A NON-PARAMETRIC TEST FOR NON-INDEPENDENT NOISES AGAINST A BILINEAR DEPENDENCE

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1 REVSTAT Statistical Journal Volume 3, Number, November 5, A NON-PARAMETRIC TEST FOR NON-INDEPENDENT NOISES AGAINST A BILINEAR DEPENDENCE Authors: E. Gonçalves Departamento de Matemática, Universidade de Coimbra, Portugal esmerald@mat.uc.pt P. Jacob Université de Montpellier II, France jacob@math.univ-montp.fr N. Mendes-Lopes Departamento de Matemática, Universidade de Coimbra, Portugal nazare@mat.uc.pt Received: March 5 Revised: June 5 Accepted: August 5 Abstract: A new methodology, based on the asymptotic separation of probability laws, was introduced by Gonçalves, Jacob and Mendes-Lopes in the development of the statistical inference of bilinear models, namely in the construction of a consistent decision procedure for the simple bilinear ones. This paper presents a generalisation of that study by introducing in the procedure a smoother decision statistics. The aim of this decision method is to discriminate between an error process and a simple bilinear model. So, we use it as a consistent test, its consistence being obtained by establishing the asymptotic separation of the sequences of probability laws defined by each hypothesis. The convergence rate of the procedure is studied under the truthfulness of the error process hypothesis. An exponential decay is obtained. Key-Words: time series; asymptotic separation; bilinear models; test. AMS Subject Classification: 6M1, 6F3.

2 156 E. Gonçalves, P. Jacob and N. Mendes-Lopes

3 Non-Parametric Test for Non-Independent Noises Against a Bilinear Dependence INTRODUCTION In Gonçalves, Jacob and Mendes-Lopes a new methodology of statistical decision to discriminate between an error process and a diagonal simple bilinear model was presented. This methodology was inspired in an asymptotic separation result obtained, in 1976, by Geffroy, which appeared particularly useful to construct consistent tests and estimators for detecting a signal in a white noise Pieczinsky, 1986, Moché, Let X X t, t Z be a real stochastic process whose law belongs to a set of parametric laws P θ, θ Θ, with Θ {θ 1, θ }. Following Geffroy 1976, we say that the two laws P θ1 and P θ are asymptotically separated if there exists a sequence of Borel sets of R T, A T, T N, such that Pθ T A T 1 T + Pθ T 1 A T, T + where P T θ denotes the probability law of X 1, X,..., X T. In this way, a consistent decision rule was defined and studied in Gonçalves, Jacob and Mendes-Lopes to separate the hypothesis H : X follows an error process against H 1 : X follows a diagonal bilinear model. With the aim of improving the rate of convergence of the decision procedure we present, in this paper, a generalisation of that study in which a smoother statistics is considered in the definition of the sequence of acceptance regions A T T N. In fact, unlike what we have considered in that pioneer study, the statistics here considered is, in general, a continuous function of the sample.. GENERAL PROPERTIES AND HYPOTHESES Let us consider the diagonal bilinear model X X t, t Z defined by 1 X t ϕx t 1 ε t 1 + ε t, where ϕ is a real number and ε ε t, t Z a real stochastic process. We are going to construct a decision procedure to discriminate between the hypotheses H : ϕ against H 1 : ϕ >, fixed. Let us denote the process X X t, t Z distribution and the corresponding expectation by P ϕ and E ϕ respectively, when the parameter of the model is equal to ϕ.

