U (ϑ,t+o(1. Here ϑ ϑ ˆϑ ε and + D(ˆϑ ε,s,x s εσ(s,x s ds 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, σ(s,x s ds [ S (ˆϑ ε,s, X s S (ˆϑ ε,s,x s εσ(s,x s Ṡ(ϑ,s,x s σ(s,x s dw s, ] (X s x s (ˆϑ ε ds Ṡ(ϑ,s,x s σ(s,x s ds +o(1 X xs xs s X s (ˆϑ ε 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, the similar arguments provide us the uniform w.r.t. t [,T] convergence 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

24 We have to show that K ε (t K(ϑ,t, where K(ϑ,t Let us denote Y ε (ˆϑε,t,X t h(ϑ,sdw s Ṡ Z (ˆϑε,t,X t R (ˆϑε,t,X t h(ϑ,sdw s h(ϑ,s h(ϑ,s ds. [ ] Ṡ (ˆϑε,s,X s S(ϑ,s,X s D(ˆϑ ε,s,x s σ(s,x s 2 ds, (ˆϑε,s,X s S(ϑ,s,X s σ(s,x s 2 ds. Then we can write [ ] K ε (t ε 1 Y ε (ˆϑε,t,X +Z (ˆϑε t,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 (ˆϑε ϑ,ṡ ( ϑ,s,xs [ ][ ] + S (ˆϑ ε,s, X s S (ˆϑ ε,s,x s X s x s (ˆϑ ε (ˆϑε ϑ,ṡ ( ϑ,s,xs +O ( ε 2. Therefore ε 1 Y ε (ˆϑε,t,X t Further, ε (Z (ˆϑε 1,t,X t where the matrix Ż ( ϑ,t,x t Xt x Xs (ˆϑε ϑ ε Ṡ(ˆϑε,s,X s Ṡ (ˆϑε,s,X s σ(s,x s 2 ds Z ( ϑ,t,x t ˆϑ ε ϑ Ż ( ϑ,t,x t +o(1 ε S(ϑ,t,y σ(t,y 2 dy x S(ϑ,s,X s S(ϑ,s,X s σ(s,x s 2 ds S s(ϑ,s,yσ(s,y 2 S(ϑ,s,yσ s(s,y σ(s,y 2 dyds. 24

25 Here S( is the matrix of the second derivatives w.r.t. ϑ. We have uniforme w.r.t. t convergence of X t to x t, hence Ż( ϑ,t,x t Ż( ϑ,t,x t. sup t T Note that for any continuosly differentiable w.r.t. s function g(s,x we have the relation xt xs g(t,y dy g(s,x s S(ϑ,s,x s ds g s(s,ydyds x x because and g(s,x s S(ϑ,s,x s ds g(s,x s dx s g(t,x s dx s 1I {v: xv>x s} g(s,x s dx s g(v,x s dv dx s v Hence the matrix Ż(ϑ,t,xt for all t [,T]. We have by Itô formula s g(v,x s dv dx s v xv x g v(v,ydydv. Z(ϑ,t,X t ε R(ϑ,t,Xt ε Ṡ(ϑ,s,X s εσ(s,x s 2dX s + ε 2 Ṡ(ϑ,s,X s S(ϑ,s,X s εσ(s,x s 2 ds σ(s,x s 2 M xx (ϑ,s,x sds Ṡ(ϑ,s,X s σ(s,x s dw s +O(ε. Ṡ(ϑ,s,X s S(ϑ,s,X s εσ(s,x s 2 ds Therefore, we obtain the convergence K ε (t ε (R (ˆϑε 1,t,X t Q (ˆϑε,t,X t ε (Y (ˆϑε 1,t,X t +Z (ˆϑε,t,X t K(ϑ,t. 25

26 Further, the matrix N ε (ˆϑ ε,s converges uniformly on s [,T] to the matrix N (ϑ,s. Therefore, we have uniform on s [,T ν] for ν > convergence of N ε (ˆϑ ε,s 1 + to N (ϑ,s 1. Introduce the random function y ε (s ε 1 hε (ˆϑ ε,s Nε (ˆϑε,s 1 It is shown that we have the convergence where + sup y ε (s y(ϑ,s, s T ν [ ] R (ˆϑε,s,X s Q (ˆϑε,s,X s. y(ϑ,s h(ϑ,s N(ϑ,s 1 K(ϑ,s. Hence we have as well the convergence for all t [,1 W ε (t U (ϑ,t+ h(ϑ,s N (ϑ,s 1 K(ϑ,sds w(t. By the similar way it can be shown that for any t 1 <... < t k T we have the convergence of the vectors (W ε (t 1,...,W ε (t k (w(t 1,...,w(t k. Further, direct but cumbersome calculations allow 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 the 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 [13] for similar integral. 26

