Supplemental Material to Comparison of inferential methods in partially identified models in terms of error in coverage probability

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1 Supplemetal Material to Compariso of iferetial methods i partially idetified models i terms of error i coverage probability Federico A. Bugi Departmet of Ecoomics Duke Uiversity federico.bugi@duke.edu. September 22, 2014 Abstract This supplemet provides all the ecessary results to show that a stadardized versio of the statistics cosidered i Bugi (2014) satisfy the same coclusios regardig the rate of covergece of the error i the coverage probability. To establish these fidigs, we require stregtheig the fiite momet requiremets from fiite fourth absolute momets to slightly over sixth absolute momets. Thaks to a aoymous referee for suggestig me to explore this extesio. assistace. Ay ad all errors are my ow. Takuya Ura provided excellet research 1

2 1 Itroductio Bugi (2014) cosiders the problem of iferece i a partially idetified momet (i)equality models based o the criterio fuctio approach. Accordig to Assumptios CF.1-CF.2, the sample aalogue criterio fuctios are of the followig form Q (θ) = G([ m,j (θ)] J j=1), (1.1) where G is a o-stochastic fuctio. The fact that G is o-stochastic implies that the sample momet coditios are ot allowed to be stadardized, i.e., divided by the sample stadard deviatio. This could be cosidered a importat limitatio relative to the existig literature. The objective of this supplemet is to show that the results i Bugi (2014) ca be exteded to allow for stadardized sample momet coditios, i.e., criterio fuctios of the form Q (θ) = G([ m,j (θ)/ˆσ,j (θ)] J j=1), (1.2) where ˆσ,j 2 (θ) 1 (m(w i, θ) m,j (θ)) 2 is the sample variace of m j (W i, θ) for j = 1,..., J. The strategy to establish these results follows the oe used i Bugi (2014). The results i that paper are based o several represetatio theorems that express a statistic of iterest as a well-uderstood radom variable plus a error term, which is show to coverge to zero at a sufficietly fast rate. These type of represetatio results are obtaied for the sample statistic ad each of the approximatig statistics (bootstrap, asymptotic approximatio, subsamplig 1, ad subsamplig 2). This supplemet shows that stadardizatio adds a ew compoet to the error term of each of these represetatio results. As a cosequece, all of the results i Bugi (2014) exted for the criterio fuctios i Eq. (1.2) as log as this additioal compoet coverges to zero at a appropriate rate. This supplemet shows that occurs, provided that we stregthe the fiite momet requiremets from fiite fourth absolute momets to slightly over sixth absolute momets. We itroduce the followig otatio i additio to the oe used i Bugi (2014). For all j = 1,..., J, where the superscript s i v s,j (θ), ṽs,j 2 Assumptios σ 2 j (θ) V [m j (Z, θ)] ˆσ,j(θ) 2 1 (m j (Z i, θ) m,j (θ)) 2 σ,j(θ) 2 1 (m j (Z i, θ) E[m j (Z i, θ)]) 2 v s,j(θ) ( m,j (θ) E[m j (Z, θ)])/σ j (θ), ṽ s,j(θ) ( m,j (θ) E[m j (Z, θ)])/ˆσ,j (θ), v s,,j (θ) ( m,j(θ) m,j (θ) ) /ˆσ,j (θ), (θ), ad vs,,j (θ) deotes that the sample statistic is stadardized. I order to develop the results i this supplemet, we replace Assumptios A.5, CF.1, ad CF.2 i Bugi (2014) by the followig alterative assumptios. Assumptio A.5. E[ m(z, θ) 6+δ ] < for some δ > 0. 2

3 Assumptio CF.1. The populatio criterio fuctio is Q(θ) = G([E[m j (Z, θ)]/σ j (θ)] J j=1 ), where G : R J + R is a o-stochastic ad o-egative fuctio that is strictly icreasig i every coordiate, weakly covex, cotiuous, homogeeous of degree β > 0, ad satisfies G(y) = 0 if ad oly if y = 0 J. Assumptio CF.2. The populatio criterio fuctio is Q(θ) = G([E[m j (Z, θ)]/σ j (θ)] J j=1 ), where G : R J + R is oe of the followig two fuctios: (a) G(x) = J j=1 ϖ jx j or (b) G(x) = maxϖ j x j J j=1, where ϖ R J + is a arbitrary vector of positive costats. Assumptio A.5 stregthes the fiite momet requiremets from fiite fourth absolute momets i Assumptio A.5 to slightly over sixth absolute momets. By virtue of Assumptios CF.1-CF.2, the properly scaled sample aalogue criterio fuctio is ow give by Eq. (1.2). 3 Represetatio results with stadardizatio This sectio establishes represetatio results for the sample statistic ad each of the approximatig statistics. We ote that all the proofs ad several itermediate results are collected i the appedix of this supplemet. We begi with the represetatio result for the stadardized sample statistic. Theorem 3.1 (Represetatio result - Sample statistic). Assume Assumptios A.1-A.3, CF.1, ad that θ satisfies Assumptios A.4 ad A.5. Let ρ rak(v [m(z, θ)]). 1. If θ Θ I, the Q (θ) = H( Ȳ) + δ, where (a) for ay C <, P ( δ > C 1/2 ) = o( 1/2 ). (b) Ȳ : Ω R ρ is a zero mea sample average of i.i.d. observatios from a distributio with o-sigular variace-covariace matrix V = I ρ ad fiite fourth momets, (c) H : R ρ R is cotiuous, o-egative, weakly covex, ad homogeeous of degree β. H(y) = 0 implies for some o-zero vector b R ρ, b y 0. For ay µ > 0, ay h µ > 0, ay C 2 > 0, ay positive sequece g 1, ad ay positive sequece ε 1 with ε = o(1), H 1 (h ε ) y C 2 g H 1 (h) η where η = O( g ε ), uiformly i h. (d) If we add Assumptio CF.2, the for ay µ > 0, ay h µ > 0 ad ay sequece ε 1 with ε = o(1), H 1 (h ε ) H 1 (h) γ where γ O(ε ), uiformly i h. 2. If θ It(Θ I ), the lim ifq (θ) = 0 a.s. As a ext step, we cosider the stadardized versio of each of the approximatig statistics i Bugi (2014): bootstrap, asymptotic approximatio, subsamplig 1, ad subsamplig 2. I each case, oe ca stadardize resamplig momet iequalities usig: (a) the sample stadard deviatio or (b) the resamplig stadard deviatio. Both of these optios give the same formal results but the latter requires slightly loger argumets. For the sake of brevity, we express our approximatio i terms of the first optio. I particular: For the bootstrap, we replace Bugi (2014, Eq. (3.2)) with Q (θ) = G([ ( m,j(θ) m,j (θ) ) /ˆσ,j (θ)] 1[ m,j (θ)/ˆσ,j (θ) τ / ] J j=1) = G([v,s,j (θ)] 1[ m,j (θ)/ˆσ,j (θ) τ / ] J j=1), where m (θ) is the sample mea for the bootstrap sample. 3

