Weighted Cramér-von Mises-type Statistics

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1 Weighte Cramér-von Mises-type Statistics LSTA, Université Paris VI 175 rue u Chevaleret F 7513 Paris, France ( p@ccr.jussieu.fr) Paul Deheuvels Abstract. We consier quaratic functionals of the multivariate uniform empirical process. Making use of Karhunen-Loève expansions of the corresponing limiting Gaussian processes, we obtain the asymptotic istributions of these statistics uner the assumption of inepenent marginals. Our results have irect applications to tests of gooness of fit an tests of inepenence by Cramér-von Mises-type statistics. AMS 2 classification: 6F5, 6F15, 6G15, 62G3. Keywors: Cramér-von Mises tests, tests of gooness of fit, tests of inepenence, weak laws, empirical processes, Karhunen-Lo eve ecompositions, Gaussian processes, Bessel functions. 1 Introuction an Premiminaries. 1.1 Introuction. In this paper, we survey some recent results ([14, 15, 13]) relate to quaratic functionals of the form t 2β1 1...t 2β α 2 n,(t 1,...,t )t 1...t, (1) where α n, is an appropriate version of the uniform empirical process on [, 1] (see (36) in the sequel for explicit efinitions). We first establish conitions on the β 1,..., β, uner which the statistic in (1) converges to a quaratic functional of a Gaussian process, of the form t 2β1 1...t 2β B 2 (t 1,...,t )t 1...t, (2) with B enoting a tie-own Brownian brige. Secon, we will characterize the istribution of the ranom variable in (2), through a Karhunen-Loève expansion of the corresponing weighte Gaussian process. This problem has been initiate by Cramér [1] (see, e.g., Nikitin [26], Scott [32] an the references therein). In higher imensions, we refer to Blum, Kiefer an Rosenblatt [6], Cotterill an Csörgő [8, 9], Deheuvels [13], Dugué [17, 18, 19], Hoeffing [2], Kiefer [24], Martynov [27], an Smirnov [33, 35,

2 24 Deheuvels 34]. Quaratic functionals of Gaussian processes have been stuie by Biane an Yor [5], Donati-Martin an Yor [16], Pitman an Yor [28, 29, 3, 31], an Yor [37, 38]. The results of Deheuvels an Martynov [14], an Deheuvels, Peccati an Yor [15], Deheuvels [13], give the core of the present survey paper. The theory of Bessel functions plays here an essential role an we refer to Bowman [7] an Watson [36] for etails. In 1.2 an 1.3, we give some preliminaries. We escribe the univariate case in 2.1 an the multivariate case, with 2, in Some Preliminaries on Gaussian Processes. Let {X(t) : t [, 1] } enote a centere Gaussian process, with 1. We set s = (s 1,..., s ) R an t = (t 1,..., t ) R, an set R(s,t) = E ( X(s)X(t) ) for s,t [, 1]. (3) We will are concerne with the quaratic functional [,1] X 2 (t)t, (4) where t is the Lebesgue measure. We will work uner the assumption that ( ) < E X 2 (t)t = R(t,t)t <. (5) [,1] [,1] The conition (5) entails that, almost surely, X( ) L 2( [, 1] ) belongs to the class of Hilbert space value centere Gaussian processes (see, e.g., 1 in Lifshits [25]). By the Cauchy-Schwarz inequality, for each s,t [, 1], R(s,t) 2 = E ( X(s)X(t) ) 2 E ( X(s) 2 ) E ( X(t) 2) = R(s,s)R(t,t). When combining this last inequality with (5), we obtain that { } 2 R 2 L := R(s,t) 2 st R(t,t)t <, (6) 2 [,1] [,1] [,1] so that R L 2( [, 1] [, 1] ). Uner (6), the Freholm transformation y( ) L 2( [, 1] ) ỹ( ), efine by ỹ(t) = R(s,t)y(s)s for t [, 1], (7) [,1] is a continuous linear mapping of L 2( [, 1] ) onto itself. The conition (6) also implies the existence of a convergent orthonormal sequence [c.o.n.s.],

