Bootstrap independence test for functional linear models
Bootstrap independence test for functional linear models
Bootstrap independence test for functional linear models
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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 <strong>Bootstrap</strong>).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 <strong>test</strong> 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) < ∞ <strong>for</strong>certain 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) <strong>for</strong> 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, <strong>for</strong> 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