Bootstrap independence test for functional linear models

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Finally note that, for almost all ω ∈ Ω,S ∗ n = √ nT W ∗n + (Y w − µ Y ) 1 √ n+ (X w − µ X ) 1 √ nn∑i=1n∑(Xi w − µ X )ε ∗ ii=1(Yi w − µ Y )ε ∗ i + (X w − µ X )(Y w − µ Y ) √ 1nn∑ε ∗ i .i=1Lemma 2, together with the SLLN above–mentioned, guarantees the convergence in probability to0 of the last three summands, and thus the result is reached in virtue of Slutsky’s Theorem.The “wild” bootstrap approach proposed can be applied by means of the following algorithm.Algorithm 2 (Wild Bootstrap).Step 1. Compute the value of the statistic T n (or the value T n /σ n ).Step 2. Draw {ε ∗ i }n i=1 a sequence of i.i.d. random elements ε, and compute a n = ‖TnW ∗ ‖ (orb n = ‖TnW ∗ ‖/σn, ∗ in this case σn ∗ is computed like in Step 2 of the Naive Bootstrap algorithm).Step 3. Repeat Step 2 a large number of times B ∈ N in order to obtain a sequence of values{a l n} B l=1 (or {bl n} B l=1 ).Step 4. Approximate the p–value of the test by the proportion of values in {a l n} B l=1greater than orequal to ‖T n ‖ (or by the proportion of values in {b l n} B l=1 greater than or equal to ‖T n‖/σ n ).3 Bootstrap calibration vs. asymptotic theoryFor simplicity, suppose from now on that b = 0 and X of zero–mean in (1), that is, suppose thatthe regression model is given byY = 〈Θ, X〉 + ε.Furthermore, ∆(h) = E (〈X, h〉Y ) and, analogously, Γ(h) = E (〈X, h〉X). In such case, if we assumethat ∑ ∞j=1 (∆(v j)/λ j ) 2 < +∞ and Ker(Γ) = {0}, thenΘ =∞∑j=1∆(v j )λ jv j ,being {(λ j , v j )} j∈N the eigenvalues and eigenfunctions of Γ (see Cardot, Ferraty, and Sarda (2003)).A natural estimator for Θ is the FPCA estimator based on k n functional principal components givenbyˆΘ kn =k n ∑j=1∆ n (ˆv j )ˆλ jˆv j ,where ∆ n is the empirical estimation of ∆, that is, ∆ n (h) = (1/n) ∑ ni=1 〈X i, h〉Y i , and {(ˆλ j , ˆv j )} j∈Nare the eigenvalues and the eigenfunctions of Γ n , the empirical estimator of Γ: Γ n (h) = (1/n)∑ ni=1 〈X i, h〉X i .Different statistics can be used for testing the lack of dependence between X and Y . Bearing inmind the expression (5), one can think about using an estimator of ‖Θ‖ 2 = ∑ ∞j=1 (∆(v j)/λ j ) 2 inorder to test these hypotheses. In an alternative way, the expression (6) can be a motivation fordifferent class of statistics based on the estimation of ‖∆‖ ′ .10
One asymptotic distribution free based on the latter approach was given by Cardot, Ferraty, Mas,and Sarda (2003). They proposed as test statisticT 1,n = kn(ˆσ −1/2 −2 || √ )n∆ n Â n || 2 − k n , (8)where Ân(·) = ∑ k n ˆλ−1/2j=1 j〈·, ˆv j 〉ˆv j and ˆσ 2 is a consistent estimator of σ 2 . Cardot, Ferraty, Mas, andSarda (2003) showed that, under H 0 , T 1,n converges in distribution to a centered Gaussian variablewith variance equal to 2. Hence, H 0 is rejected if |T 1,n | > √ 2z α (z α the α–quantile of a N (0, 1)),and accepted otherwise. Besides, Cardot, Ferraty, Mas, and Sarda (2003) also proposed anothercalibration of the statistic distribution based on a permutation mechanism.On the other hand, taking into account that ||Θ|| 2 = ∑ ∞j=1 (∆(v j)/λ j ) 2 , one can use the statisticwhich limit distribution is not known.T 2,n =k n ∑j=1(∆ n (ˆv j )ˆλ j) 2, (9)Finally, a natural competitive statistic is the one proposed throughout Section 2.3∥ 1n∑∥∥∥∥T 3,n =(X i −∥n¯X)(Y i − Ȳ ) , (10)i=1which we will denote by “F–test” from now on since it is the natural generalization of the well–known F–test in the finite–dimensional context. Another possibility is to consider the studentizedversion of (10)∥T 3s,n = 1ˆσ1n∑∥∥∥∥ (X i −∥n¯X)(Y i − Ȳ ) , (11)where ˆσ 2 is the empirical estimation of σ 2 .i=1In general, for the statistics such as (8), (9), (10) and (11), the calibration of the distributioncan be obtained by using bootstrap. Furthermore, in the previous section, “naive” and “wild”bootstrap were shown to be consistent for the F–test, that is, the distribution of T 3,n and T 3s,ncan be approximated by their corresponding bootstrap distribution, and H 0 can be rejected whenthe statistic value does not belong to the interval defined for the bootstrap acceptation region ofconfidence 1 − α. The same calibration bootstrap can be applied to the tests based on T 1,n andT 2,n , although the consistence of the bootstrap procedure in this cases have not been proved in thiswork.4 Simulation and real data applicationsIn this section a simulation study and an application to a real dataset illustrate the performance ofthe asymptotic approach and the bootstrap calibration from a practical point of view.4.1 Simulation studyWe have simulated ns = 500 samples, each being composed of n ∈ {50, 100} observations fromthe functional linear model Y = 〈Θ, X〉 + ε, being X a Brownian motion and ε ∼ N (0, σ 2 ) withsignal–to–noise ratio r = σ/ √ E(〈X, Θ〉 2 ) ∈ {0.5, 1, 2}.11
Page 1 and 2: Bootstrap independence test for fun
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Bootstrap independence test for functional linear models

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