3 Weighte Cramér-von Mises-type Statistics 25 {λ k, e k ( ) : 1 k < K} with the following properties. {λ k : 1 k < K} are positive constants an K {2,..., } a possibly infinite inex, with λ 1... λ k... >. (8) The {e k ( ) : 1 k < K} are orthonormal in L 2( [, 1] ), an fulfill { 1 if k = l, e k (t)e l (t)t = [,1] if k l. The function R may be ecompose into the series R(s,t) = λ k e k (s)e k (t), (9) convergent in L 2( [, 1] ). This entails that R L 2 = R(s,t) 2 st = [,1] [,1] λ 2 k <. (1) The λ k (resp. e k ( )) are the eigenvalues (resp. eigenfunctions) of the Freholm operator (7), an fulfill the relations, for each 1 k < K, ẽ k (t) = R(s,t)e k (s)s = λ k e k (t). (11) [,1] The Karhunen-Loève [KL] ecomposition of X( ), (see, e.g., Kac an Siegert [23, 22], Kac [21], Ash an Garner [4], an Aler [2]) ecomposes X( ) into X(t) = Y k λk e k (t), (12) where {Y k : 1 k < K} are inepenent an ientically istribute [i.i..] normal N(, 1) ranom variables. Uner (5), the series in (12) is convergent in mean square, since this conition is equivalent to ( ) < E X 2 (t)t = λ k <. (13) [,1] This, in turn, reaily implies that, as k K with k < K, ( { k } 2t ) E X(t) Y m λm e m (t) = λ k. [,1] m>k m=1 The conition (5) (13) is strictly stronger than (1). It implies that the quaratic functional (4) can be ecompose into the sum of the series X 2 (t)t = λ k Yk 2. (14) [,1] The latter is almost surely convergent if an only if (5) hols. Therefore, we will assume, from now on, that this conition is satisfie.

4 26 Deheuvels 1.3 A General Convergence Theorem. With R(, ) as in (3), we consier inepenent replicæ ξ 1 ( ), ξ(2),... of a general stochastic process ξ( ), fulfilling (H.1 2 3) below. (H.1) ξ( ) L 2( [, 1] ) ; (H.2) E ( ξ(t) ) = for all t [, 1] ; (H.3) E ( ξ(s)ξ(t) ) = R(s,t) for all s,t [, 1]. Uner (H.1 2 3) (see, e.g., Ex. 14, p. 25 in Araujo an Giné [3]), as n, the convergence in istribution n ζ n ( ) := n 1/2 ξ i ( ) hols if an only if (5) (13)) is satisfie, namely, when E ( ξ 2 (t) ) t = R(t,t)t <. [,1] [,1] i=1 We have therefore the following theorem. X( ), (15) Theorem 1 Uner (5) an (H.1 2 3), we have, as n, the convergence in istribution ζn 2 (t)t λ k Yk 2. (16) [,1] Proof. Uner (5) (or equivalently (13)), it follows from (15) that ζn(t)t 2 X 2 (t)t, [,1] [,1] which, in turn, reuces (16) to a irect consequence of (15). Below, we provie some useful statistical applications of Theorem 1. 2 Weighte Empirical Processes. 2.1 The Univariate Case ( = 1). Let U 1, U 2,... be i.i.. uniform [, 1] ranom variables. For n 1, set F n (t) = 1 n 1I {Un t}, (17) n i=1 for the empirical istribution function [f] base upon U 1,...,U n, an let α n (t) = n 1/2{ F n (t) t } for t [, 1], (18)

5 Weighte Cramér-von Mises-type Statistics 27 enote the uniform empirical process. Fix β R, an set, for n 1, ξ n (t) = t β{ 1I {Un t} t } for t [, 1]. (19) We let t = 1 for all t R, when β =. In agreement with (15), (18), (19), an the notation of 1.3, we may write n ζ n (t) = n 1/2 ξ i (t) = t β α n (t) for t [, 1]. (2) i=1 The assumptions (H.1 2 3) in 1.3 are fulfille with R efine by R(s, t) = s β t β{ s t st } for s, t [, 1]. (21) For this choice of R, (5) (13) hol if an only if 1 t 2β{ t(1 t) } t <, (22) which is equivalent to β > 1. Now, since s t st is the covariance function of a stanar Brownian brige {B(t) : t [, 1]}, the kernel R in (21) is nothing else but the covariance function of the weighte Brownian brige X(t) = t β B(t) for t (, 1]. (23) Deheuvels an Martynov [14] have given the KL ecomposition of X( ) in (23) when β 1 ν = 1/(2(1 + β)) >. For ν R, we the first Bessel function (see, e.g., in Abaramowitz an Stegun [1]) is J ν (x) = ( 1 2 x)ν k= ( 1 4 x2 ) k Γ(ν + k + 1)Γ(k + 1). (24) Whenever ν > 1, the positive zeros of J ν are isolate an form an infinite increasing sequence {z ν,k : k 1}, such that (see, e.g., Watson [36]) an, as k, < z ν,1 < z ν,2 <..., (25) z ν,k = { k (ν 1 2 )} + o(1). (26) Given this notation, Theorem 1.4 in [14] asserts that, whnever β > 1, the KL representation of X(t) = t β B(t) is given by X(t) = t β B(t) = Y k λk e k (t), (27) k=1