4 For the asymptotic approximatio, we replace Bugi (2014, Eq. (4.2)) with Q AA [ ] (θ) = G ζ i (m j (Z i, θ) m,j (θ)) /ˆσ,j (θ) where ζ i is a i.i.d. sample with ζ i N(0, 1), idepedet of Z i. For subsamplig 1, we replace Bugi (2014, Eq. (5.2)) with 1[ m,j (θ)/ˆσ,j (θ) τ / ] Q SS1 (θ) = G([ b ( m SS,b,j(θ) m,j (θ))/ˆσ,j (θ)] 1[ m,j (θ)/ˆσ,j (θ) τ / ] J j=1), where m SS,b (θ) is the sample mea for the subsamplig sample. For subsamplig 2, we replace Bugi (2014, Eq. (5.4)) with Q SS2 (θ) = G([ b m SS,b,j(θ)/ˆσ,j (θ)] J j=1). where m SS,b (θ) is the sample mea for the subsamplig sample. We ow establish the aalogous represetatio results for each of these stadardized approximatio methods. It is relevat to poit out that the proofs of each of these build heavily o the correspodig represetatio results i Bugi (2014). Theorem 3.2 (Represetatio result - Bootstrap). Assume Assumptios A.1-A.3, CF.1, ad that θ satisfies Assumptios A.4-A.5. Let ρ rak(v [m(z, θ)]). 1. If θ Θ I the Q (θ) = H( Ȳ ) + δ, where (a) for ay C <, P ( δ > C 1/2 X ) = o( 1/2 ) a.s. (b) Ȳ X : Ω R ρ is a zero (coditioal) mea sample average of i.i.d. observatios from a distributio with a (coditioal) variace-covariace matrix ˆV which is o-sigular a.s. ad fiite (coditioal) fourth momets a.s., ad ˆV I p O p ( 1/2 ). (c) H : R ρ R is the same fuctio as i Theorem If θ It(Θ I ), the lim ifq (θ) = 0 a.s. Theorem 3.3 (Represetatio result - AA). Assume Assumptios A.1-A.3, CF.1, ad that θ satisfies Assumptios A.4-A.5. Let ρ rak(v [m(z, θ)]). 1. If θ Θ I the Q AA (θ) = H( Ȳ AA ) + δ AA, where (a) for ay C <, P ( δ AA > C 1/2 X ) = o( 1/2 ) a.s. (b) Ȳ AA X : Ω R ρ is a zero (coditioal) mea sample average of i.i.d. observatios from a distributio with a (coditioal) variace-covariace matrix ˆV which is o-sigular a.s. ad fiite (coditioal) fourth momets a.s., ad ˆV I p O p ( 1/2 ). (c) H : R ρ R is the same fuctio as i Theorem If θ It(Θ I ), the lim ifq AA (θ) = 0 a.s. J j=1, 4

5 Theorem 3.4 (Represetatio result - SS1). Assume Assumptios A.1-A.3, CF.1, ad that θ satisfies Assumptios A.4-A.5. Let ρ rak(v [m(z, θ)]). 1. If θ Θ I the Q SS1 (θ) = H( b Ȳ SS ) + δss1, where (a) lim ifp (δ SS1 b = 0 X, ) 1[ v (m θ ) τ ] ad lim ifδ SS1 = 0, a.s. (b) Ȳ SS X : Ω R ρ is a zero (coditioal) mea sample average of b observatios sampled without replacemet from a distributio with a (coditioal) variace-covariace matrix ˆV which is o-sigular a.s. ad fiite (coditioal) fourth momets a.s., ad ˆV I ρ O p ( 1/2 ). (c) H : R ρ R is the same fuctio as i Theorem If θ It(Θ I ), the lim ifq SS1 (θ) = 0 a.s. Theorem 3.5 (Represetatio result - SS2). Assume Assumptios A.1-A.3, CF.1, ad that θ satisfies Assumptios A.4-A.5. Let ρ rak(v [m(z, θ)]). 1. If θ Θ I, the Q SS2 (θ) = H( b Ȳ SS ) + δss2, where (a) for some C > 0, P ( δ SS2 b > C, (l l )b / X ) = o(b 1/2 ) a.s. (b) Ȳ SS X : Ω R ρ is a zero (coditioal) mea sample average of b observatios sampled without replacemet from a distributio with a (coditioal) variace-covariace matrix ˆV which is o-sigular a.s. ad fiite (coditioal) fourth momets a.s., ad ˆV I ρ O p ( 1/2 ). (c) H : R ρ R is the same fuctio as i Theorem If θ It(Θ I ), the lim ifq SS2 (θ) = 0 a.s. Remark 3.1. Notice that the represetatio result for the sample statistic (i.e. Theorem 3.1) is the oly oe of these results that requires slightly more tha fiite sixth absolute momets i Assumptio A.5. All other represetatio results i this supplemet ca be established oly usig fiite fourth absolute momets. 4 Coclusio This supplemet provides all the ecessary results to show that a stadardized versio of the statistics cosidered by Bugi (2014) satisfy the same coclusios regardig rate of covergece of the error i the coverage probability. Our strategy is to establish that all the represetatio results used i Bugi (2014) for o-stadardized statistics ca also be established for stadardized statistics, both for the sample statistic ad for all approximatig statistics (bootstrap, asymptotic approximatio, subsamplig 1, ad subsamplig 2). To establish these result we employ slightly loger formal argumets ad we stregthe the fiite momet requiremets from fiite fourth absolute momets to slightly more tha fiite sixth absolute momets. Usig the represetatio results i this supplemet, oe ca repeat the argumets i Bugi (2014) to establish the exact same rates of covergece of the error i the coverage probability. 5

