Testing equality of autocovariance operators for functional time series

Transcription

1 Testing equality of autocovariance operators for functional time series Dimitrios PILAVAKIS, Efstathios PAPARODITIS and Theofanis SAPATIAS Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, CY 678 icosia, CYPRUS. arxiv: v [math.st] 24 Jan 209 Astract We consider strictly stationary stochastic processes of Hilert space-valued random variales and focus on tests of the equality of the lag-zero autocovariance operators of several independent functional time series. A moving lock ootstrap-ased testing procedure is proposed which generates pseudo random elements that satisfy the null hypothesis of interest. It is ased on directly ootstrapping the time series of tensor products which overcomes some common difficulties associated with applications of the ootstrap to related testing prolems. The suggested methodology can e potentially applied to a road range of test statistics of the hypotheses of interest. As an example, we estalish validity for approximating the distriution under the null of a fully functional test statistic ased on the Hilert-Schmidt distance of the corresponding sample lag-zero autocovariance operators, and show consistency under the alternative. As a prerequisite, we prove a central limit theorem for the moving lock ootstrap procedure applied to the sample autocovariance operator which is of interest on its own. The finite sample size and power performance of the suggested moving lock ootstrap-ased testing procedure is illustrated through simulations and an application to a real-life dataset is discussed. Some key words: Autocovariance Operator; Functional Time Series; Hypothesis Testing; Moving Block Bootstrap. Introduction Functional data analysis deals with random variales which are curves or images and can e expressed as functions in appropriate spaces. In this paper, we consider functional time series X n = {X, X 2,..., X n } steming from a strictly stationary stochastic process X = X t, t Z of Hilert space-valued random functions X t τ, τ I, which are assumed to e L 4 -m-approximale, a dependence assumption which is satisfied y large classes of commonly used functional time series models; see, e.g., Hörmann and Kokoszka 200. We would like to infer properties of a group of K pilavakis.dimitrios@ucy.ac.cy stathisp@ucy.ac.cy fanis@ucy.ac.cy Corresponding author

2 independent functional processes ased on oserved stretches from each group. In particular, we focus on the prolem of testing whether the lag-zero autocovariance operators of the K processes are equal and consider fully functional test statistics which evaluate the difference etween the corresponding sample lag-zero autocovariance operators using appropriate distance measures. As it is common in the statistical analysis of functional data, the limiting distriution of such statistics depends, in a complicate way, on difficult to estimate characteristics of the underlying functional stochastic processes like, for instance, its entire fourth order temporal dependence structure. Therefore, and in order to implement the testing approach proposed, we apply a moving lock ootstrap MBB procedure which is used to estimate the distriution of the test statistic of interest under the null. otice that for testing prolems related to the equality of second order characteristics of several independent groups, in the finite or infinite dimensional setting, applications of the ootstrap to approximate the distriution of a test statistic of interest are commonly ased on the generation of pseudo random oservations otained y resampling from the pooled mixed sample consisting of all availale oservations. Such implementations lead to the prolem that the generated pseudo oservations have not only identical second order characteristics ut also identical distriutions. This affects the power and the consistency properties of the ootstrap in that it restricts its validity to specific situations only; see Lele and Carlstein 990 for an overview for the case of independent and identically distriuted i.i.d. real-valued random variales and Remark 3.2 in Section 3 elow for more details in the functional setting. To overcome such prolems, we use a different approach which is ased on the oservation that the lag-zero autocovariance operator C 0 = EX t µ X t µ is the expected value of the tensor product process {Y t = X t µ X t µ, t Z}, where µ = EX t denotes the expectation of X t. Therefore, the testing prolem of interest can also e viewed as testing for the equality of expected values mean functions of the associated processes of tensor products. The suggested MBB procedure works y first generating functional pseudo random elements via resampling from the time series of tensor products of the same group and then adjusting the mean function of the generated pseudo random elements in each group so that the null hypothesis of interest is satisfied. We stress here the fact that the proposed method is not designed having any particular test statistic in mind and it is, therefore, potentially applicale to a wide range of test statistics. As an example, we estalish validity of the proposed MBB-ased testing procedure in estimating the distriution of a particular fully functional test statistic under the null, which is ased on the Hilert-Schmidt norm etween the sample lag-zero autocovariance operators, and show its consistency under the alternative. As a prerequisite, we prove a central limit theorem for the MBB procedure applied to the sample version of the autocovariance operator C h = EX t µ X t+h µ, h Z, of an L 4 -m-approximale stochastic process, which is of interest on its own. Our results imply that the suggested MBB-ased testing procedure is not restricted to the case of testing for equality of the lag-zero autocovariance operator only ut it can e adapted to tests dealing with the equality of any finite numer of autocovariance operators C h for lags h different from zero. Asymptotic and ootstrap ased inference procedures for covariance operators for two or more 2