4 158 E. Gonçalves, P. Jacob and N. Mendes-Lopes We suppose that C 1 : ε ε t, t Z is a strictly stationary and ergodic process. C : E log ε t < + and E log ε t + log ϕ <. Under these conditions, model 1 has a strictly stationary and ergodic solution, P ϕ -a.s. unique, given by X t ε t + + n1 n 1 ϕ n ε t n j ε t 1 j a.s., t Z. If, in adition, we have C 3 : E log X t < + and E log X t + log ϕ <, then model 1 is invertible and ε t X t + + n1 n 1 ϕ n X t n j X t 1 j a.s., t Z. Under conditions C 1, C and C 3 we deduce, in view of the two equalities above, that X t ε t, X t and ε t denoting the σ-fields generated by X t, X t 1,... and ε t, ε t 1,... respectively. Hereafter we assume these general hypotheses concerning the stationarity, ergodicity, and invertibility of model 1. We also define the process Y Y t, t Z by n 1 Y t X t X t + ϕ n X t n X t 1 j a.s.. n1 This process is also strictly stationary and ergodic. We will denote it by Y t ϕ, if its dependence on the parameter ϕ is to be emphazised. We note that X t ϕy t 1 + ε t, according to 1. Otherwise, taking into account that E log ε t < + and E log X t < +, we have Y t ϕ, a.s., ϕ. j 3. A CONSISTENT TEST We are going to construct a decision procedure to distinguish, in model 1, the hypotheses H : ϕ against H 1 : ϕ > from T observations of the process X, denoted by x 1, x,..., x T.

5 Non-Parametric Test for Non-Independent Noises Against a Bilinear Dependence 159 The procedure we are proposing is based, as referred above, on the notion of asymptotic separation of two families of probability laws Geffroy 198, Moché 1989 and it generalises recent works as, for instance, Gonçalves, Jacob, Mendes- Lopes, Gonçalves, Martins, Mendes-Lopes 1. First of all, we establish the asymptotic separation of the families of probability laws associated to the hypotheses under investigation by presenting a sequence of Borel sets of R T, A T, T N, called separation regions, such that P T A T 1 T + P T A T T + denoting by Pϕ T the probability law of X 1,..., X T when the parameter is equal to ϕ. We will accept H : ϕ against H 1 : ϕ > if x 1,..., x T A T. The separation regions that we are going to propose are inspired in previous works. In those papers the test takes into account the number of times that u u v > when u, v y t 1, x t, y t denoting the particular value of Y t, t 1,..., T. So, the set D {u, v R : u >, v < } u {u, v R : u <, v > } u { } u, v R : u u v > is very important in the construction of a convergent test for the same hypotheses. The generalization here studied consider a test statistic which is defined following the same basical idea but using a smoother function, eventually a continuous one. From the definition of D we have y t 1, x t D y t 1 >, y t 1 x t > or y t 1 <, y t 1 x t <. So, if we consider a distribution function F of a symmetrical law we have y t 1, x t D F y t 1 1, F y t 1 x t 1 or F y t 1 1, F y t 1 x t 1 F y t 1 1] ] F y t 1 x t 1. The study here presented takes into account this product. Moreover, a great degree of generality is achieved as the distribution function considered in

6 16 E. Gonçalves, P. Jacob and N. Mendes-Lopes the first factor may be different from that appearing in the second one. So, let us define gu, v Gu 1] ] F u v 1, u, v R, where F and G are distribution functions of symmetrical laws with decreasing densities on R +. Let us consider the following regions { T A T x 1, x,..., x T R T : g } y t 1, x t. t In what follows, we take g t gy t 1, x t and g T 1 T T t g t, and we assume the hypothesis: C 4 : the conditional distribution of ε t given ε t 1 is symmetrical. We have the following result: Lemma 3.1. i Under the hypothesis ϕ, lim g T E g >. T ii Under the hypothesis ϕ >, lim g T E g <. T Proof: By the ergodic theorem we have lim T g T E ϕ g a.s., with E ϕ g E ϕ g Y 1, X E ϕ G Y 1 1] F ] Y 1 X 1. Let us now study the sign of the limit under each one of the hypotheses H and H 1. In what follows, we take Y 1 Y 1. Under ϕ we have X ε and so ] E g E G Y 1 1] F Y 1 ε 1 ] } E G Y 1 1] E {F Y 1 ε 1 ε 1 ] } E G Y 1 1] I {Y1 >}E {F Y 1 ε 1 ε 1 + ] } +E G Y 1 1] I {Y1 <}E {F Y 1 ε 1 ε 1.