6 28 Deheuvels where {Y k : k 1} are i.i.. normal N(, 1) ranom variables, an λ k = { 2ν z ν,k } 2, k = 1, 2,..., (28) { e k (t) = t 1 Jν (z 2ν 1 ν,k t 1 2ν ) } 2 ν Jν 1 (z ν,k ) for < t 1. (29) Refer to Deheuvels an Martynov [14] for etails. We get the theorem: Theorem 2 For any β > 1, setting ν = 1/(2(1+β)), we have, as n, the convergence in istribution 1 t 2β α 2 n(t)t 1 t 2β B 2 (t)t = k=1 { 2ν z ν,k } 2Y 2 k, (3) where {Y k : k 1} is an i.i.. sequence of normal N(, 1) ranom variables. Proof. In view of (28) (29), it is a irect consequence of Theorem The Multivariate Case ( 2). We now let 2. When s = (s 1,..., s ) R an t = (t 1,..., t ) R, we enote by s t the fact that s j t j for j = 1,...,k, an set, accoringly, s t = ( s 1 t 1,...,s t ). Letting U = (U(1),...,U()) [, 1] be uniformly istribute on [, 1], we let U n = (U n (1),..., U n ()) [, 1], n = 1, 2,... be i.i.. replicæ of U. For each n 1, the empirical f base upon U 1,...,U n is enote by We enote by F n (t) = 1 n n 1I {Ui t}, (31) i=1 F(t) = P ( U t ) = the (exact) istribution function of U, an set t j, (32) α n (t) = n 1/2( F n (t) F(t) ) for t [, 1], (33) for the corresponing uniform empirical process. Making use of 1.3, we obtain that the following convergence in istribution hols. As n, α n ( ) B( ), (34)

7 Weighte Cramér-von Mises-type Statistics 29 where {B(t) : t [, 1] } is a stanar multivariate Brownian brige. Namely, B( ) is a centere Gaussian process, with covariance function E ( B(s)B(t) ) = E ( α n (s)α n (t) ) = { } { } sj t j sj t j. (35) The KL ecomposition of B( ), with covariance function as in (35), is not known explicitly for 2. A more tractable tie-own empirical process α n, ( ) is as follows. Set α n, (t) = α n (t) + 1 j<l 1 j t j α n (t 1,..., t j 1, 1, t j+1,..., t ) t j t l α n (t 1,..., t j 1, 1, t j+1,..., t l 1, 1, t l+1,...,t ) (1) t 1... t α n (1,...,1). (36) In (36), α n (1,..., 1) =, but this term is state for convenience. In view of 1.3, we obtain the following convergence in istribution. As n, α n, ( ) B ( ), (37) where {B (t) : t [, 1] } is a tie-own multivariate Brownian brige. Namely, B ( ) is a centere Gaussian process, with covariance function E ( B (s)b (t) ) = { } sj t j s j t j. (38) We have the following easy consequence of the results of Deheuvels an Martynov [14] (see also Deheuvels, Peccati an Yor [15]). Theorem 3 Let β 1,..., β be constants such that β j > 1 for j = 1,...,. Set ν j = 1/(2(1 + β j )) > for j = 1,...,. Then, the Karhunen-Loève ecomposition of the centere Gaussian process is given by X(t) = X(t) = t β tβ B (t) for t (, 1], (39)... k 1=1 λk1,...,k Y k1,...,k e k1,...,k (t), (4) k =1