27 4 Examples Example 1. We consider the simplest case which allow us to have the ADF GoF test for each ε, i.e., no need to study statistics as ε. Note that the similar situation is discussed in [6]. Suppose that the observed diffusion process (under hypothesis is Then we have Further Hence dx t ϑdt+εdw t, X, t 1. (25 h(ϑ,t 1, I(ϑ 1, N (ϑ,t 1 t, ˆϑ ε X 1, ε (ˆϑε 1 ϑ W 1 N (,1. 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 t B(t. W ε (t ε 1 (X t X 1 t+ε 1 (1 s 1 [X s X 1 s] ds and under hypothesis we have Therefore W ε (t B(t+ ε B(s 1 s W ε (t 2 dt ds w(t. w(t 2 dt and for the test ˆΨ ε 1I { ε>c α} K α we have the equality { } E ϑˆψε P B(t 2 dt > c α α Example 2. Let us 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 the functions H(t,x and σ(t,x satisfy the conditions of regularity. In this case we can take h ε (ϑ,t h ε (t, i.e.; this 27

28 vector-function does not depend on ϑ. Hence the stochastic integral is well defined and the test has a simplified form. We have h ε (t H(t,X t σ(t,x t, H(t,x t (ϑh(t,x t (ϑ Nε (ϑ,s s σ(t,x t (ϑ 2 ds, [ ] du ε (t dx ˆϑ ε,h(t,x t (ˆϑ ε + ˆϑ ε,h t εσ(t,x t x (t,x t (X t x t (ˆϑ ε dt, εσ(t,x t 1 s W ε (t U ε (t+ H(s,X s Nε (ˆϑε,s H(v,X v du ε (v ds and so on. References [1] Dachian, S. and Kutoyants, Yu.A. (27 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, [2] Darling, D. A. (1955 The Cramér-Smirnov test in the parametric case. Ann. Math. Statist., 26, 1-2. [3] Freidlin, M. I. and Wentzell, A. D.(1998 Random Perturbations of Dynamical Systems. 2nd Ed., Springer, N.Y. [4] Hitsuda, M. (1968 Representation of Gaussian processes equivalent to the Wiener process. Osaka J. Math., 5, [5] Iacus S. and Kutoyants Yu. A. (21 Semiparametric hypotheses testing for dynamical systems with small noise, Math Methods Statist, 1, 1, [6] Khmaladze, E. (1981 Martingale approach in the theory of goodness-offit tests. Theory Probab. Appl., 26, [7] Kleptsyna, M., Kutoyants Yu. A. (213 On asymptotically distribution free tests with parametric hypothesis for ergodic diffusion processes. To appear in Statist Inf Stoch Processes, arxiv: [8] Kutoyants, Yu.A. (1994 Identification of Dynamical Systems with Small Noise, Kluwer, Dordrecht. 28

29 [9] Kutoyants, Yu. A., (211 On goodness-of-fit tests for perturbed dynamical systems. Journal of Statistical Planning and Inference, 141, [1] Kutoyants, Yu. A., (213 On asymptotic distribution of parameter free tests for ergodic diffusion processes. To appear in Statist Inf Stoch Processes, arxiv: [11] Kutoyants, Yu. A., (213 On ADF goodness-of-fit tests for stochastic processes. To appear in Proceedings Applied Stochastic Modeling and Data Analysis, International Conference (ASMDA213. [12] Liptser, R. and Shiryaev, A.(21 Statistics of Random Processes. Vols. I+II, 2nd Ed., Springer. [13] Maglapheridze, N, Tsigroshvili, Z. P. and van Pul, M. (1998 Goodnessof-fit tests for parametric hypotheses on the distribution of point processes, Mathematical Methods of Statistics, 7, [14] Shepp, L. (1966 Radon-Nykodym derivatives of Gaussian measures. Ann. Math. Statist., 37, 2, [15] Yoshida, N. (1993. Asymptotic expansion of Bayes estimators for small diffusions. Probab. Theory Relat. Fields, 95, [16] Yoshida, N.(1996 Asymptotic expansions for perturbed systems on Wiener space: maximum likelihood estimators, J.Multivariate Analysis, 57,