6 Appedix A Appedix A.1 Proofs of theorems Proof of Theorem 3.1. Part 1. We show the result by slightly modifyig the argumets i Bugi (2014, Part 1, Theorem A.1). Let Σ V [m(z, θ)] ad, thus, ρ = rak(v [m(z, θ)]), ad let D Diag(Σ) ad Ω D 1/2 ΣD 1/2. By defiitio, there are (J ρ) coordiates of D 1/2 m(z i, θ) that ca be expressed as a liear combiatio of the remaiig ρ coordiates of D 1/2 m(z i, θ) for all i = 1,..., (a.s.). We refer to these ρ coordiates as the fudametal coordiates. Without loss of geerality, we ca rearrage D 1/2 m(z i, θ) s.t. the fudametal coordiates are the last ρ oes. This implies that there is a matrix A R (J ρ) ρ s.t. D 1/2 m(z i, θ) = [A, I ρ] m j(z i, θ)/σ j(θ) ρ j=1 for all i = 1,...,, a.s. (A.1) Let Ω ρ deote the variace-covariace matrix of m j(z i, θ)/σ j(θ) ρ j=1, which is ecessarily positive defiite ad let Ω 1/2 ρ deote the iverse of its square root. The, we defie Y i Ω 1/2 ρ (m j(z i, θ) E[m j(z i, θ)])/σ j(θ) ρ j=1 for all i = 1,..., ad the matrix B [A, I ρ] Ωρ 1/2 R J ρ. By these defiitios ad by Eq. (A.1), we the coclude that BY i = D 1/2 (m(z i, θ) E[m(Z i, θ)]) for all i = 1,...,. Accordig to this defiitio, v(θ) s = BȲ, E[Yi] = 0ρ, V (Yi) = Ip, ad E[ Yi c ] < for all c > 0 s.t. E[ m(z, θ) c ] <. Moreover, m(z i, θ) are i.i.d., ad so Y i are also i.i.d. Let B j R 1 ρ deote the j th row of B. The fuctio H(y) : R ρ R is defied as H(y) G([B jy] 1[E[m j(z, θ)] = 0] J j=1). The same argumets i Bugi (2014, Theorem A.1) ca be used to show that this fuctio has all the desired properties. By defiitio, H( Ȳ) G([Bj Ȳ ] 1[E[m j(z, θ)] = 0] J j=1)) = G([v s,j(θ)] 1[E[m j(z, θ)] = 0] J j=1)) ad δ Q (θ) H( Ȳ). I tur, δ = δ,1 + δ,2 where δ,1 Q (θ) G( [ ṽ s,j(θ) ] 1[E[mj(Z, θ)] = 0]J j=1), δ,2 G( [ ṽ s,j(θ) ] 1[E[mj(Z, θ)] = 0]J j=1) G( [ v s,j(θ) ] 1[E[mj(Z, θ)] = 0]J j=1). (A.2) We cosider these two terms i separate steps. Step 1 shows that shows that P ( δ,1 > 0) = o( 1/2 ) by a slightly modifyig the argumets i Bugi (2014, Theorem A.1) ad Step 2 shows that P ( δ,2 > C 1/2 ) = o( 1/2 ) for all C <. The combiatio of these two steps completes the proof of this part. Step 1: Argumet for δ,1. Sice θ Θ I Θ I the E[m(Z, θ)] 0 J. Let S j 1,..., J : E[m j(z, θ)] = 0, S 1,..., J/S (which may be empty), η mi j S E[m j(z, θ)]/σ j(θ) > 0 if S ad η 0 if S =. For ay j = 1,..., J, cosider the followig argumet. First, otice that: [ m,j(θ)/ˆσ,j(θ)] = [ m,j(θ)/ˆσ,j(θ)] 1[E[m j(z, θ)] = 0] + [ m,j(θ)/ˆσ,j(θ)] 1[E[m j(z, θ)] > 0] Secod, otice that: [ṽ,j(θ)] 1[E[m j(z, θ)] = 0]. [ m,j(θ)/ˆσ,j(θ)] = [ m,j(θ)/ˆσ,j(θ)] 1[E[m j(z, θ)] = 0] + [ m,j(θ)/ˆσ,j(θ)] 1[E[m j(z, θ)] > 0] [ṽ,j(θ)] s 1[E[m j(z, θ)] = 0]+ = [v,j(θ) s + E j[m j(z, θ)]/σ j(θ)] 1[E[m j(z, θ)] > 0](σ j(θ)/ˆσ,j(θ)) [ṽ s,j(θ)] 1[E[m j(z, θ)] = 0] + [v s,j(θ) + η] 1[E[m j(z, θ)] > 0](σ j(θ)/ˆσ,j(θ)). From both of these, we extract the followig coclusios. If S = the δ,1 = 0 ad so P ( δ,1 > 0) = 0 = o( 1/2 ). If S the otice that mi j S v s,j(θ) + η 0 δ,1 = 0. Therefore, P ( δ,1 > 0) P (mi j S v s,j(θ) + 6