8 3 Deheuvels where an λ k1,...,k = e k1,...,k (t) = =: { 2νj } 2 =: L(ν j, k j ), (41) z νj,k j [ t 1 2ν 1 j 2 j 1 { 2ν Jνj (z νj,k t j j ) νj J νj 1(z νj,k) } ] E(ν j, t j ). (42) Proof. By (38) the covariance function of X(t) in (39) is given by R(s,t) = s βj j tβj j { } sj t j s j t j =: R(s j, t j ). (43) Therefore, via (28) (29), λ k1,...,k is an eigenvalue of the Freholm operator (7) pertaining to e k1,...,k ( ). To conclue, we show that all eigenvalues are so obtaine. For this, we combine (1) with (43), to write that = R(s,t) 2 st =... [,1] j 1=1 j =1 {... L(ν j, k j ) 2} =... [,1] j 1=1 j =1 k k 1=1 1 1 R(s j, t j ) 2 s j t j λ 2 k 1,...,k. k =1 This shows that there is no other remaining eigenvalue of (7). The next theorem is an easy consequence of the preceing results. Theorem 4 Let β 1,..., β be constants such that β j > 1 for j = 1,...,. Set ν j = 1/(2(1 + β j )) > for j = 1,...,. Then, we have, as n, = [,1] t 2β t 2β α 2 n,(t)t... k 1=1 { k =1 [,1] t 2β t 2β B 2 (t)t { 2νj z νj,k j } 2 } Y 2 k 1,...,k, (44) where {Y k1,...,k : k 1 1,...,k 1} is an i.i.. array of normal N(, 1) ranom variables. The limiting istribution in Theorem 4 coincies with that of the Blum- Kiefer-Rosenblatt statistic (see, e.g., [6]), when = 2 an β 1 =... = β =.

9 Weighte Cramér-von Mises-type Statistics 31 Conclusion. For 2, the eigenvalues λ k1,...,k in the KL ecomposition (41) (42) are multiple. This reners the numerical computation of the limit istribution of the test statistic in (44) more elicate than in the univariate case. This problem will be investigate elsewhere. References 1.Abramowitz, M. an Stegun, I. A. (1965). Hanbook of Mathematical Integrals. Dover, New York. 2.Aler, R. J. (199). An Introuction to Continuity, Extrema an Relate Topics for General Gaussian Processes. IMS Lecture Notes-Monograph Series 12. Institute of Mathematical Statistics, Haywar, California. 3.Araujo, A. an Giné, E. (198). The Central Limit Theorem for Real an Banach Value Ranom Variables. Wiley, New York. 4.Ash, R. B. an Garner, M. F. (1975). Topics in Stochastic Processes. Acaemic Press, New York. 5.Biane, P. an Yor, M. (1987). Variations sur une formule e P. Lévy. Ann. Inst. Henri Poincaré Blum, J. R., Kiefer, J. an Rosenblatt, M. (1961). Distribution-free tests of inepenence base on the sample istribution function. Ann. Math. Statist Bowman, F. (1958). Introuction to Bessel Functions. Dover, new York. 8.Cotterill, D. S. an Csörgő, M. (1982). On the limiting istribution an critical values for multivariate Cramér-von Mises statistic. Ann. Math. Statist Cotterill, D. S. an Csörgő, M. (1985). On the limiting istribution an critical values for the Hoeffing, Blum, Kiefer, Rosenblatt inepenence criterion. Statist. Decisions Cramér, H. (1938). Sur un nouveau théorème limite e la théorie es probabilités. Actualités Scientifiques et Inustrielles Hermann, Paris. 11.Csörgő, M. (1979). Strong approximation of the Hoeffing, Blum, Kiefer, Rosenblatt multivariate empirical process. J. Multivariate. Anal Deheuvels, P. (1981). An asymptotic ecomposition for multivariate istributionfree tests of inepenence. J. Mult. Anal Deheuvels, P. (25). Weighte multivariate Cramér-von Mises-type statistis. Afrika Statistica. (to appear). 14.Deheuvels, P. an Martynov, G. (23). Karhunen-Loève expansions for weighte Wiener processes an Brownian briges via Bessel functions. Progress in Probability , Birkhäuser, Basel. 15.Deheuvels, P., Peccati, G. an Yor, M. (24). On quaratic functionals of the Brownian sheet an relate processes. Prépublication n 91. Laboratoire e Probabilités & Moèles Aléatoires. 4p. Paris. 16.Donati-Martin, C. an Yor, M. (1991). Fubini s theorem for ouble Wiener integrals an the variance of the Brownian path. Ann. Inst. Henri Poincaré Dugué, D. (1967). Fonctions caractéristiques intégrales browniennes. Rev. Roum. Math. Pures Appl

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