3 populations of i.i.d. functional data have een extensively discussed in the literature; see, e.g., Panaretos et al. 200, Fremdt et al. 203 for tests ased on finite-dimensional projections, Pigoli et al. 204 for permutation tests ased on distance measures and Paparoditis and Sapatinas 206 for fully functional tests. otice that testing for the equality of the lag-zero autocovariance operators is an important prolem also for functional time series since the associated covariance kernel c 0 u, v = CovX t u, X t v of the lag-zero autocovariance operator C 0 descries, for u, v I I, the entire covariance structure of the random function X t. Despite its importance, this testing prolem has een considered, to the est of our knowledge, only recently y Zhang and Shao 205. To tackle the aforementioned prolems associated with the implementaility of limiting distriutions, Zhang and Shao 205 considered tests ased on projections on finite dimensional spaces of the differences of the estimated lag-zero autocovariance operators. otice that similar directional tests have een previously considered for i.i.d. functional data; see Panaretos et. al. 200 and Fremdt et al Although projection-ased tests have the advantage that they lead to manageale limiting distriutions, and can e powerful when the deviations from the null are captured y the finite-dimensional space projected, such tests have no power for alternatives which are orthogonal to the projection space. Moreover, and apart from eing free from the choice of tuning parameters and consistent for a roader class of alternatives, fully functional tests also allow for an interpretation of the test results; we refer to Section 4 for an example. The paper is organised as follows. In Section 2, the asic assumptions on the underlying stochastic process X are stated and the asymptotic validity of the MBB procedure applied to estimate the distriution of the sample autocovariance operator is estalished. In Section 3, the proposed MBBased procedure for testing equality of the lag-zero autocovariance operators for several independent functional time series is introduced. Theoretical justifications for approximating the null distriution of a particular fully functional test statistic are given and consistency under the alternative is otained. umerical simulations are presented in Section 4 in which the finite sample ehaviour of the proposed MBB-ased testing methodology is investigated. A real-life data example is also discussed in this section. Auxiliary results and proofs of the main results are deferred to Section 5 and to the supplementary material. 2 Bootstrapping the autocovariance operator 2. Preliminaries and Assumptions We consider a strictly stationary stochastic process X = {X t, t Z}, where the random variales X t are random functions X t ω, τ, τ I, ω Ω, t Z, defined on a proaility space Ω, A, P and take values in the separale Hilert space of squared-integrale R-valued functions on I, denoted y L 2 I. The expectation function of X t, EX t L 2 I, is independent of t, and it is denoted y µ. We define f, g = I fτgτdτ, f 2 = f, f and the tensor product etween f and g y f g = f, g. For two Hilert Schmidt operators Ψ and Ψ 2, we denote y Ψ, Ψ 2 = Ψ e i, Ψ 2 e i the inner product which generates the Hilert Schmidt norm Ψ = Ψ e i 2, where {e i, i =, 2,...} 3

4 is any orthonormal asis of L 2 I. If Ψ and Ψ 2 are Hilert Schmidt integral operators with kernels ψ u, v and ψ 2 u, v, respectively, then Ψ, Ψ 2 = I I ψ u, vψ 2 u, vdudv. We also define the tensor product etween the operators Ψ and Ψ 2 analogous to the tensor product of two functions, i.e., Ψ Ψ 2 = Ψ, Ψ 2. ote that Ψ Ψ 2 is an operator acting on the space of Hilert Schmidt operators. Without loss of generality, we assume that I = [0, ] the unit interval and, for simplicity, integral signs without the limits of integration imply integration over the interval I. We finally write L 2 instead of L 2 I, for simplicity. To descrie more precisely the dependence structure of the stochastic process X, we use the notion of L p -m-approximaility; see Hörmann and Kokoszka 200. A stochastic process X = {X t, t Z} with X t taking values in L 2, is called L 4 -m-approximale if the following conditions are satisfied: i X t admits the representation X t = fδ t, δ t, δ t 2,... for some measurale function f : S L 2, where {δ t, t Z} is a sequence of i.i.d. elements in L 2. ii E X 0 4 < and E Xt X t,m 4 /4 <, 2 m where X t,m = fδ t, δ t,..., δ t m+, δ m t,t m, δm t,t m,... and, for each t and k, δm t,k independent copy of δ t. is an The rational ehind this concept of weak dependence is that the function f in is such that the effect of the innovations δ i far ack in the past ecomes negligile, that is, these innovations can e replaced y other, independent, innovations. For the stochastic process X considered in this paper, we somehow strengthen 2 to the following assumption. Assumption. X is L 4 -m-approximale and satisfies lim m E X t X t,m 4 /4 = 0. m Since E X t 2 <, the autocovariance operator at lag h Z exists and is defined y C h = E[X t µ X t+h µ]. Having an oserved stretch X, X 2,..., X n, the operator C h is commonly estimated y the corresponding sample autocovariance operator, which is given y n n h X t X n X t+h X n, if 0 h < n, Ĉ h = n n+h X t h X n X t X n, if n < h < 0, 0, otherwise, 4