7 η < 0). Thus, the proof of this step is completed by showig that: P (mi v,j(θ) s + η < 0) = o( 1/2 ). j S To show this, otice that for ay λ (1/4, 1/2) we have that: P (mi v,j(θ) s + η < 0) = P ( j Sv,j(θ) s < η ) P ( v,j(θ) s > η ) j S j S j S[1[ λ > η] + P ( v s,j(θ) > λ )] = o( 1/2 ), where the rate of covergece follows from η > 0 (S 1 ) ad Lemma ) A.2. Step 2: Argumet for δ,2. We show that P ( δ,2 > C 1/2 = o( 1/2 ) for all C <. This part of the proof is the oly geuiely ew part of the argumet relative to Bugi (2014, Theorem A.1). Fix c > 0 arbitrarily small so that δ > (4 + 2δ)2c + 4 ad so O( 1/2 δ/4+(1+δ/2)2c ) = o( 1/2 ). The, P ( δ,2 > C 1/2 ) = P ( δ,2 > C 1/2 J j=1 v,j(θ) s c/2β ˆσ,j(θ) σ j(θ) 1/2 2c ) +P ( δ,2 > C 1/2 J j=1 v,j(θ) s > c/2β ˆσ,j(θ) σ j(θ) > 1/2 2c ) J J P ( v,j(θ) s > c/2β ) + P ( ˆσ,j(θ) σ j(θ) > 1/2 2c ), j=1 j=1 where the iequality holds by Lemma A.4 for all sufficietly large. The first sum is o( 1/2 ) by Lemma A.2 (with λ = c/2β) ad the secod sum is o( 1/2 ) by Lemma A.8 (with λ = 2c). This completes the proof of this step. Part 2. If θ It(Θ I) the E[m j(z, θ)] > 0 for all j = 1,..., J. Defie η = mi j=1,...,je[m j(z, θ)]/σ j(θ) > 0. Cosider ay positive sequece ε 1 s.t. l l /ε = o(1) ad ε / = o(1). Suppose that the evet v s (m θ ) ε occurs. The, for all large eough, Q (θ) G([( ε / + η) ] J j=1) = 0. Therefore, lim if v s (m θ ) ε lim ifq (θ) = 0. By the LIL, P (lim if v s (m θ ) ε ) = 1 ad the result follows. Proof of Theorem 3.2. Part 1. We show the results by slightly modifyig the argumets i Bugi (2014, Part 1, Theorem A.2). Let the matrices Σ R J J, D Diag(Σ), A R (J ρ) ρ, Ω ρ R ρ ρ, ad B [A, I ρ] Ω 1/2 ρ R J ρ be defied as i proof of Theorem 3.1. Also, defie ˆΣ ˆV [m(z, θ)] ad ˆD Diag(ˆΣ). Sice bootstrap samples are costructed from the origial radom sample, it has to be the case that the coordiates of m(z i, θ) ca be arraged ito the same ρ fudametal ad (J ρ) o-fudametal coordiates described i Theorem 3.1. I particular, the bootstrap sample satisfies Eq. (A.1). The, by the argumet i Theorem 3.1, we defie Yi Ωρ 1/2 (m j(zi, θ) m,j(θ))/ˆσ j(θ) ρ j=1 for all i = 1,...,, which ca be show to satisfy BYi = ˆD 1/2 (m(zi, θ) m (θ)) for all i = 1,...,. Accordig to this defiitio of Y i, v s, (θ) = BȲ, E[Yi X ] = 0 ρ, V [Yi X ] = ˆV where ˆV = Ω 1/2, where ˆΩ ρ deotes the sample correlatio of m j(z, θ) ρ j=1. By the SLLN, ˆΣ Σ = o(1) a.s. which ρ ˆΩρΩ 1/2 ρ implies that ˆD D = o(1) a.s. ad ˆΩ ρ Ω ρ = o(1) a.s. By this ad the CMT, it the follows that ˆV I ρ = o(1) a.s., implyig that ˆV is o-sigular, a.s. By a similar argumet, the SLLN implies that E( Y i c X ) < for all c > 0 s.t. E[ m(z, θ) c ] <. Moreover, Y i X are i.i.d. because m(z i, θ) X are i.i.d. Fially, if we add Assumptio A.5, the CLT ad Slutzky s theorem imply that (ˆΩ ˆΩ) = O p(1) ad, so, ˆV I ρ O p( 1/2 ). Let H(y) : R ρ R be defied as i Theorem 3.1. The same argumets i Bugi (2014, Theorem A.1) ca be used to show that this fuctio has all the desired properties. By defiitio, it the follows that H( Ȳ ) G([B j Ȳ ] 1[E[m j(z, θ)] = 0] J j=1)) = G([v,s,j (θ)] 1[E[mj(Z, θ)] = 0]J j=1)) ad δ Q (θ) H( Ȳ ). To coclude the proof, it suffices to show that this error term satisfies P ( δ > 0 X ) = o( 1/2 ). Sice θ Θ I Θ I the E[m j(z, θ)] 0 j = 1,..., J. Let S j 1,..., J : E[m j(z, θ)] > 0, 7

8 S 1,..., J/S (which may be empty), η mi j S E[m j(z, θ)]/σ j(θ) > 0 if S ad η 0 if S =, σ 2 L(θ) mi j=1,...,j σ 2 j (θ), σ 2 H(θ) max j=1,...,j σ 2 j (θ), C L (1 + σ 2 L(θ)/(2σ 2 H)) 1/2, ad C H (1 σ 2 L(θ)/(2σ 2 H)) 1/2 > 0. Cosider the followig derivatio: v s (m θ ) τ /C H ˆσ 2 (θ) σ 2 (θ) σ 2 L(θ)/2 J j=1 ṽ s,j(m θ ) (σ j(θ)/ˆσ,j(θ))/(τ /C H) C L σ j(θ)/ˆσ,j(θ) C H J j=1 ( m,j(θ) E[m j(z, θ)]) /ˆσ,j(θ) τ C L σ j(θ)/ˆσ,j(θ) C H j S m,j(θ)/ˆσ,j(θ) τ j S m,j(θ)/ˆσ,j(θ) C L η τ j S m,j(θ)/ˆσ,j(θ) τ j S m,j(θ)/ˆσ,j(θ) > τ = Q (θ) = 0, where all iclusios are based o elemetary argumets ad the last iclusio holds for all sufficietly large. From here, it the follows that P (δ = 0 X ) 1/2 (P ( v s (m θ ) > τ /C H X ) + P ( ˆσ 2 (θ) σ 2 (θ) > σ 2 L(θ)/2 X ) 1) 1/2 = (1[ v s (m θ ) > τ /C H] + 1[ ˆσ 2 (θ) σ 2 (θ) > σ 2 L(θ)/2]) 1/2. where the equality uses the fact that, coditioally o X, the evets v s (m θ ) > τ /C H ad ˆσ 2 (θ) σ 2 (θ) > σ 2 L(θ)/2 are o-stochastic. To complete the proof, it suffices to show that P (lim if v s (m θ ) > τ /C H) = 1 ad P (lim if ˆσ 2 (θ) σ 2 (θ) < σ 2 L(θ)/2) = 1. These two results follow from the LIL ad the SLLN, respectively. Part 2. If θ It(Θ I) the E[m j(z, θ)] > 0 for all j = 1,..., J. Let η mi j=1,...,j E[m j(z, θ)]/σ j(θ) > 0, σ 2 L(θ) mi j=1,...,j σ 2 j (θ), σ 2 H(θ) max j=1,...,j σ 2 j (θ), C L (1 + σ 2 L(θ)/(2σ 2 H)) 1/2, ad C H (1 σ 2 L(θ)/(2σ 2 H)) 1/2 > 0. By the same argumet as i part 1, we have the followig derivatio: v s (m θ ) τ /C H ˆσ 2 (θ) σ 2 (θ) σ 2 L(θ)/2 J j=1 ṽ s,j(m θ ) (σ j(θ)/ˆσ,j(θ))/(τ /C H) C L σ j(θ)/ˆσ,j(θ) C H J j=1 ( m,j(θ) E[m j(z, θ)]) /ˆσ,j(θ) τ C L σ j(θ)/ˆσ,j(θ) C H J j=1 m,j(θ)/ˆσ,j(θ) C L η τ j S m,j(θ)/ˆσ,j(θ) > τ = Q (θ) = 0, where all iclusios are based o elemetary argumets ad the last iclusio holds for all sufficietly large. From here, it the follows that lim if v s (m θ ) τ /C H lim if ˆσ 2 (θ) σ 2 (θ) σ 2 L(θ)/2 = lim if v s (m θ ) τ /C H ˆσ 2 (θ) σ 2 (θ) σ 2 L(θ)/2 lim ifq (θ) = 0. To complete the proof, it suffices to show that P (lim if v s (m θ ) > τ /C H) = 1 ad P (lim if ˆσ 2 (θ) σ 2 (θ) < σ 2 L(θ)/2) = 1. These two results follow from the LIL ad the SLLN, respectively. Proof of Theorem 3.3. This proof follows closely the argumets used to prove Theorem 3.2. The oly differece is that we replace Y i with Yi AA defied by Yi AA Ω 1/2 ρ ζ i(m j(zi, θ) m,j(θ))/ˆσ j(θ) ρ j=1. Proof of Theorem 3.4. This proof follows closely the argumets used to prove Theorem 3.2. The oly differece is that we replace Y i with Yi SS b SS defied by Yi Ω 1/2 ρ (m j(zi SS, θ) m,j(θ))/ˆσ j(θ) ρ j=1. Proof of Theorem 3.5. We show the results by slightly modifyig the argumets i Bugi (2014, Theorem A.15). Sice the structure of this proof is differet from the oe used to prove Theorems 3.3 or 3.4, we cover the mai differeces. Part 1. Defie δ SS 2 Q SS 2 (θ) G([vs,SS b (m, j,θ)] 1[E[m j(z, θ)] = 0] J j=1), where v s,ss b( m SS,j(θ) m,j(θ))/ˆσ,j(θ). Parts (b)-(c) follow from Theorem 3.4 so we focus o part (a). (m j,θ) 8

