Bootstrap independence test for functional linear models

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for all h ∈ H and thus E(F 2 (X)) = E(‖X‖ 2 ) < ∞.Consider the bounded and linear (so continuous) operator δ from H to l ∞ (F) given by δ(h)(f) =δ h (f) = f(h) for all h ∈ H and all f ∈ F and denote by R(δ) ⊂ l ∞ (F) its range. As ‖δ(h)‖ = ‖h‖for all h ∈ H then, there exists δ −1 : R(δ) → H, so that δ −1 is continuous. In addition, as R(δ)is closed, Dugundji Theorem allows us to consider a continuous extension δ −1 : l ∞ (F) → H (seefor instance Kosorok (2008), Lemma 6.16 and Theorem 10.9). Thus, following the typical empiricalprocess notation, the empirical process (1/ √ n) ∑ ni=1 (δ X i− P) indexed in F is directly connectedwith (1/ √ n) ∑ ni=1 (X i − E(X)) by means of the continuous mapping δ −1 and vice–versa.Some consequences of this formulation applied to the work developed by Giné and Zinn (1990) leadto the results collected in following lemma.Lemma 1. Let ξ be a measurable mapping from a probabilistic space denoted by (Ω, σ, P ) to aseparable Hilbert space (H, 〈·, ·〉) with corresponding norm ‖ · ‖ so that E(‖ξ‖ 2 ) < ∞. Let {ξ i } n i=1 bea sequence of i.i.d. random elements with the same distribution as ξ, and let {ξi ∗}n i=1 be i.i.d. from{ξ i } n i=1 . Then1. √ n(ξ ∗ − ξ) converges in law to Z ξ a.s.–P2. ξ ∗ converges in probability to E(ξ) a.s.–P3. ‖ξ ∗ ‖ 2 converges in probability to E(‖ξ‖ 2 ) a.s.–PProof. To prove item 1 note that the CLT for separable Hilbert–valued random elements (see,for instance, Laha and Rohatgi (1979)) together with the Continuous Mapping Theorem appliedto δ guarantees that F ∈ CTL(P ). Thus, Theorem 2.4 of Giné and Zinn (1990) ensures thatn 1/2 ( ˆP n (w)−P n (w)) converges in law to a Gaussian process on F, G = δ(Z ξ ) a.s.–P . Consequently,by applying again the Continuous Mapping Theorem √ n(ξ ∗ − ξ) = δ −1 (n 1/2 ( ˆP n (w) − P n (w))) convergesin law to Z ξ = δ −1 (G).Items 2 and 3 can be checked in a similar way by applying Theorem 2.6 of Giné and Zinn (1990).Note that item 1 is also a direct consequence of Remark 2.5 of Giné and Zinn (1990); neverthelessit was proven based on Theorem 2.4 to illustrate the technique.The following theorem establishes the asymptotic consistency and correctness of the “naive” bootstrapapproach.Theorem 3. Under the conditions of Theorem 1, we have that √ nTnN∗ converges in law toZ (X−µX )(Y −µ Y ) a.s.–P . In addition, σn ∗2 converges in probability to σY 2a.s.–P .= σ2 + E(〈X − µ X , Θ〉 2 )Proof. First of all consider the bootstrap statisticS ∗ n = 1 √ nn∑i=1( (X∗i − µ X)(Y∗i − µ Y)−(X − µX)(Y − µY) )and note that { ( Xi ∗ − µ )X)(Y∗i − µ Y }ni=1 are i.i.d. H–valued random elements chosen at randomfrom the “bootstrap population” { ( )( )X i − µ X Yi − µ Y }ni=1 . Then, item 1 in Lemma 1 guaranteesthat Sn ∗ converges in law to Z (X−µX )(Y −µ Y ) a.s.–P .On the other hand, Sn ∗ equals √ nTnN∗ plus the following terms1√ n√ n(X ∗ − X) √ n(Y ∗ − Y ) + √ n(X ∗ − X)(Y ∗ − µ Y ) + (X ∗ − µ X ) √ n(Y ∗ − Y ).8
Items 1 and 2 in Lemma 1, together with Slutsky’s Theorem, ensure that these three terms convergein probability to 0 a.s.–P , and consequently the convergence in law stated in the theorem is proven.Finally, the convergence of σ ∗2n holds in virtue of items 2 and 3 in Lemma 1.The “naive” bootstrap approach is described in the following algorithm.Algorithm 1 (Naive Bootstrap).Step 1. Compute the value of the statistic T n (or the value T n /σ n ).Step 2. Draw {(Xi ∗, Y i ∗)}ni=1 , a sequence of i.i.d. random elements chosen at random from theinitial sample (X 1 , Y 1 ), . . . , (X n , Y n ), and compute a n = ‖TnN∗ ‖ (or b n = ‖TnN∗ ‖/σn).∗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 )Analogously, let {ε ∗ i }n i=1 be i.i.d. centered real random variables so that E( (ε ∗ i )2) = 1 and∫ ∞0 (P (|ε 1| > t) 1/2 ) < ∞ (to guarantee this last assumption, it is enough that E ( (ε ∗ i )d) < ∞ forcertain d > 2), and consider the “wild” bootstrap statisticT W ∗n = 1 √ nn∑ (Xi − X )( Y i − Y ) ε ∗ i .i=1In order to analyze the asymptotic behavior of the “wild” bootstrap statistic, the following lemmawill be fundamental. It is a particularization of a result due to Ledoux, Talagrand and Zinn (cf.Giné and Zinn (1990), and Ledoux and Talagrand (1988)). See also the Multiplier Central LimitTheorem in Kosorok (2008) for the empirical process indexed by a class of measurable functionscounterpart.Lemma 2. Let ξ be a measurable mapping from a probabilistic space denoted by (Ω, σ, P ) to aseparable Hilbert space (H, 〈·, ·〉) with corresponding norm ‖ · ‖ so that E(‖ξ‖ 2 ) < ∞. Let {ξ i } n i=1 bea sequence of i.i.d. random elements with the same distribution as ξ, and let {W i } n i=1 be a sequenceof i.i.d. random variables (in the same probability space and independent of {ξ i } n i=1 ) with E(W i) = 0and ∫ ∞0 (P (|W 1| > t) 1/2 ) < ∞, then the following are equivalent1. E(‖ξ‖ 2 ) < ∞ (and consequently √ n(ξ − E(ξ)) converges in law to Z ξ ).2. For almost all ω ∈ Ω, (1/ √ n) ∑ ni=1 W iξ i (ω) converges in law to Z ξ .As a consequence, the asymptotic consistency and correctness of the “wild” bootstrap approach isguaranteed by the following theorem.Theorem 4. Under the conditions of Theorem 1, we get that √ nTnW ∗Z (X−µX )(Y −µ Y ) a.s.–P .converges in law toProof. According to Lemma 2, for almost all ω ∈ Ω,S ∗ n = 1 √ nn∑i=1(Xwi − µ X)(Ywi− µ Y)ε∗iconverges in law to Z (X−µX )(Y −µ Y ). Moreover (Y w −µ Y ) and (X w −µ X ) converges to 0 (by SLLN).9
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Bootstrap independence test for functional linear models

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