7 Non-Parametric Test for Non-Independent Noises Against a Bilinear Dependence 161 When Y 1 >, we have G Y 1 1 > and E {F ] } Y 1 ε 1 ε1 > using the symmetry of the law ] of } -ε t given ε t 1 ; if Y 1 < then G Y 1 1 < and E {F Y 1 ε 1 ε1 <. So, E g >. Under ϕ > we have Y 1 X 1 ε 1, X X 1 ε 1 + ε and then E g E G X 1 ε 1 1] F E G X 1 ε 1 1] E {F X 1ε 1 X 1 ε 1 ε X 1ε 1 ε 1 ] 1 ] } ε 1 E G X 1 ε 1 1] I {X1 ε 1 >}E {F X 1ε 1 ε +E G X 1 ε 1 1] I {X1 ε 1 <}E {F As previously, G X 1 ε 1 1 > and E {F X 1ε 1 ε 1 ] } 1 ε 1 ] } ε 1 ] } X 1ε 1 ε 1 ε1 <, when X 1 ε 1 > ; on the other ] hand, } if X 1 ε 1 <, G X 1 ε 1 1 < and E {F X 1ε 1 ε 1 ε1 >. Then E g <.. We immediately deduce, by the bounded convergence theorem, the following result: Corollary 3.1. i If ϕ, P g T 1, as T +. ii If ϕ >, P g T, as T +. Taking into account the definition of regions A T, we conclude that the probability laws of process X t, t Z defined by the hypotheses H : ϕ and H 1 : ϕ > are asymptotically separated. So, A T is the acceptance region of a consistent test for these hypotheses. 4. CONVERGENCE RATE OF THE DECISION PROCEDURE The convergence rate of the decision procedure, presented in the previous paragraph as a test, may be evaluated when we consider, in the acceptance regions A T, the true value of Y t, i.e., Y t ϕ, and we assume that the null hypothesis is true. Let us denote these borelians by A T ϕ.

8 16 E. Gonçalves, P. Jacob and N. Mendes-Lopes We are going to evaluate the convergence rate of P AT ϕ. We have P AT ϕ T P g t < t ] T E exp g t t { ]} T E E exp g t ε T 1 t T 1 ] E {exp } g t E exp gt ε T 1 t Firstly, we study E gt ε t 1 ], t Z. E gt ε t 1 ] G ε t 1 1 ] { E F } ε t 1 ε t ε t 1] 1. Let us suppose that ε verifies the following condition C 5 : ε t η t 1 Z t, t Z where η t is a measurable and strictly positive function of ε t, ε t 1,... with < m η t M and Z t, t Z is a sequence of independent and identically distributed real random variables, with distribution function F and density f that we suppose symmetrical and decreasing on R +. We also assume that Z t is independent of ε t 1. So, E F ε t 1 ε t ε t 1] E F ε t 1 η t 1 Z t ε t 1] F ε t 1 η t 1 u fu du F ε t 1 Mu fu du. Choosing the function G v F v Mu fu du, we note that, by lemma 5.1 in the appendix, G is the distribution function of a law with a symmetrical density, decreasing on R +. Moreover, we obtain E F ε t 1 ε t ε t 1] G ε t 1

9 Non-Parametric Test for Non-Independent Noises Against a Bilinear Dependence 163 and so E gt ε t 1 ] G ε t 1 1 ]. From Hoeffding inequality Hoeffding 1953, E e g t ε t 1 ] e E g t ε t 1 ]+ 1 Gε t 1 1] e 1 Gε t 1 1]. Then T 1 P A T E {exp g t exp t T E {exp g t t 1 ] } G ε T 1 1 E exp g T 1 exp 1 Gε T 1 1 ] ] } ε T. From lemma 5.5 see appendix, we have the following inequality, for every t Z, E {exp g t 1 exp 1 ] } G ε t 1 1 ε t ] E exp gt 1 ε t E exp 1 ] ] G ε t 1 1 ε t. In fact, i given ε t, g t 1 G ε t 1 ] F ] ε t t 1 x 1 has the form of the function hx c R a dx 1] presented in lemma 5. see the appendix, as x t 1 η t Z t 1 under H and c G ε t 1 1 >, R F, a ε t 1 >, and d η t >. ii On the other hand, 1 G d x 1 ] 1 G d x G d x ] is a symmetrical function, increasing on R +, null in the origin and bounded. As Z t 1 is independent of ε t, the inequality takes the form ] E exp h Z t 1 g Z t 1 ] ] E exp h Z t 1 E exp g Z t 1. We can then write, with u T exp P AT ϕ g t, T { ] E u T E exp gt 1 ε T E exp 1 ] ]} G ε T 1 1 εt t