9 For arbitrary δ (0, 1/2), let S j 1,..., J : E[m j(z, θ)] = 0, S 1,..., J/S, η mi j S E[m j(z, θ)]/σ j(θ) > 0, σ 2 L(θ) mi j=1,...,j σ 2 j (θ), λ max j=1,...,j(1 + δ) 2V [m j(z, θ)], ad let A defied as follows: A v s,ss (m j,θ) b (1 δ)/2 J j=1 v s (m j,θ ) λ l l J j=1 ˆσ 2 (θ) σ 2 (θ) σ 2 L(θ)/2. (A.3) For ay C > 0, otice that: be bp ( δ SS 2 > C (l l )b / X ) b P (A c X ) + b P ( δ SS 2 > C (l l )b / A X ). (A.4) The proof is completed by showig that the two terms o the RHS of Eq. (A.4) are o(1), a.s. We begi with the first term i the RHS of Eq. (A.4). Cosider the followig derivatio: bp (A c X ) = = J j=1 J j=1 bp ( v s,ss b (m, j,θ) > b (1 δ)/2 X ) + J j=1 bp ( v(m s j,θ ) > λ l l X )+ J j=1 bp ( ˆσ,j(θ) 2 σj 2 (θ) > σl(θ)/2 X 2 ) bp ( v s,ss b (m, j,θ) > b (1 δ)/2 X ) + J j=1 b1( v(m s j,θ ) > λ l l )+, b1( ˆσ,j(θ) 2 σj 2 (θ) > σl(θ)/2), 2 J j=1 where the first equality follows from elemetary argumets ad the secod equality follows from the fact that v (m j,θ ) X is determiistic. Fix j = 1,..., J ad ε > 0 arbitrarily. By the LIL, lim if v (m j,θ ) λ l l J j=1 a.s. ad so P (lim b 1( v (m j,θ ) > λ l l ) = 0) = 1. By the SLLN, lim if ˆσ 2,j(θ) σ 2 j (θ) σ 2 L(θ)/2 a.s. ad so P (lim b 1( ˆσ 2,j(θ) σ 2 j (θ) > σ 2 L(θ)/2) = 0) = 1. Next, cosider the followig derivatio: P (lim ifp ( v s,ss b (m, j,θ) > b (1 δ)/2 X ) b ε) ( ) lim ifp ( v s,ss b P (m, j,θ) > b (1 δ)/2 X )b (δ 1) (1 + ε) lim ifb (δ 1/2) (1 + ε) ε P (lim ifp ( v s,ss b (m, j,θ) > b (1 δ)/2 X )b (δ 1) (1 + ε)) +P (lim ifb (δ 1/2) (1 + ε) ε) 1 = P (lim ifp ( v s,ss b (m, j,θ) > b (1 δ)/2 X )b (δ 1) (1 + ε)) P (lim if1 1 + ε) = 1, where the two first two iequalities follow from elemetary argumets, the followig equality follows from b (δ 1/2) = o(1), ad the third iequality is show later i Eq. (A.5). To complete this argumet, cosider the followig derivatio: E(v s,ss = = b (m, j,θ) 2 X ) ( ( q 1 q s=1 b 1 q 1 q s=1 b 1 b a=1 b (mj(xss i,s, θ) m,j(θ)) 2 /ˆσ 2,j(θ)+ b b=1,b a (mj(xss a,s, θ) m,j(θ))(m j(x SS 1 (mj(xi, θ) m,j(θ))2 /ˆσ 2,j(θ)+ ) b,s, θ) m,j(θ))/ˆσ,j(θ) 2 ) +2 1 ( 1) 1 a=1 b=1,b a (mj(xa, θ) m,j(θ))(mj(x b, θ) m,j(θ))/ˆσ 2,j(θ) where the first equality holds by expadig squares, the followig equality holds by the fact that we are samplig without replacemet, ad the fial iequality holds by verifyig that, i the previous lie, the first term equals oe ad the secod term is o-positive by the egative associated produced by samplig without replacemet (see Joag-Dev ad Proscha (1983, Sectio 3.2(a))). Chebyshev s iequality the implies that 1 E(v s,ss b (m, j,θ) 2 X ) P ( v s,ss b (m, j,θ) > b (1 δ)/2 X )b (1 δ), (A.5) which completes the proof for the first term o the RHS of Eq. (A.4). 1, 9