5 where X n = /n n X t is the sample mean function. The limiting distriution of n Ĉ h C h can e derived using the same arguments to those applied in Kokoszka na Reimherr 203 to investigate the limiting distriution of n Ĉ 0 C 0. More precisely, it can e shown that, for any fixed lag h, h Z, under L 4 -approximaility conditions, n Ĉ h C h Zh, where Z h is a Gaussian Hilert- Schmidt operator with covariance operator Γ h given y Γ h = s= E[X µ X +h µ C h X +s µ X +h+s µ C h ]; see also Mas 2002 for a related result if X is a Hilertian linear processes. 2.2 A Bootstrap CLT for the empirical autocovariance operator In this section, we formulate and prove consistency of the MBB for estimating the distriution of n Ĉh C h for any fixed lag h, h Z, in the case of weakly dependent Hilert space-valued random variales satisfying the L 4 -approximaility condition stated in Assumption. The MBB procedure was originally proposed for real-valued time series y Künsch 989 and Liu and Singh 992. Adopted to the functional set-up, this resampling procedure first divides the functional time series at hand into the collection of all possile overlapping locks of functions of length. That is, the first lock consists of the functional oservations to, the second lock consists of the functional oservations 2 to +, and so on. Then, a ootstrap sample is otained y independent sampling, with replacement, from these locks of functions and joining the locks together in the order selected to form a new set of functional pseudo oservations. However, to deal with the prolem of estimating the distriution of the sample autocovariance operator Ĉh, we modify the aove asic idea and apply the MBB directly to the set of random elements Y n h = {Ŷt,h, t =, 2,..., n h}, where Ŷt,h = X t X n X t+h X n. As mentioned in the Introduction, this has certain advantages in the testing context which will e discussed in the next section. The MBB procedure applied to generate the pseudo random elements Y,h, Y 2,h,..., Y n h,h is descried y the following steps. Step : Let = n, < n h, e an integer and denote y B t = {Ŷt,h, Ŷt+,h,..., Ŷt+,h} the lock of length starting from the tensor operator Ŷt, where t =, 2,..., and = n h + is the total numer of such locks availale. Step 2 : Let k e a positive integer satisfying k < n h and k n h and define k i.i.d. integer-valued random variales I, I 2,..., I k selected from a discrete uniform distriution which assigns proaility / to each element of the set {, 2,..., }. Step 3 : Let Bi = B I i, i =, 2,..., k, and denote y {Yi +,h, Y i +2,h,..., Y i,h } the elements of Bi. Join the k locks in the order B, B 2,..., B k together to otain a new set of functional pseudo oservations. The MBB generated sample of pseudo random elements consists then of the set Y,h, Y 2,h,..., Y n h,h. 5

6 ote that if we are interested in the distriution of the sample autocovariance operator Ĉh for some fixed lag h, n < h < 0, then the aove algorithm can e applied to the time series of operators Y n+h = {Ŷt,h, t = h +, h + 2,..., n}, where Ŷt,h = X t h X n X t X n, t = h +, h + 2,..., n, with minor changes. Hence, elow, we only focus on the case of 0 h < n. Given a stretch Y,h, Y 2,h,..., Y n h,h of pseudo random elements generated y the aove MBB procedure, a ootstrap estimator of the autocovariance operator is given y the sample mean Ĉh = n h Yt,h n. The proposal is then to estimate the distriution of nĉh C h y the distriution of the ootstrap analogue nĉ h E Ĉ h, where E Ĉ h is conditionally on X n the expected value of Ĉ h. Assuming, for simplicity, that n h = k, straightforward calculations yield E Ĉ h = n h n h Ŷ t,h j n j= Ŷj,h + Ŷn h j+,h. 3 The following theorem estalishes validity of the MBB procedure suggested for approximating the distriution of nĉh C h. Theorem 2.. Suppose that the stochastic process X satisfies Assumption. For 0 h < n, let Y,h, Y 2,h,..., Y n h,h e a stretch of functional pseudo random elements generated as in Steps -3 of the MBB procedure and assume that the lock size = n satisfies + n /3 = o as n. Then, as n, dl nĉ h E Ĉ h X n, L nĉh C h 0, in proaility, where d is any metric metrizing weak convergence on the space of Hilert-Schmidt operators acting on L 2 and LX denotes the law of the random element X. 3 Testing equality of lag-zero autocovariance operators In this section, we consider the prolem of testing the equality of the lag-zero autocovariance operators for a finite numer of functional time series and use a modified version of the propopsed MBB procedure. This modification leads to a MBB-ased testing procedure which generates functional pseudo oservations that satisfy the null hypothesis that all lag-zero autocovariance operators are equal. Since this procedure is designed without having any particular statistic in mind, it can potentially e applied to a road range of possile test statistics which are appropriate for the particular testing prolem considered. To make things specific, consider K independent, L 4 -m-approximale functional time series, denoted in the following y X K,M = {X i,t, i =, 2..., K, t =, 2,..., n i }, where n i denotes the length of the i-th time series. Let C i,0, i =, 2..., K, e the lag-zero autocovariance operator of the 6

7 i-th functional time series, i.e., C i,0 = E[X i,t µ i X i,t µ i ], where µ i = EX i,t. Also, denote y M = K n i the total numer of oservations. The null hypothesis of interest is then and the alternative hypothesis is H 0 : C,0 = C 2,0 =... = C K,0 4 H : k, m {, 2,..., K} with k m such that C k,0 C m,0. By considering the operator processes {Y i,t = X i,t µ i X i,t µ i, t Z}, denoting y µ Y i written as and the alternative hypothesis as i =, 2..., K, and = EY i,t the expectation of Y i,t, the null hypothesis of interest can e equivalently H 0 : µ Y = µy 2 =... = µy K 5 H : k, m {, 2,..., K} with k m such that µ Y k µy m. Consequently, the aim of the ootstrap is to generate a set of K pseudo random elements Y K,M = {Y i,t, i =, 2..., K, t =, 2,..., n i} which satisfy the null hypothesis 5, that is, the expectations E Yi,t should e identical for all i =, 2,..., K. This leads to the MBB-ased testing procedure descried in the next section. 3. The MBB-ased Testing Procedure Suppose that, in order to test the null hypothesis 5, we use a real-valued test statistic T M, where, for simplicity, we assume that large values of T M argue against the null hypothesis. Since we focus on the tensor operators Y i,t, t =, 2,..., n i, i =, 2..., K, it is natural to assume that the test statistic T M is ased on the tensor product of the centered oserved functions, that is on Ŷ i,t = X i,t X i,ni X i,t X i,ni, i =, 2..., K, t =, 2,..., n i, where X i,ni is the sample mean function of the i-th population, i.e, X i,ni = /n i n i X i,t. Suppose next, without los of generality, that the null hypothesis 5 is rejected if T M > d M,α, where, for α 0,, d M,α denotes the upper α-percentage point of the distriution of T M under H 0. We propose to approximate the distriution of T M under H 0 y the distriution of the ootstrap quantity TM, where the latter is otained through the following steps. Step : Calculate the pooled mean Y M = M K n i Ŷ i,t. Step 2 : For i =, 2,..., K, let i = i n {, 2,..., n } e the lock size used for the i-th functional time series and let i = n i i +. Calculate Ỹ i,ξ = i +ξ i t=ξ 7 Ŷ i,t, ξ =, 2,..., i

