- (2) This can be proved analogously to the derivation of the distribution of the Ç̄2 statistic (see Kudo 1963; Nüesch 1966). (3) This follows from (2) upon observing that Ç̃2(V ) is supported only on the nonnegative reals and that {[c, ∞) : c > 0} is a generating class.
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- ≤ C4B2 N,T,4(log N)/ √ T. Thus, by Lemma A.2 in Chernozhukov, Chetverikov, and Kato (2014) for every r > 0 and a universal constant K2 P max 1≤i≤N Uit(h, h0 ) − EP [Uit(h, h0 )] ≥ 2CB2 N,T,4(log N)/ √ T + r ≤e−Tr2/(48K2 βB4 N,T,4) + K216K2 βr−2 T−1 B4 N,T,4. Taking r = C1T−(1−c)/2B2 N,T,4 for 0 < c < 1 and C1 = 4( √ K2 + √ 3)Kβ ∨ C then yields P T−1 PT t=1 (dit(h)dit(h0) − EP [dit(h)dit(h0)]) Ã2 i si,T (h)si,T (h0) > C1B2 N,T,4T−(1−c)/2 log N ! ≤ 2T−c . By Hölder’s inequality EP " max 1≤i≤N max 1≤t≤T dit(h) Ãisi,T (h) 2 # ≤ v u u tEP max 1≤i≤N max 1≤t≤T dit(h) Ãisi,T (h) 4 ! ≤ √ T4KβB2 N,T,4.
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- By Lemma A.2 of Chernozhukov, Chetverikov, and Kato (2014), we thus have P max 1≤i≤N max h∈G\{g(i)} 1 T T X t=1 ait(h) ≥ CDT,N,2 p log((G − 1)N) √ T + BT,N,4 log((G − 1)N) T ! + t ! ≤e−t2/(3(D2 T,N,2/T) + K t2 1 T B2 T,N,4D2 T,N,2, for any t > 0. Taking t = T−(1−c)/2DT,N,2BT,N,4 and arranging the terms, we have P max 1≤i≤N max h∈G\{g(i)} 1 T T X t=1 ait(h) ≥ CDT,N,2BT,N,4T−(1−c)/2 log((G − 1)N) ! ≤ CT−c . We thus have P max 1≤i≤N max h∈G\{g(i)} 1 T T X t=1 ãit(h) > r 2 − r2 16 ! ≤ CT−c , by Assumption (9).
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- is therefore a sparse-convex set, as defined in Chernozhukov, Chetverikov, and Kato (2016). Let Zit(h) = dit(h)/(Ãisi,T (h)).
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- Lemma E.1 part (iii) and following the same argument of Lemma A.5 of Chernozhukov, Chetverikov, and Kato (2014). Following the argument in the proof of Step 2 of Theorem 4.2 of Chernozhukov, Chetverikov, and Kato (2014) under (8), (9) and that (10) and (11) implies U N,T,2 p log((G − 1)N/β) ≤ CT−1/6, it holds that P max 1≤i≤N max h∈G\{g(i)} √ T(E( dU i (h)) − dU i (h)) S̃U i,T (h) > (1 − r)cSNS β,N − CU N,T,2 ! ≤ β + CT−c . For the second term (34), let ait(h) = 2(dU it(h)−EP (dU it(h)))(EP (dU it(h))−EP ( dU i (h)))/(ÃisU i,T (h))2. The second term is P max 1≤i≤N max h∈G\{g(i)} 1 T T X t=1 ãit(h) > r 2 − r2 16 ! ≤P max 1≤i≤N max h∈G\{g(i)} 1 T T X t=1 ait(h) > 1 − r 2 r 2 − r2 16 ! + P max 1≤i≤N max h∈G\{g(i)} (S̃U i,T (h))2 (ÃisU i,T (h))2 − 1 > r 2 ! ,
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- Proof. (1) In the derivation of the weights (see e.g., Nüesch 1966) the weights correspond to probabilities of events that partition the sample space. To prove the asserted upper bounds use the representation from (4) and write w(p, p, V ) = P(Y2(∅) > 0) ≤1 − P(there is j = 1, . . . , p such that Y2,j(∅) ≤ 0) ≤1 − max j=1,...,p P(Y2,j(∅) ≤ 0) = 1 2 . For the other weights, the bound can be proved in a similar way.
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- Proof. Applying Theorem 1 in Soms (1976) with n = 2 yields the inequality 1 − Fν(x) ≥ (1 + x2 /ν) 1 − ν (ν + 2)x2 fν(x)/x. Now, x2 > 2 implies 1 − Fν(x) > 1 − 1 2 fν(x)/x. Lemma D.12. Let ξ1, . . . , ξT be independent centered random variables with E(ξ2 t ) = 1 and E(|ξt|2+ν) < ∞ for all 1 ≤ t ≤ T where 0 < ν ≤ 1. Let ST = PT t=1 ξt, V 2 T = PT t=1 ξ2 t and DT,ν = (T−1 PT t=1 E(|ξt|2+ν))1/(2+ν). Then uniformly in 0 ≤ x ≤ Tν/(2(2+ν))/DT,ν, Pr(ST /VT ≥ x) 1 − Φ(x) − 1 ≤ KT−ν/2 D2+ν T,ν (1 + x)2+ν .