10 We ow cosider the secod term RHS of Eq. (A.4). Coditio o X ad assume that A occurs. The, Q SS 2 (θ) = G([vs,SS b (m, j,θ) + b /v(m s j,θ ) + b E[m j(z, θ)]/ˆσ,j(θ)] J j=1) G([v s,ss b (m, j,θ) λ (l l )b / + b E[m j(z, θ)]/ˆσ,j(θ)] J j=1) [v s,ss G b (m, j,θ) λ J (l l )b /] 1[E[m j(z, θ)] = 0]+ [ b 1/2 δ λ (l l )b / + b 2η] 1[E[m j(z, θ)] > 0] = G([v s,ss (m j,θ) λ (l l )b /] 1[E[m j(z, θ)] = 0] J j=1) G([v s,ss (m j,θ)] 1[E[m j(z, θ)] = 0] J j=1 + λ (l l )b /1 J), j=1 (A.6) (A.7) where the first equality is elemetary, the secod iequality holds because G is icreasig ad A implies that v(m s j,θ ) λ l l, the secod iequality follows from the fact that A implies that v s,ss b (m, j,θ) > b (1 δ)/2, the ext equality follows for all large eough as η, λ, δ > 0, ad the fial iequality holds by elemetary argumets. By usig similar argumets, we ca establish a aalogous lower boud for Q s,ss 2 (θ). As a cosequece, if A occurs, Q SS 2 (θ) [ G([v s,ss (m j,θ)] 1[E[m j(z, θ)] = 0] J j=1 λ (l l )b /1 J), G([v s,ss (m j,θ)] 1[E[m j(z, θ)] = 0] J j=1 + λ (l l )b /1 J) By Assumptio CF.2, x R J ad ε > 0, D > 0 s.t. G(x + ε) G(x) D ε. If we set C = Dλ > 0, δ SS 2 b =, QSS 2 (θ) G([vs,SS b (m, j,θ)] 1[E[m j(z, θ)] = 0] J j=1) max r 1,1 G([v s,ss (m j,θ)] 1[E[m j(z, θ)] = 0] J j=1 + rλ (l l )b /1 J) G([v s,ss (m j,θ)] 1[E[m j(z, θ)] = 0] J j=1) D λ (l l )b / 1 J = Dλ (l l )b /. This completes the argumet for the secod term o the RHS of Eq. (A.4). Part 2. Sice Q SS 2 b (θ) 0, it suffices to show that, b P (Q SS 2 (θ) > 0 X) = o(1). For arbitrary δ (0, 1/2), let η mi j=1,...,j E[m j(z, θ)]/σ j(θ) > 0 ad let A be defied as i Eq. (A.3). By elemetary argumets bp (Q SS 2 b (θ) > 0 X), b P (A c X ) + b P (Q SS 2 (θ) > 0 A X). We ca ow repeat argumets used i part 1 to argue that both terms o the RHS are o(1), a.s. O the oe had, the same argumet as i part 1 implies that b P (A c X ) = o(1) a.s. O the other had, if A occurs, the argumet used i Eq. (A.6) implies that Q SS 2 (θ) G([ b1/2 δ λ (l l )b / + η2 b ] J j=1). Sice η, δ > 0, the RHS expressio is equal to zero for all large eough. This the implies that b P (Q SS 2 (θ) > 0 A X) = o(1), completig the proof. A.2 Proofs of itermediate results Lemma A.1. Assume Assumptio A.1 ad that θ Θ satisfies E[ m(z, θ) 2+ψ ] < for some ψ > 0. For all j = 1,..., J, [ E m j(z i, θ) E[m j(z i, θ)] 2+ψ] <, E [ (m j(z i, θ) E[m j(z i, θ)]) 2 σ 2 j (θ) 1+ψ/2] <. ] Proof. Fix j = 1,..., J arbitrarily ad let σ 2 σ 2 j (θ) ad M i m j(z i, θ) for all i = 1,...,. By Assumptio A.5 ad Holder s iequality, E[ M i ] 2+ψ E[ M i 2+ψ ] <. Aother applicatio of Holder s iequality yields M i E[M i] 2+ψ M i 2+ψ + E[M i] 2+ψ ad, so, E M i E[M i] 2+ψ E M i 2+ψ + E[M i] 2+ψ <. This proves the first result. 10

11 Defie Y i (M i E[M i]) 2 /σ 2 1 ad φ ψ/2. By defiitio, [ E[ (M i E[M i]) 2 σ 2 1+φ ] = σ 2+φ (Mi E[Mi])2 1+φ] E 1 = σ 2+φ E[ Y i 1+φ ]. σ 2 So, it suffices to show that E[ Y i 1+φ ] <. For the remaider of the proof let β > 1 be arbitrarily chose. Notice that E[ Y i 1+φ ] = E[ Y i 1+φ 1[ Y i > β]] + E[ Y i 1+φ 1[ Y i β]] E[ Y i 1+φ 1[ Y i > β]] + β 1+φ. So, it suffices to show that E[ Y i 1+φ 1[ Y i > β]] <. Sice β > 1, it follows that (M i E[M i]) 2 Y i > β > β + 1 σ 2 Y i (Mi E[Mi])2 σ 2 ad, therefore, E[ Y i 1+φ 1[ Y i > β]] E [ E [ Mi E[Mi] σ Mi E[Mi] σ 2+ψ [ (Mi E[M i]) 2 ] ] 1 > β + 1 σ 2 ] 2+ψ = 1 σ 2+ψ E[ Mi E[Mi] 2+ψ ] <. This proves the secod result ad completes the proof. Remark A.1. Lemma A.1 ca be used with ψ = 2 uder Assumptio A.5 or with ψ > 4 uder Assumptio A.5. Lemma A.2. Assume Assumptio A.1 ad that θ Θ satisfies Assumptio A.5. For all j = 1,..., J ad c > 0, P [ v s,j(θ) > c ] = o( 1/2 ). Proof. For ay ψ > 0, cosider the followig argumet based o Chebyshev s iequality: P [ v s,j(θ) > c ] (2+ψ)c = P [ v s,j(θ) 2+ψ > (2+ψ)c ] (2+ψ)c E[ v,j(θ) s 2+ψ ] = E 2+ψ [m j(z, θ) E[m j(z, θ)]] 1/2 σ j(θ) = σ (2+ψ) j (θ) 1 ψ/2 E 2+ψ [m j(z, θ) E[m j(z, θ)]] [( )] σ (2+ψ) j (θ) 1 ψ/2 E m j(z, θ) E[m j(z, θ)] 2+ψ [ σ (2+ψ) j (θ) ψ/2 E m j(z, θ) E[m j(z, θ)] 2+ψ]. From this, Lemma A.1 with ψ = 2, ad Assumptio A.5, we coclude that where we have used that c > 0 ad ψ 1. P [ v s,j(θ) > c ] = O( ψ/2 (2+ψ)c ) = o( 1/2 ), 11