8 Step 3 : For simplicity assume that n i = k i i and for i =, 2,..., K, let q i, qi 2,..., qi k i e i.i.d. integers selected from a discrete proaility distriution which assigns the proaility / i to each element of the set {, 2,..., i }. Generate ootstrap functional pseudo oservations Y i,t, t =, 2,..., n i, i =, 2,..., K, as Y i,t = Y M + Ŷ i,t Ỹi,ξ, ξ = i if t mod i = 0 and ξ = t mod i otherwise, where Ŷ i,ξ+s i = Ŷi,q i s+ξ, s =, 2..., k i and ξ =, 2,..., i Step 4 : Let T M e the same statistic as T M ut calculated using, instead of the Ŷi,t s the ootstrap pseudo random elements Y i,t, t =, 2,..., n i, i =, 2,..., K. Given X K,M, denote y D M,T the distriution of T M. Then for α 0,, the null hypothesis H 0 is rejected if T M > d M,α, where d M,α denotes the upper α-percentage point of the distriution of T M, i.e., PT M > d M,α = α. otice that the distriution D M,T is unknown ut it can e evaluated y Monte-Carlo. Before estalishing validity of the descried MBB procedure some remarks are in order. Oserve that the mean Ỹi,ξ calculated in Step 2, is the conditional on X K,M expected value of Ŷ for i,qs i +ξ ξ = i if t mod i = 0 and ξ = t mod i otherwise. This motivates the definition Y i,t = Y M + Ŷ i,t Ỹi,ξ, t =, 2,..., n i, i =, 2,..., K, used in Step 3 of the MBB algorithm. This definition ensures that the generated pseudo random elements Y i,t, t =, 2,..., n i, i =, 2,..., K, satisfy the null hypothesis 5. In fact, it is easily seen that the pseudo random elements Y i,t have conditional on X K,M an expected value which is equal to Y M, that is E Y i,t = Y M for all t =,..., n i and i =,..., K. 3.2 Validity of the MBB-ased Testing Procedure Although the proposed MBB-ased testing procedure is not designed having any specific test statistic in mind, estalishing its validity requires the consideration of a specific class of statistics. In the following, and for simplicity, we focus on the case of two independent population, i.e., K = 2. In this case, a natural approach to test equality of the lag-zero autocovariance operators is to consider a fully functional test statistic which evaluates the difference etween the empirical lag-zero autocovariance operators, for instance, to use the test statistic T M = n n 2 M Ĉ,0 Ĉ2,0 2 = n n 2 M Y,n Y 2,n2 2, where Y i,ni = /n i n i Ŷi,t, i =, 2, and M = n + n 2. The following lemma delivers the asymptotic distriution of T M under H 0. 8

9 Lemma 3.. Let H 0 hold true, Assumption e satisfied and assume that, as min{n, n 2 }, n /M θ 0,. Then, T M d Z0 2 where Z 0 = θz,0 θz 2,0 and Z i,0, i =, 2, are two independent mean zero Gaussian Hilert- Schmidt operators with covariance operators Γ i,0, i =, 2, given y Γ i,0 = E[X i, µ i X i, µ i C i,0 X i, µ i X i, µ i C i,0 ] + 2 E[X i, µ i X i, µ i C i,0 X i,s µ i X i,s µ i C i,0 ]. s=2 As it is seen from the aove lemma, the limiting distriution of T M depends on the difficult to estimate covariance operators Γ i,0, i =, 2, which descrie the entire fourth order structure of the underlying functional processes X i, making the implementation of this asymptotic result for calculating critical values of the T M test a difficult task. Theorem 3. elow shows that the MMBased testing procedure estimates consistently the limiting distriution Z 0 2 of the T M test and consequently that it can e applied to estimate the critical values of interest. For this, we apply the MBB-ased testing procedure introduced in Section 3. to generate {Y i,t, t =, 2,....n i}, i {, 2}, and use the ootstrap pseudo statistic T M = n n 2 M Y,n Y 2,n 2 2, where Y i,n i = /n i n i Y i,t, i =, 2. We then have the following result. Theorem 3.. Let Assumption e satisfied and assume that min{n, n 2 }, n /M θ 0,. Also, for i {, 2}, let the lock size i = i n satisfies i + i n /3 i = o, as n i. Then, sup P TM x X K,M P H0 T M x 0, in proaility, x R where P H0 X denotes the distriution function of the random variale X when H 0 is true. Remark 3.. If H is true, that is if C,0 C 2,0 = EY,t EY 2,t > 0, then it is easily seen that T M under the conditions on n and n 2 stated in Lemma 3.. This, together with Theorem 3. and Slutsky s theorem, imply consistency of the T M test ased on ootstrap critical values otained using the distriution of T M, i.e., the power of the test approaches unity, as n, n 2. Remark 3.2. The advantage of our approach to translate the testing prolem considered to a testing prolem of equality of mean functions and to apply the ootstrap to the time series of tensor operators Y i,t, t =, 2,..., n i, i =,..., K, is manifested in the generality under which validity of the MBBased testing procedure is estalished in Theorem 3.. To elaorate, a MBB approach which would select locks from the pooled mixed set of functional time series in order to generate ootstrap pseudo elements which satisfy the null hypothesis, will lead to the generation of K new functional 9