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- Proof. Let Uj ∼ Ç2 j , j = 1, . . . , p. For notational convenience, write cN = cQLR α,N (V ). By Lemma D.13, cN is bounded from above by the (1 − α/N)-quantile of Up. Lemma 1 in Laurent and Massart (2000) implies that, for each x ≥ 0, P (Up − p ≥ 2 √ px + 2x) ≤ exp(−x). Suppose that N ≥ N0 ≥ α−1. Choosing x = log(N/α) in the above inequality yields P Up ≥ p + 2 p log(N/α) √ p + p log(N/α) ≤ α N . For N large enough, p + 2 p log(N/α) √ p + p log(N/α) < 2a log(N/α). This establishes the upper bound on cN . Let x = p
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- Proof. This lemma is first proved by Jing, Shao, and Wang (2003). Here we use the version by Chernozhukov, Chetverikov, and Kato (2014, Lemma A.1), which is based on de la Pena, Lai, and Shao (2009, Theorem 7.4).
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- This allows us to choose C1G1/2B3 N,T,4 as the sequence of constants in assumption (M.2) in Chernozhukov, Chetverikov, and Kato (2016). Lastly, we verify assumption (E.2) in Chernozhukov, Chetverikov, and Kato (2016). To this end, note that EP " max 1≤i≤N max h∈G\{g0 i } Zit(h)/(G1/4 B3 N,T,4) 4 # ≤ X h∈G EP max 1≤i≤N Zit(h)/(G1/4 B3 N,T,4) 4
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- This verifies assumption (M.1â€Â) in Chernozhukov, Chetverikov, and Kato (2016). Next, by Hölder’s inequality there is a constant C1 ≥ 1 depending only on Kβ such that 1 T T X i=1 EP [|Zit|3 ] ≤ C B4 N,T,4 3/4 ≤ C1G1/2 B3 N,T,4, 1 T T X i=1 EP [|Zit|4 ] ≤ CB4 N,T,4 ≤ (C1G1/2 B3 N,T,4)2 .
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- Thus, by Lemma A.3 in Chernozhukov, Chetverikov, and Kato (2014) for a universal constant K EP " max 1≤i≤N 1 T T X t=1 dit(h) Ãisi,T (h) # ≤ K T−1/2 p log N + 2T−3/4 p KβBN,T,4 log N .
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- Together with (26) this implies (25). It remains to prove (26). Note that EP [V 2 it ] ≤ B8 N,T,8 and EP [max1≤i≤T max1≤i≤N V 2 it ] ≤ TB8 N,T,8. By Lemma A.3 in Chernozhukov, Chetverikov, and Kato (2014) there is a universal constant K such that EP " max 1≤i≤N 1 T T X t=1 (V 2 it − EP [V 2 it ]) # ≤ KB4 N,T,8 log N √ T .
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- where b1 and b2 depend only on κ, Ä, a and p̄. As in Zhilova (2015), note that kÆik ai = sup γ∈Rpi :kγk=a−1 i γ0 Æi. We employ an approximation argument based on the inequality P max 1≤i≤N kÆik ai − cN ≤ ≤P max 1≤i≤N max γj∈Γi γ0 jÆi − cN ≤ 2 + P max 1≤i≤N sup γ∈Rpi :kγk=a−1 i min γj∈Γi |(γ − γj)0 Æi| > ! ≡A1 + A2. To bound A1, note that each γ0 jÆi is a normal random variable with standard deviation bounded between ā−1 and a−1. This follows from our assumptions about the covering Γi and E (γ0 jÆi)2 = γ0 jE[ÆiÆ0 i]γj = kγjk2 = a−2 i .
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- where the inequality holds because (S̃U i,T (h))2 ≥ (1−r/2)(ÃisU i,T (h))2 if 1−(S̃U i,T (h))2/(ÃisU i,T (h))2 > −r/2. The second term is bounded by CT−c by Lemma A.5 of Chernozhukov, Chetverikov, and Kato (2014) (Note that the statement of Lemma A.5 of Chernozhukov, Chetverikov, and Kato (2014) is about Ã̂j/Ãj (in their notation) but their proof is based on Ã̂2 j /Ã2 j ). For the first term, observe that T X t=1 EP ((ait(h)/T)2 ) = 1 T2 T X t=1 var(dU it(h)) (ÃisU i,T (h))4 (EP (dU it(h)) − EP ( dU i (h)))2 ≤ 1 T2 T X t=1 (EP (dU it(h)) − EP ( dU i (h)))2 (ÃisU i,T (h))2 , and T X t=1 E max 1≤i≤N max h∈G\{g(i)} (ait(h)/T)2 ≤ 1 T2 T X t=1 B2 T,N,4 max 1≤i≤N max h∈G\{g(i)} (EP (dU it(g0 i , h)) − EP ( dU i (g0 i , h)))2 /(ÃisU i,T (h))2 ≤ 1 T GB2 T,N,4D2 N,T,2.
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- Zhang, Xinyang and Guang Cheng (2017). “Guassian approximation for high dimensional vector under physical dependenceâ€Â. In: Bernoulli. forthcoming.
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Zhilova, Mayya (2015). “Simultaneous likelihood-based bootstrap confidence sets for a large number of modelsâ€Â. Working paper. Appendix A. Proofs of mains results In the proofs, we drop the g argument for ease of notation and write, e.g., dit(h) instead of dit(g, h) (or dit(g0 i , h)). The g argument is made explicit in the statements of the lemmas. Here, we provide proofs of Theorem 1 – Theorem 3. All supporting lemmas and the proof of Theorem 4 are given in the Supplementary Appendix. For our proof of the QLR procedure we analyze the limiting distribution of the QLR statistic, which we call the Ç̃2-distribution. Let V denote a nonsingular covariance matrix, and let X ∼ N(0, V ). The Ç̃2(V ) distribution is given by the distribution of the random variable W = min t≤0 (X − t)0 V −1 (X − t).