Bootstrap independence test for functional linear models

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Corollary 1. Under the conditions of Theorem 1, if the null hypothesis H 0 : ‖∆‖ ′ = 0 is satisfied,then √ nT n converges in law to Z (X−µX )ε (with covariance operator σ 2 Γ), and consequently, ‖ √ nT n ‖converges in law to ‖Z (X−µX )ε‖.In contrast to Theorem 1 in Cardot, Ferraty, Mas, and Sarda (2003), the result in Corollary 1 isestablished directly on the Hilbert space H instead of on its dual space. In addition, no assumptionof centered X random elements or null intercept term is necessary. Nevertheless these two assumptionscould be easily removed in that paper in order to establish a dual result of Corollary 1.Furthermore, in view of Corollary 1, the asymptotic null distribution of ‖ √ nT n ‖ is not explicitlyknown. This is the reason why no further research on how to use in practice this statistic (or itsdual one) for checking if Θ 2 equals 0 is carried out in Cardot, Ferraty, Mas, and Sarda (2003).Instead, an alternative statistic that is used in the simulation section for comparative purposes isconsidered. Nevertheless, it is still possible to use ‖ √ nT n ‖ as a core statistic in order to solve thistest in practice by means of bootstrap techniques.One natural way of using the asymptotic result of Corollary 1 for solving the test under studyis as follows. Consider a consistent (at least under H 0 ) estimator σn 2 of σ 2 (for instance, thesample variance of Y could be used, or perhaps the one introduced by Cardot, Ferraty, Mas, andSarda (2003), provided that its theoretical behavior is analyzed). Then, according to Slutsky’sTheorem ‖ √ nT n ‖/σ n converges in law under H 0 to the norm of Z X . As its covariance operator Γ isunknown, it can be approximated by the empirical one Γ n . And thus, ‖Z X ‖ can be approximatedby ‖Z n ‖, being Z n a centered Gaussian element in H with covariance operator Γ n . Of course, thedistribution of ‖Z n ‖ is still difficult to compute directly, nevertheless one can make use of the CLTand approximate its distribution by Monte Carlo method by the distribution of1m∑(Xi ∗ − X)∥m∥i=1for a large value of m, being {Xi ∗}m i=1 i.i.d. random elements chosen at random from the fixedpopulation (X 1 , . . . , X n ). Obviously, this method is a precursor of the bootstrap procedures.In order to complete the asymptotic study of the statistic ‖ √ nT n ‖, its behavior under local alternativesis going to be analyzed. To this purpose, let us consider Θ ∈ H so that ‖Θ 2 ‖ > 0, and givenδ n > 0 consider the modified random sampleY ni= 〈X i , δ n√ nΘ〉 + b + ε i ,for all i ∈ {1, . . . , n}. Then, the null hypothesis is not verified. However, if δ n / √ n → 0, then‖(δ n / √ n)Θ‖ → 0, that is, H 0 is approached with “speed” δ n / √ n. In these conditions,E ( (X i − µ Xi )(Y ni− µ Y ni) ) = δ n√ ∆(Θ), nand thus the following theorem that establishes the behavior of the statistic under the consideredlocal alternatives can be easily deduced.Theorem 2. Under the conditions of Theorem 1 and with the above notation, if δ n → ∞ andδ n / √ n → 0 as n → ∞ then(∥ ∥∥∥∥1n∑ (P √n Xi − X ) ( )Yin − Y n)∥ ∥ ∥∥∥≤ t → 0as n → ∞, for all t ∈ R.i=16
2.4 Bootstrap proceduresThe difficulty of using the previously proposed statistic to solve the hypothesis test by means ofasymptotic procedures suggests the development of appropriated bootstrap techniques. The asymptoticconsistency of a bootstrap approach is guaranteed if the associated bootstrap statistic convergesin law to a non–degenerated distribution irrespectively of H 0 being satisfied or not. In addition, inorder to ensure its asymptotic correctness, this limit distribution must coincide with the asymptoticone of the testing statistic provided that H 0 holds.Consequently, the asymptotic limit established in Corollary 1 plays a fundamental role for definingappropriate bootstrap statistics. In this way, recall that1√ nn∑ ((Xi − X )( Y i − Y ) − E ( (X − µ X )(Y − µ Y ) ))i=1converges in law to Z (X−µX )(Y −µ Y ), irrespectively of H 0 being satisfied or not and, in addition, ifH 0 is satisfied then Γ (X−µX )(Y −µ Y ) = σ 2 Γ. Thus, this is a natural statistic to be mimicked by abootstrap one. Note that,(1nn∑ (Yi − Y ) ) (2 1n∑ ( ) ) √ Xi − µ X , (7)ni=1converges in law to (σ 2 +E(〈X −µ X , Θ〉 2 )Z X , whose covariance operator is (σ 2 +E(〈X −µ X , Θ〉 2 )Γ.Particularly, when H 0 is satisfied, this operator reduces again to σ 2 Γ. Consequently, another possibilityconsists in mimicking this second statistic by means of a bootstrap one, improving theapproximation suggested in the previous subsection. Note that the left term in the product inequation (7) could be substituted by any other estimator under H 0 of σ 2 that converges to a finiteconstant if H 0 does not hold. Anyway, this second approximation could lead to worst results underthe null hypothesis, because the possible dependency between X and ε is lost (as the resamplewould focus only on the X information).Two possibilities for mimicking the statistics which were above–mentioned are going to be explored,namely a “naive” paired bootstrap and a “wild” bootstrap approach. In order to achieve this goal, let{(Xi ∗, Y i ∗)}ni=1 be a collection of i.i.d. random elements drawn at random from (X 1, Y 1 ), . . . , (X n , Y n ),and let us consider the following “naive” paired bootstrap statisticT N∗n = 1 √ nn∑i=1i=1((X∗i − X ∗)( Y ∗i − Y ∗) − ( X i − X )( Y i − Y )) .− Y ∗ ) 2 , the empir-In addition, let us consider σn 2 = (1/n) ∑ ni=1 (Y i − Y ) 2 and σn∗2 = (1/n) ∑ nical estimator of σY2 under H 0 and its corresponding bootstrap version.i=1 (Y i∗The asymptotic behavior of the “naive” bootstrap statistic will be analyzed through some resultson bootstrapping general empirical measures obtained by Giné and Zinn (1990). It should be notedthat the bootstrap results in that paper refer to empirical process indexed by a class of functions F,that particularly extend to the bootstrap about the mean in separable Banach (and thus Hilbert)spaces. In order to establish this connection, it is enough to chooseF = {f ∈ H ′ |‖f‖ ′ ≤ 1}(see Giné (1997) and Kosorok (2008), for a general overview of indexed empirical process). F isimage admissible Suslin (considering the weak topology). In addition, F (h) = sup f∈F |f(h)| = ‖h‖7
Page 1 and 2: Bootstrap independence test for fun
Page 3 and 4: esults. A simulation study and a re
Page 5: Remark 2. Note that another usual a
Page 9 and 10: Items 1 and 2 in Lemma 1, together
Page 11 and 12: One asymptotic distribution free ba
Page 13 and 14: Nevertheless, in order to simplify
Page 15 and 16: 4.2 Data applicationFor the real da
Page 17: González-Rodríguez, G., Colubi, A

Bootstrap independence test for functional linear models

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