10 pseudo time series, which asymptotically will imitate correctly the pooled second and the fourth order moment structure of the underlying functional processes. As a consequence, the limiting distriution of T M as stated in Lemma 3. and that of the corresponding MBB analogue will coincide only if Γ = Γ 2. This oviously restricts the class of processes for which the MBB procedure is consistent. In the more simple i.i.d. case, a similar limitation exists y the condition B = B 2 imposed in Theorem of Paparoditis and Sapatinas 206. otice that this limitation can e resolved y applying also in the i.i.d. case the asic ootstrap idea proposed in this paper. That is, to first translate the testing prolem to one of testing equality of means of samples consisting of the i.i.d. tensor operators and then to apply an appropriate i.i.d. ootstrap procedure. 4 umerical Results In this section, we investigate via simulations the size and power ehavior of the MBB-ased testing procedure applied to testing the equality of lag zero autocovariance operators and we illustrate its applicaility y considering a real life data-set. 4. Simulations In the simulation experiment, two groups of functional time series are generated from the functional autoregressive FAR model X t u = ψu, vx t v dv + δx t 2 u + B t u, 6 or from the functional moving average FMA model, X t u = ψu, vb t v dv + δb t 2 u + B t u. 7 The kernel function ψ, in the aove models is equal and it is given y +v 2 /2 ψu, v = e u2, u, v [0, ] 2, 4 e t2 dt while the B t s are generated as i.i.d. Brownian ridges. All curves were approximated using T = 2 equidistant points τ, τ 2,..., τ 2 in the unit interval I and transformed into functional ojects using the Fourier asis with 2 asis functions. Functional time series of length n = n 2 = 200 are then generated and testing the null hypothesis H 0 : C,0 = C 2,0 is considered using the T M test investigated Section 3.2. All ootstrap calculations are ased on B = 000 ootstrap replicates, R = 000 model repetitions have een considered and a range of different lock sizes have een used. Since n = n 2 we set for simplicity = = 2. Regarding the selection of we mention the following. As an inspection of the proof of Theorem 2. shows, the MBB estimator of the distriution of interest also delivers a lag-window type estimator 0

11 of the covariance operator Γ 0 of the limiting Gaussian process Z 0 using implicitly the Bartlett lagwindow with truncation lag the lock size ; see also equation 3. Viewing the choice of as the selection of the truncation lag in the aforementioned lag window type estimator, allows for the use of some results availale in the literature in order to select. To elaorate, the choice of the truncation lag in the functional set-up has een discussed in Horváth et al. 206 and Rice and Shang 207, where different procedures to select this parameter have een investigated. In our context, we found the simple rule proposed y Rice and Shang 207 quite effective according to which the lock length is set equal to the smallest integer larger or equal to n /3. In the case of n = n 2 = 200 oservations, this rule leads to the value = 6. This choice of as well as of some other values of the lock length have een considered in our simulations. The T M test has een applied using three standard nominal levels α = 0.0, 0.05 and 0.0. otice that δ = 0 corresponds to the null hypothesis while to investigate the power ehavior of the test we set δ = 0 for the first functional time series and allow for δ {0.2, 0.5, 0.8} for the second and for each of the two different models considered. The results otained for different values of the lock size using the FAR model 6 as well as the FMA model 7 are shown in Tale. As it is seen from this tale, the MBB ased testing procedure retains the nominal level with good size results, especially for = 6 and for oth dependence structures considered. Furthermore, the power of the T M test increases as the deviations from the null increase and reaches high values for the large values of the deviation parameter δ considered. 4.2 A Real-Life Data Example In this section, the ootstrap ased T M test testing is applied to a real-life data set which consists of daily temperatures recorded in 5 minutes intervals in icosia, Cyprus, i.e., there are 96 temperature measurements for each day. Sample A and Sample B consist of the daily temperatures recorded in Summer /06/2007-3/08/2007 and Summer /06/2009-3/08/2009 respectively. The measurements have een transformed into functional ojects using the Fourier asis with 2 asis functions. All curves are rescaled in order to e defined in the interval I = [0, ]. Figure shows the estimated lag-zero autocovariance kernels ĉ i u, v = n ni i X i,tu X i ux i,t v X i v, u, v I I, associated with the lag-zero autocovariance operators for the temperature curves of the summer 2007 i = and of the summer 2009 i = 2. We are interested in testing whether the covariance structure of the daily temperature curves of the two summer periods is the same. The p-values of the MBB-ased T M test using B = 000 ootstrap replicates and for a selection of different lock sizes = = 2, are 0.06 = 3, 0.05 = 4, = 5 and = 6. otice that in this example, n = n 2 = 92 and that, for this sample size, the value of = 5 is the one chosen y the simple selection rule discussed in the previous section. As it is evident from these results, the p-values of the MBB-ased test are quite small and lead to a rejection of H 0, for instance at the commonly used 5% level. To see were the differences etween the temperatures in the two summer periods come from and to etter interpret the test results, Figure 2 presents a contour plot of the estimated squared differences