12 Lemma A.3. Assume Assumptio CF.1 ad that θ Θ I satisfies Assumptio A.4. The, δ,2 i Eq. (A.2) satisfies [ ( ) ] δ,2 1 max j=1,...,j ˆσ,j (θ) σ j (θ) β 1 G([v,j(θ)] 1[E[m j(z, θ)] = 0] J j=1), mi j=1,...,j σ j (θ) Proof. For each j = 1,..., J, defie v L,j(θ) miṽ s,j(θ), v s,j(θ) ad v H,j(θ) maxṽ s,j(θ), v s,j(θ). Sice θ Θ I, E[m(Z, θ)] 0 J. Defie S 0, (θ) j = 1,..., J : E[m j(z, θ)] = 0 v L,j(θ) < 0 (which may be empty). First, cosider the case whe S 0, (θ) =. I this case, either E[m j(z, θ)] > 0 or 0 v L,j(θ) v H,j(θ) for all j = 1,..., J. Therefore, [ṽ s,j(θ)] 1[E[m j(z, θ)] = 0] J j=1 = [v s,j(θ)] 1[E[m j(z, θ)] = 0] J j=1 = 0 J, ad, thus, δ,2 = 0, ad the statemet holds. Secod, cosider the case whe S 0, (θ). Defie α max j S0, (θ)v L,j(θ)/v H,j(θ) 1. Now cosider the followig derivatio: α β G([v H,j(θ)] 1[E[m j(z, θ)] = 0] J j=1) = G([αv H,j(θ)] 1[E[m j(z, θ)] = 0] j S0, (θ), 0 j S0, (θ)) G([v L,j(θ)] 1[E[m j(z, θ)] = 0] j S0, (θ), 0 j S0, (θ)) = G([v L,j(θ)] 1[E[m j(z, θ)] = 0] J j=1) where the first equality follows from homogeeity of degree β ad the fact that [αv H,j(θ)] 1[E[m j(z, θ)] = 0] = 0 for all j S 0, (θ), the first iequality follows from the mootoicity of G ad αv H,j(θ) < v L,j(θ)/v H,j(θ)v H,j(θ) = v L,j(θ) for all j S 0, (θ), ad the fial equality follows from the fact that [v L,j(θ)] 1[E[m j(z, θ)] = 0] = 0 for all j S 0, (θ). From here, cosider the followig derivatio: δ,2 G([v L,j(θ)] 1[E[m j(z, θ)] = 0] J j=1) G([v H,j(θ)] 1[E[m j(z, θ)] = 0] J j=1) (α β 1)G([v,j(θ)] H 1[E[m j(z, θ)] = 0] J j=1) [ ( ) β ˆσ,j(θ) max max j=1,...,j σ, σ j(θ) 1] G([v,j(θ)] s 1[E[m j(z, θ)] = 0] J j=1) j(θ) ˆσ,j(θ) ) β ˆσ (max j=1,...,j max,j (θ) σ j (θ) σ L + 1, 1 (θ) 1 1 ˆσ,j (θ) σ j (θ) σ L (θ) G([v,j(θ)] s 1[E[m j(z, θ)] = 0] J j=1) [ ( ) β maxj=1,...,j ˆσ,j(θ) σj(θ) 1 1] G([v,j(θ)] s 1[E[m j(z, θ)] = 0] J j=1), σ L(θ) where all iequalities are elemetary ad based o the defiitio of α ad the mootoicity of G. Lemma A.4. Assume Assumptio CF.1 ad that θ Θ I satisfies Assumptio A.4. The, c > 0, N N s.t. N, δ,2 i Eq. (A.2) satisfies max ˆσ,j(θ) σj(θ) 1/2 2c j=1,...,j max j=1,...,k vs,j(θ) c/2β Proof. First, assume that max j=1,...,j ˆσ,j(θ) σ j(θ) 1/2 2c. δ,2 1/2 c. This implies that max j=1,...,j ˆσ,j(θ) σ j(θ) / mi j=1,...,j σ j(θ) 1/2 for all sufficietly large. For x [0, 1/2] cosider the fuctio f(x) = (1 x) β 1. By the itermediate value theorem, there is x [0, 1/2] s.t. f(x) < β2 (β+1) x. Therefore, [ ( 1 ) β maxj=1,...,j ˆσ,j(θ) σj(θ) 1] mi j=1,...,j σ j(θ) f(x) = f(0) + f ( x)x = β(1 x) (β+1) x ad, so, β2 (β+1) max ˆσ,j(θ) σj(θ) j=1,...,j β2(β+1) 1/2 2c. 12

13 Secod, assume that max j=1,...,j v,j(θ) c/2β. If we combie this with the mootoicity ad the homogeeity of G we deduce that G([v s,j(θ)] 1[E[m j(z, θ)] = 0] J j=1) c/2 G(1 J j=1). By combiig these two steps with Lemma A.3, we coclude that max j=1,...,j ˆσ,j(θ) σ j(θ) 1/2 2c ad max j=1,...,j v s,j(θ) c/2β imply that, for all sufficietly large, δ,2 β2 (β+1) 1/2 2c c/2 G(1 J j=1) β2 (β+1) G(1 J j=1) 1/2 1.5c. The RHS is less tha 1/2 c for all sufficietly large, ad this completes the proof. Lemma A.5. Assume Assumptio A.1 ad that θ Θ satisfies E[ m(z, θ) 2+ψ ] < for some ψ > 0. For all j = 1,..., J ad ay sequece a 1 = o(1), P ( σ,j(θ) σ j(θ) > σa )(3σ j(θ)a ) 1+ψ/2 ψ/2 E[ (m j(z i, θ) E[m j(z i, θ)]) 2 σ 2 j (θ) 1+ψ/2 ]. Proof. Cosider the followig argumet: P ( σ,j(θ) σ j(θ) > σ j(θ)a ) = P ( σ,j(θ) σ j(θ) > σ j(θ)a σ,j(θ) σ j(θ) < σ j(θ)a ) = P ( σ,j(θ) > σ j(θ) (1 + a ) σ,j(θ) < σ j(θ) (1 a )) ( ( σ 2,j (θ) σj 2 (θ)) > σj 2 (θ)(2a + a 2 ) ) = P ( σ 2,j (θ) σj 2 (θ)) < σj 2 (θ)( 2a + a 2 ) P ( ( σ 2,j(θ) σ 2 j (θ)) > σ 2 j (θ)(2a + a 2 ) ) +P ( ( σ 2,j(θ) σ 2 j (θ)) < σ 2 j (θ)( 2a + a 2 ) ) P ( ( σ 2,j(θ) σ 2 j (θ)) > 3σ 2 j (θ)a ) + P ( ( σ 2,j (θ) σ 2 j (θ)) > 3σ 2 j (θ)a ) P ( σ 2,j(θ) σ 2 j (θ) > 3σ 2 j (θ)a ), where all relatioships are elemetary ad we have used that a = o(1) implies that there is large eough s.t. 2a + a 2 < 3a ad 2a + a 2 > 3a. From here we coclude that P ( σ,j(θ) σ j(θ) > σ j(θ)a )(3σ j(θ)a ) 1+ψ/2 P ( σ 2,j(θ) σ 2 j (θ) > 3σ 2 j (θ)a )(3σ j(θ)a ) 1+ψ/2 where we have used Markov s ad Holder s iequalities. E( σ,j(θ) 2 σj 2 (θ) 1+ψ/2 ) ( = E (mj(zi, θ) E[mj(Zi, θ)])2 σ 2 j (θ) ) 1+ψ/2 ( = ψ/2 E (mj(zi, θ) E[mj(Zi, ) θ)])2 σj 2 (θ) 1+ψ/2 ( ψ/2 E (mj(zi, θ) E[mj(Zi, ) θ)])2 σj 2 (θ) 1+ψ/2 [ = ψ/2 E (m j(z i, θ) E[m j(z i, θ)]) 2 σj 2 (θ) 1+ψ/2], Lemma A.6. Assume Assumptio A.1 ad that θ Θ satisfies E[ m(z, θ) 2+ψ ] < for some ψ > 0. For all j = 1,..., J ad ay sequece a 1 = o(1), P ( σ,j(θ) ˆσ,j(θ) > a )(3σ j(θ)a ) 1+ψ/2 1 ψ E[ m j(z i, θ) E[m j(z i, θ)] 2+ψ ] + ψ/2 E[ (m j(z i, θ) E[m j(z i, θ)]) 2 σ 2 j (θ) 1+ψ/2 ]. 13