10 164 E. Gonçalves, P. Jacob and N. Mendes-Lopes { E E exp { E E exp g t ε T E exp 1 ] ] ]} G ε T 1 1 εt T 1 t g t ε T E exp 1 G m ZT ] ] ]} 1 1 εt. T 1 t But E exp 1 G m ZT ] ] 1 1 εt is constant as Z t 1 is independent of ε T, t Z. So, P AT ϕ E exp 1 ]] G m ZT 1 1 E exp ] g t ε T. T 1 t Using recursively the procedure leading to ] ] T T 1 E exp g t ce exp g t t t we obtain P AT ϕ {E exp 1 G m Z 1 T 1 ]]} where Z is a random variable with the same law of Z t. Finally, we may state the following result: Theorem 4.1. Let X X t, t Z be a real stochastic process satisfying the model 1 subject to the general conditions C 1, C and C 3. If the error process satisfies condition C 5 and the function G is defined by G v F v Mu fudu then the proposed decision rule satisfies P A T ϕ 1 {E exp 1 G m Z 1 ]]} T 1, T N. 5. APPENDIX The convergence rate study has been developped assuming absolute continuity and symmetry of the distribution laws involved. So, in this appendix we establish several lemmas concerning distribution functions of symmetrical densities.

11 Non-Parametric Test for Non-Independent Noises Against a Bilinear Dependence 165 Lemma 5.1. Let f be a symmetrical density decreasing on R + with distribution function F. Let a and b be fixed real numbers, with a >. Then the function G defined by G v F av bu f u du is the distribution function of a law with symmetrical density g decreasing on R +. Proof: As we can differentiate under the integral sign Métivier, 197, p. 156 we obtain d dv G v af av bu f u du. Then, as f is symmetrical, d dv G v d dv G v. af av bu f u du af av + bu f u du af av by f y dy af av bu f u du Denoting d dv G g, g is a symmetrical function. Let us prove that g is a density function and G the distribution function of density g. From Fubini, we obtain dv af av bu f u du fu af av bu dv du. But Then af av bu dv dv af z 1 a dz 1. af av bu f u du 1.

12 166 E. Gonçalves, P. Jacob and N. Mendes-Lopes On the other hand, again from Fubini, x x gvdv af av bu du dv G v. x fu fu af av bu dv ax bu F ax bu fudu ] du af z 1 a dz ] du From the definition of g and as f is decreasing on R +, it is obvious that g is decreasing on R +. Lemma 5.. Let h x c R a dx 1], x R, where c, a, d are positive numbers and R is the distribution function of a symmetrical and decreasing on R + density, r. Let Hx e hx. Then Hx + H x is increasing on R +. Proof: We have d Hx + H x] d e hx + e h x] dx dx h xe hx + h xe h x] cdr a dx e hx cdr a + dx e h x. Let us show that this derivative is non negative. As c and d are positive, it is enough to show that { r a dx r a + dx, x e hx e h x, x. As a > and d > and r is decreasing on R +, we have r a dx r a + dx, for every x such that a dx >. But, as r is symmetrical, r is increasing on R and if a dx < we have as dx a < dx + a. r a dx r dx a r a + dx, Moreover, as r is a symmetrical density, the function R x 1 R x R x

13 Non-Parametric Test for Non-Independent Noises Against a Bilinear Dependence 167 is odd and obviously increasing on R +. As c and d are positive we can conclude, by an analogous way, that for every x c R a dx 1 ] c R a + dx 1 ] that is and, in consequence, hx h x e hx e h x. Lemma 5.3. Let ϕ and f be two symmetrical densities and a > such that ϕ > f on, a and ϕ < f on ]a, +. Let T be a positive and increasing function, defined on R +. Then ϕ x T x dx < f x T x dx. Proof: We have ϕ x f x] T x dx ϕ x f x] T x dx +,a ]a,+ ϕ x f x] T x dx < T a ϕ x f x] dx + T a + ϕ x f x] dx,a ]a,+ as T is an increasing function and where T a denotes the left limit and T a + the right limit on a. As the first quantity is positive, we have ϕ x f x] T x dx < T a + ϕ x f x] dx, ],+ taking into account that ϕ and f are symmetrical densities. Lemma 5.4. Let h be the function of lemma 5., ϕ and f the probability densities of lemma 5.3 and Y and Z real random variables with densities f and ϕ, respectively. Then E e hz] E e hy ].