12 Block Size, = δ α FAR FAM Tale : Empirical size and power of the T M test using ootstrap critical values. ĉ u, v ĉ 2 u, v 2 for different values of u, v in the plane [0, ] 2. ote that the Hilert-Schmidt distance Ĉ,0 Ĉ2,0 appearing in the test statistic T M can e approximated y the discretized quantity L 2 L L j= ĉ u i, v j ĉ 2 u i, v j 2, where L = 96 is the numer of equidistant time points in the interval [0, ] used and at which the temperature measurements are recorded. Large values of ĉ u i, v j ĉ 2 u i, v j 2 i.e., dark gray regions in Figure 2 contriute strongly to the value of the test statistic T M and pinpoint to regions where large differences etween the corresponding lag-zero autocovariance operators occur. Taking into account the symmetry of the covariance kernel c,, Figure 2 is very informative. It shows that the main differences etween the two covariance operators are concentrated etween the time regions 4.00am to 6.00am and 0.30am to 8.00pm of the daily temperature curves, with the strongest contriutions to the test statistic eing due to the 2

13 Figure : Estimated lag-zero autocovariance kernels of the temperature curves: Summer 2007 left panel and summer 2009 right panel :00 6: : : : :00 04:00 08:00 2:00 6:00 20:00 Figure 2: Contour plot of the estimated differences ĉ u i, v j ĉ 2 u i, v j 2 for i, j {, 2,..., 96}. largest differences recorded around 4.00 to 4.30 in the morning and 6.00 to 8.00 in the evening. 3

14 5 Appendix : Proofs In the following we assume, without loss of generality, that µ = 0 and we consider the case h = 0 only. Furthermore, we let C0 = n n X t X t, Z t = X t X t C 0, Ẑt = X t X t C0, Z t = X t X t, Z t,m = X t,m X t,m C 0, Z t = X t X t and Ẑ t = X t X t C0. Also, we denote y Z t u, v the kernel of the integral operator Z t, i.e., Z t u, v = X t ux t v c 0 u, v, where c 0 u, v = E[X t ux t v], and y Z t,m u, v the kernel of the integral operator Z t,m, i.e., Z t,m u, v = X t,m ux t,m v c 0 u, v. We first fix some notation and present two asic lemmas which will e used in the proofs. Towards this note first that we repeatedly use the fact that, y stationarity, E X t,m X t p = E X 0,m X 0 p and E X t,m p = E X t p = E X 0 p for p and for all t Z. Also note that Kokoszka and Reimherr 203 proved that the L 4 -m-approximaility of X implies that the tensor product process {X t X t, t Z} is L 2 -m-approximale. For X t,m X t,m the m-dependent approximation of X t X t, we, therefore, have /2 E X t X t X t,m X t,m 2 <. 8 m= Furthermore, since X 0 X t = X 0 X t for all t Z, and using Cauchy-Schwarz s inequality, we get, for all t Z, E X t X t X t,m X t,m 2 2E X t X t X t,m 2 + 2E X t X t,m X t,m 2 4E X t 4 /2 E X t X t,m 4 /2. Therefore, y Assumption, we get, for all t Z, lim m E X t X t X t,m X t,m 2 /2 m 2E Xt 4 /4 lim me X t X t,m 4 /4 = 0 9 m and y the same arguments, E[X 0 X t ] = E[X 0 X t X t,t ] E X 0 2 /2 E X0 X 0,t 2 /2 E X 0 2 /2 E X0 X 0,t 4 /4. Therefore, the L 4 -m-approximaility assumption implies that t Z E[X 0 X t ] <. To prove Theorem 2., we estalish elow Lemma 5. and Lemma 5.2. Their proofs are given in the supplementary material. Lemma 5.. Let g e a non-negative, continuous and ounded function defined on R, satisfying g 0 =, g u = g u, g u for all u, g u = 0, if u > c, for some c > 0. Assume that for any fixed u, g u as n. Suppose that the process X satisfies Assumption and that = n is a sequence of integers such that + n /3 = o as n. Then, as n, where ˆΓ s = n s= + g sˆγ s s= n s Ẑt Ẑt+s for 0 s and ˆΓ s = n 4 E[Z 0 Z s ] = o p, n+s Ẑt s Ẑt for + s < 0.

15 Lemma 5.2. Let g e a non-negative, continuous and ounded function satisfying the conditions of Lemma 5.. Suppose that X satisfies Assumption and that = n is a sequence of integers such that + n /2 = o as n. Then, as n, s= + g s n n s Z t u, vz t+ s u, vdudv P s= E Z 0 u, vz s u, vdudv. Proof of Theorem 2.. By the triangle inequality and Theorem 3 of Kokoszka and Reimherr 203, the assertion of the theorem is estalished if we show that, as n, n Ĉ 0 E Ĉ 0 Z 0, 0 in proaility, where Z 0 is a mean zero Gaussian Hilert Schmidt operator with covariance operator given y Γ 0 = E[Z Z ] + 2 Using Theorem of Horváth et al. 203, we get E[Z Z s ]. n Ĉ0 E Ĉ 0 = n ] [X t X t E X t X t X n X t E X t X t E X t X n n Also note that = n n n [Zt E Zt ] + O P / n. n [Zt E Zt ] = k = k k k Ŷ t, s=2 Z t +i E Zt +i with an ovious notation for Ŷ t, t =, 2,..., k. Recall that due to the lock ootstrap resampling scheme, the random variales Ŷ t, t =, 2,..., k, are i.i.d. Therefore to prove 0, it suffices y Lemma 5 of Kokoszka and Reimherr 203, to prove that, i k k Ŷ t d, y 0, σ 2 y for every Hilert Schmidt operator y acting on L 2, and that ii lim n E k k Ŷ t 2 exists and is finite. 5