14 Proof. First, cosider the followig argumet: P ( σ,j(θ) ˆσ,j(θ) > a ) (3σ j(θ)a ) 1+ψ/2 = P ( σ 2,j(θ) ˆσ 2,j(θ) 1+ψ/2 > (3σ j(θ)a ) 1+ψ/2 )(3σ j(θ)a ) 1+ψ/2 E( σ,j(θ) 2 ˆσ,j(θ) 2 1+ψ/2 ) ( = E [(mj(zi, θ) m,j(θ))2 (m j(z i, θ) E[m j(z i, θ)]) 2 ] 1+ψ/2 ) = E( m,j(θ) E[m j(z i, θ)] 2+ψ ) 1 ψ E( m j(z i, θ) E[m j(z i, θ)] 2+ψ ), where every relatioship is elemetary. Now, cosider the followig derivatio: P ( σ,j(θ) ˆσ,j(θ) > a ) = P (ˆσ,j(θ) > a + σ,j(θ) ˆσ,j(θ) < a + σ,j(θ)) = P (ˆσ,j(θ) 2 σ,j(θ) 2 > a 2 + 2a σ,j(θ) ˆσ,j(θ) 2 σ,j(θ) 2 < a 2 2a σ,j(θ)) P (ˆσ,j(θ) 2 σ,j(θ) 2 > a 2 (1 + 2σ j(θ)) + 2a σ j(θ) ˆσ,j(θ) 2 σ,j(θ) 2 < a 2 a 2 (1 + 2σ j(θ)) 2a σ j(θ)) +P ( σ,j(θ) ˆσ,j(θ) > σ j(θ)a ) P ( ˆσ,j(θ) 2 σ,j(θ) 2 > 3a 2 σ j(θ)) + P ( σ,j(θ) ˆσ,j(θ) > σ j(θ)a ), where all relatioships are elemetary ad we have used that a = o(1) implies that there is large eough s.t. a 2 (1 + 2σ j(θ)) + 2a σ j(θ) < 3a σ j(θ) ad a 2 (1 + 2σ j(θ)) 2a σ j(θ) > 3a σ j(θ). From here we coclude that P ( σ,j(θ) ˆσ,j(θ) > a )(3σ j(θ)a ) 1+ψ/2 P ( ˆσ,j(θ) 2 ˆσ,j(θ) 2 > 3a σ j(θ))(3σ j(θ)a ) 1+ψ/2 + P ( σ,j(θ) σ j(θ) > σ j(θ)a )(3σ j(θ)a ) 1+ψ/2 1 ψ E( m j(z i, θ) E[m j(z i, θ)] 2+ψ ) + ψ/2 E[ (m j(z i, θ) E[m j(z i, θ)]) 2 σj 2 (θ) 1+ψ/2 ], where we have used Lemma A.5. Lemma A.7. Assume Assumptio A.1 ad that θ Θ satisfies E[ m(z, θ) 2+ψ ] < for some ψ > 0. For all j = 1,..., J ad ay sequece a 1 = o(1), P ( ˆσ,j(θ) σ j(θ) > 2a )(3σ j(θ)a ) 1+ψ/2 1 ψ E[ m j(z i, θ) E[m j(z i, θ)] 2+ψ ] + ψ/2 2E[ (m j(z i, θ) E[m j(z i, θ)]) 2 σj 2 (θ) 1+ψ/2 ]. Proof. By triagular iequality, ˆσ,j(θ) σ j(θ) ˆσ,j(θ) σ,j(θ) + σ,j(θ) σ j(θ) ad therefore P ( ˆσ,j(θ) σ j(θ) > 2a ) P ( ˆσ,j(θ) σ,j(θ) > a σ,j(θ) σ j(θ) > a ) P ( ˆσ,j(θ) σ,j(θ) > a ) + P ( σ,j(θ) σ j(θ) > a ). The result follows from the previous iequality ad Lemmas A.5 ad A.6. Lemma A.8. Assume Assumptio A.1 ad that θ Θ satisfies E[ m(z, θ) 2+ψ ] < for some ψ > 0. For all j = 1,..., J ad ay c > 0, P ( ˆσ,j(θ) σ j(θ) > 1/2 c ) = O( 1/2 ψ/4+(1+ψ/2)c ). Uder Assumptio A.5 the RHS expressio is o( 1/2 ) for a choice of c > 0 that is small eough. Proof. Let c (0, 1/2). By Lemmas A.1 ad A.7 applied to a = 1/2 c /2 = o(1), we coclude that P ( ˆσ,j(θ) σ j(θ) > 1/2 c ) = (3σ 1/2 c /2) 1 ψ/2 O( 1 ψ + ψ/2 ) = O( 1/2 ψ/4+(1+ψ/2)c ). 14

15 To complete the proof, we otice that O( 1/2 ψ/4+(1+ψ/2)c ) = o( 1/2 ) if ad oly if ψ > (4 + 2ψ)c + 4. By makig c arbitrarily small, the coditio ca be achieved wheever ψ > 4, i.e., uder Assumptio A.5. Refereces Bugi, F. A. (2014): A compariso of iferetial methods i partially idetified models i terms of error i coverage probability (Formerly circulated as Bootstrap Iferece i Partially Idetified Models Defied by Momet Iequalities: Coverage of the Elemets of the Idetified Set ), Accepted for publicatio i Ecoometric Theory o August 11, Joag-Dev, K. ad F. Proscha (1983): Negative associatio of radom variables with applicatios, The Aals of Statistics, 11,