14 168 E. Gonçalves, P. Jacob and N. Mendes-Lopes Proof: We have E e hz] E e hz ϕ z dz e hz ϕ z dz e h u ϕ u du e h Z]. Then, with H x e hx, H x ϕ x dx as ϕ is symmetrical. In the same way, we have E e hy ] H x ϕ x dx H x + H x ϕ x dx Hx+H x ϕx dx + H x + H x ϕ x dx, H x + H x] f x dx. Hx+H x ϕx dx As, by lemma 5., the function H x + H x is increasing on R +, we can apply lemma 5.3 to obtain ϕ x H x + H x] dx < f x H x + H x] dx, that s to say, E e hz] < E e hy ]. Lemma 5.5. Let g be a symmetrical function, increasing on R +, equal to zero in the origin and bounded. Let Y be a real random variable with a symmetrical and decreasing on R + density f. Let h be the function of lemma 5.. Then E e gy hy ] < E e gy ] E e hy ].

15 Non-Parametric Test for Non-Independent Noises Against a Bilinear Dependence 169 Proof: Let us take 1 b e gx f x dx. We note that b > 1, as e g < 1 almost everywhere. We consider ϕ x be gx f x. Then ϕ is a symmetrical density. On the other hand, as b > 1 and g, we obtain ϕ be g f > f. Moreover ϕ x f x be gx 1. As g is monotone increasing, there is a unique root a > such that ϕ > f in, a and ϕ < f in ]a, +. Let Z be a real random variable with density ϕ. From lemma 5.4 we have E e hz] hy E e ] e hx ϕ x dx e hx f x dx. As ϕ x be gx f x, we obtain b or, using the b definition, which is equivalent to e hx e gx f x dx e hx e gx f x dx e hx f x dx e hx f x dx E e gy hy ] E e gy ] E e hy ]. e gx f x dx

16 17 E. Gonçalves, P. Jacob and N. Mendes-Lopes ACKNOWLEDGMENTS The authors thank the referee for providing useful comments and suggestions. REFERENCES 1] Geffroy, J Inégalités pour le niveau de signification et la puissance de certains tests reposant sur des données quelconques, C. R. Acad. Sci. Paris, Ser. A, 8, ] Geffroy, J Asymptotic separation of distributions and convergence properties of tests and estimators. In Asymptotic Theory of Statistical Tests and Estimation I.M. Chakravarti, Ed., Academic Press, ] Gonçalves, E.; Jacob, P. and Mendes-Lopes, N A new test for arma models with a general white noise process, Test, 51, ] Gonçalves, E.; Jacob, P. and Mendes-Lopes, N.. A decision procedure for bilinear time series based on the asymptotic separation, Statistics, 33, ] Gonçalves, E.; Martins, C.M. and Mendes-Lopes, N. 1. Asymptotic separation-based tests for noise processes against first order diagonal bilinear dependence, Annales de l ISUP, 45 3, ] Granger, C.W.J. and Andersen, A An Introduction to Bilinear Time Series Models, Vandenhoeck and Ruprecht, Göttingen. 7] Hoeffding, W Probability inequalities for sums of bounded random variables, J. Amer. Stat. Assoc., 581, ] Métivier, M Notions Fondamentales de la Théorie des Probabilités, nd ed., Dunod, Paris. 9] Moché, R Quelques tests relatifs à la detection d un signal dans un bruit Gaussian, Publications de l IRMA, Univ. Lille, France, 19 III, ] Pieczinsky, W Sur diverses applications de la décantation des lois de probabilité dans la théorie générale de l estimation statistique, Thèse de Doctorat d État, Univ. Paris VI. 11] Pham, T.D. and Tran, L.T On the first order bilinear time series model, J. Appl. Prob., 18, ] Quinn, B.G Stationarity and invertibility of simple bilinear models, Stoch. Processes and their Applications, 1, 5 3.