16 To estalish assertion i, we first prove that, as n, Consider and notice that Var k k Ŷ t, y Var k k Ŷ t, y = Var Ŷ, y = E P σ 2 y. Zt E Zt, y ] 2. 2 Let = n +, Ỹt = /2 Z t + Z t Z t+, t =, 2,..., and Ỹ t = /2 Z t +i, t =, 2,..., k. Since n/ as n, in the following we will occasionally replace / y /n. otice that, E Zt, y = E Ỹ = = Ỹt, y [ n Z t, y = ˆ Cn, y Therefore, Var k Ŷt, y k = E Ẑt, y = E Ẑt, y [ [ + 2 i = E Ẑt, y [ [ + ] 2 [ [ + i ] Z i, y + Z n i+, y ] ] Z i, y + Z n i+, y ]. 3 i i ]] 2 Z i, y + Z n i+, y ] Z i, y + Z n i+, y ] Let Ŷt = /2 Ẑt + Ẑt Ẑt+, t =, 2,...,. Since, ] 2 E Ẑt, y = Ŷt, y 2 = n Ẑt, y Ẑt, y i ]] 2 Z i, y + Z n i+, y ] ]] E Ẑt, y ] 2 + O P 3 /n ]

17 we get, using 4, + i n i Ẑt, y Ẑt+i, y + Ẑt+i, y Ẑt, y ] i Ẑi, y Ẑi, y + Ẑn i+, y Ẑn i+, y] t j + i Ẑj, y Ẑj+i, y + Ẑn j+ i, y Ẑn j+, y j= Var k Ŷt, y k = n Ẑt, y Ẑt, y + i + Ẑj+i, y Ẑj, y + Ẑn j+, y Ẑn j+ i, y ], Therefore, Var k Ŷt, y k = n Ẑt Ẑt, y y + i n i Ẑt, y Ẑt+i, y + Ẑt+i, y Ẑt, y ] + O P /n + O P 2 /n + O P 3 /n 2. n i Ẑt Ẑt+i, y y + Ẑt+i Ẑt, y y ] Let g i = i in Lemma 5., and use the triangular inequality to get n Ẑ t Ẑt + n Ẑ t Ẑt + = o p. i i n i [Ẑt Ẑt+i + Ẑt+i Ẑt] n i [Ẑt Ẑt+i + Ẑt+i Ẑt] t= t= + O P 2 /n. 5 E[Z 0 Z t ], y y E[Z 0 Z t ] Therefore, and using Z 0 Z t, y y = Z 0, y Z 0, y, we get from 5, as n, Var k k Ŷ t, y P t= E[Z 0 Z t ], y y y y = Γ, y y = σ 2 y. 6 We next estalish the asymptotic normality stated in i. Since Ŷ t, y, t =, 2,..., k are i.i.d. real valued random variales, we show that Lindeerg s condition is satisfied, i.e., for every ε > 0, as 7

18 n, τ 2 k k E [ Ŷ t, y E Ŷ t, y 2 Ŷ t, y E Ŷ t, y > ετ k ] = o p, 7 where A x denotes the indicator function of the set A and τ 2 k = k Var Ŷ t, y = kvar Ŷ, y. 8 To estalish 7, and ecause of 6 and 8, it suffices to show that, for any δ > 0, as n, k [ ] P E Ŷ t, y E Ŷ t, y 2 Ŷ t, y E Ŷ t, y > ετk k > δ 0. 9 Towards this, notice first that, for any two random variales X and Y and any η > 0, E[ X + Y 2 X + Y > η] 4 [ E X 2 X > η/2 + E Y 2 Y > η/2 ] ; 20 see Lahiri 2003, p. 56. Since the random variales Ŷ t, y are i.i.d., we get using expression 3 and Markov s inequality that, as n, P k k { δ E [ {E E [ Ŷ t, y E Ŷ t, y 2 Ŷ t, y E Ŷ t, y > ετ k ] > δ [ E Ŷ, y E Ŷ, y 2 Ŷ, y E Ŷ [ = δ E Ẑt, y + Ẑt, y + [ [ = δ E Ŷt, y + i Ŷt, y + [ i [, y > ετ k ]} i Z i, y + Z n i+, y ] i Z i, y + Z n i+, y ]] ] 2 Z i, y + Z n i+, y ] Z i, y + Z n i+, y ]] [ 4δ E Ŷ, y 2 Ŷ, y > ετk /2 + E [ 4δ E Ŷ, y 2 Ŷ, y > ετk /2 + E > ετ k ] ] 2 > ετ k ]} i 2 Z i, y + Z n i+, y ] ] i Z i, y + Z n i+, y ] > ετ k /2 8 i Z i, y + Z n i+, y ] 2 ]

19 4δ E Ŷ, y 2 Ŷ, y > ετ k /2 + O3 /n 2. 2 By Lemma 4 of Kokoszka and Reimherr 203 it follows that s= E Z 0, y Z s, y converges asolutely. By Kronecker s lemma, we then get, as n, E Ŷ, y 2 = = j= s < = s < s= EẐi, y Ẑj, y ] s EẐ0, y Ẑs, y ] s EZ 0, y Z s, y ] + O/n /2 EZ 0, y Z s, y ]. Therefore, y the dominated convergence theorem, and, therefore, assertion i is proved. To estalish assertion ii, notice first that Furthermore, since E we get Z t EŶ, y 2 Ŷ, y > ετk /2 = o 22 = E k k Ỹ t = Ŷt 2 = E Ŷ 2. [ n Z t = Cn i ] [ Z i + Z n i+ ] i [ Z i + Z n i+ ], E Ŷ 2 = E Ẑt + i [ Z i + Z n i+ ] = + i 2 [ Ŷt Z i + Z n i+ ]. Since, i [ Z i + Z n i+ ] = O P 3/2 /n, it suffices to prove that the limit lim n 2 Ŷt

20 exists and it is finite. Let Y t = /2 Z t + + Z t+, t =, 2,..., and note that Ŷt 2 = Y t + C 0 C0 2. By Theorem 3 of Kokoszka and Reimherr 203, in order to prove 23, it suffices to show that exists and it is finite. We have that lim n Y t 2 = Z t, Z t + i = = t t j= n Z t, Z t + i= + i n Y t 2 24 n i Z t, Z t+i + Z t+i, Z t ] Z t, Z t + X n t+, X n t+ ] t + j Z j, Z j+t + Z n j+ t, Z n j+ n i i + Z j+t, Z j + Z n j+, Z n j+ t ] n i Z t, Z t+i + Z t+i, Z t ] + O P 2 /n Z t u, vz t+ i u, vdudv + O P 2 /n. 25 Hence, y letting g s = s / in Lemma 5.2, we get that the last term aove converges to s= E Z 0 u, vz s u, vdudv, from which we conclude that, as n, E Y 2 s= E Z 0 u, vz s u, vdudv, in proaility. Proof of Lemma 3.. Using Theorem 3 of Kokoszka and Reimherr 203 it follows that there exist two independent, mean zero, Gaussian Hilert Schmidt operators Z,0 and Z 2,0 with covariance operators Γ,0 and Γ 2,0 respectively, such that n Ĉ,0 C,0, n 2 Ĉ2,0 C 2,0 converges weakly to Z,0, Z 2,0. Since n n 2 M Ĉ,0 Ĉ2,0 = n2 M n Ĉ,0 C n 0 n2 Ĉ2,0 M C 0, where C 0 is the under H 0 common lag-zero covariance operator of the two populations, we get that, for n, n 2 and n /M θ, T M d Z0 2, 20

21 where Z 0 = θz,0 θz 2,0. Proof of Theorem 3.. Using the triangle inequality and the fact that nĉi,0 C i,0 Z i,0, i =, 2, it suffices to prove that TM converges weakly to Z 0 2, where Z 0 = θz,0 θz 2,0. This is proved along the same lines as Lemma 3. using of Theorem 2. and the independence of the pseudo-random elements Y,n and Y 2,n 2. Supplementary Material The supplementary material contains the proofs of Lemma 5. and Lemma 5.2. You can download the file from: fanis/papers/mbb-cov-supplement.pdf References [] Fremdt, S., Steineach, J.G., Horváth, L. and Kokoszka, P Testing the equality of covariance operators in functional samples. Scandinavian Journal of Statistics, Vol. 40, [2] Hörmann, S. and Kokoszka, P Weakly dependent functional data. The Annals of Statistics. Vol. 38, [3] Horváth, L., Rice, G. and Whipple, S Adaptive andwidth selection in the long run covariance estimator of functional time series. Computational Statistics and Data Analysis, Vol. 00, [4] Kokoszka, P. and Reimherr, M Asymptotic normality of the principal components of functional time series. Stochastic Processes and their Applications, Vol. 23, [5] Künsch, H.R The jackknife and the ootstrap for general stationary oservations. The Annals of Statistics, Vol. 7, [6] Lahiri, S Resampling Methods for Dependent Data. ew York: Springer-Verlag. [7] Lele, S. and Carlstein, E Two-sample ootstrap tests: when to mix? Institute of Statistics Mimeo Series, o. 203, Department of Statistics, orth Carolina State University, USA. [8] Liu, R.Y. and Singh, K Moving locks jackknife and ootstrap capture weak dependence. In Exploring the Limits of the Bootstrap R. Lepage and L. Billard, Eds., pp , ew York: Wiley. [9] Mas, A Weak convergence for the covariance operators of a Hilertian linear process. Stochastic Processes and their Applications, Vol. 99,

22 [0] Panaretos, V.M., Kraus, D. and Maddocks, J.H Second-order comparison of Gaussian random functions and the geometry of DA minicircles. Journal of the American Statistical Association, Vol. 05, [] Paparoditis, E. and Sapatinas, T Bootstrap-ased testing of equality of mean functions or equality of covariance operators for functional data. Biometrika, Vol. 03, [2] Pigoli, D., Aston, J.A.D., Dryden, I.L. and Secchi, P Distances and inference for covariance operators. Biometrika, Vol. 0, [3] Rice, G. and Shang, H.L. 207 A plug-in andwidth selection procedure for long run covariance estimation with stationary functional time series. Journal of Time Series Analysis, Vol. 38, [4] Zhang, X. and Shao, X Two sample inference for the second-order property of temporally dependent functional data. Bernoulli, Vol. 2,