Improved Nonparametric Bootstrap Tests of Lorenz Dominance

of 3.1, 3.2 3.5 numerical simulations for the matched pairs sampling framework, for the independent of the

for each p ∈ (0, 1), where (U, V ) is a pair of random variables whose joint CDF is the copula C.
The function H j,p is the antiderivative of h j,p : it satisfies H j,p (·) = • 0 h j,p dt. A generalization of Hoeffding's lemma due to Lo (2017, Thm. 3.1) -see also Cuadras (2002) and Beare (2009) -thus implies that This proves our claimed covariance formula. From this we obtain the claimed variance formulas for L 1 (p) and L 2 (p) by setting F 1 = F 2 and setting C equal to the Fréchet-Hoeffding upper bound, so that H 1,p (U ) = H 2,p (V ) almost surely. (Note that the derivation of our covariance formula was valid for any copula C, including the Fréchet-Hoeffding upper bound.) Proof of Proposition 3.1. The CDFs F 1 and F 2 are continuous under Assumption 2.1, so the pair of random variables (F 1 (X 1 ), F 2 (X 2 )) has joint CDF given by the copula C. It therefore follows from Lemma A.1 that for j = 1, 2, and The desired result follows easily.
The following lemma is used in the proof of Proposition 3.2.
Lemma A.2. Suppose that Assumptions 2.1 and 2.2 are satisfied, and that τ n → ∞ and T −1/2 n τ n → 0 as n → ∞. If H 0 is true, then for any δ > 0 we have for j = 1, 2 and p ∈ [0, 1]. Using this bound and the strong law of large numbers, it is simple to show thatṼ satisfies almost surely under Assumption 2.2(i) (independent sampling), or with the convergence to one following from the fact that τ n ν 1/2 → ∞ while T 1/2 If ε n < δ, then this occurs ifφ is everywhere within δ − ε n of φ. Since ε n → 0 almost surely, for sufficiently large n we therefore have with the convergence to one following from the fact that T

Proof of Proposition 3.2.
It is easy to see that our estimated functionals satisfy the Lipschitz conditions Therefore, Lemma S.3.6 of Fang and Santos (2019) implies that a sufficient condition for S φ and I φ to satisfy their Assumption 4 is that, for any > 0, for each h ∈ C[0, 1]. Moreover, since n 1/2 (φ * −φ) is a Borel measurable map into the separable space C[0, 1], Assumption 4 of Fang and Santos (2019) is equivalent to our Assumption 3.1; see their Remark 3.3. Thus we need only verify (A.2).
To verify the first part of (A.2), fix h ∈ C[0, 1] and > 0, and choose δ > 0 small enough that |h(p) − h(q)| < whenever |p − q| < δ. Next choose η > 0 small enough that |p − q| < δ for some q ∈ B 0 (φ) whenever p ∈ B η (φ). Observe that if then it must be the case that The first part of (A.2) now follows from Lemma A.2.
To verify the second part of (A.2), fix h ∈ C[0, 1] and > 0, and choose δ > 0 small enough that which is possible by the dominated convergence theorem. Observe that if then it must be the case that The second part of (A.2) now follows from Lemma A.2.
The following lemma is used in the proof of Proposition 3.5.
Proof of Proposition 3.5. We first observe that sinceL is Gaussian and the directional derivatives S φ and I φ are continuous and convex, Theorem 11.1 of Davydov et al. (1998) can be used to show that the CDFs of S φ (L) and I φ (L) are continuous everywhere except perhaps at zero, and that if either CDF assigns probability less than one to zero, then it is strictly increasing on (0, ∞). Thus if either CDF is not continuous and strictly increasing at its 1 − α quantile, then it must assign probability of at least 1 − α to zero.
To demonstrate claim (a), observe that if the set Ψ(φ) includes some point p 0 / ∈ {0, 1}, then with the final equality following from Lemma A.3 and the fact thatL(p 0 ) is a centered Gaussian random variable. Thus the CDF of S φ (L) can assign probability of no greater than one half to zero. Since 1 − α > 1/2, we conclude that the CDF must be continuous and strictly increasing at its 1 − α quantile. On the other hand, if Ψ(φ) does not include any point p 0 / ∈ {0, 1}, then clearly S φ (L) is degenerate at zero. To demonstrate claim (b), suppose that I φ (L) is not degenerate at zero. Since we have assumed H 0 to be satisfied, we have B + (φ) = ∅, and so B 0 (φ) must be a set of positive measure. Thus the Lebesgue density theorem ensures the existence of p 0 ∈ B 0 ∩ (0, 1) such that the set (p 0 − , p 0 + ) ∩ B 0 has positive measure for all > 0. SinceL(p) is continuous in p, ifL(p 0 ) > 0 then we must haveL > 0 on (p 0 − , p 0 + ) for some > 0, implying that (p)dp > 0.

Thus we have
with the final equality following from Lemma A.3 and the fact thatL(p 0 ) is a centered Gaussian random variable. Thus the CDF of I φ (L) can assign probability of no greater than one half to zero, and since 1 − α > 1/2, we conclude that the CDF must be continuous and strictly increasing at its 1 − α quantile.

B Further numerical simulations
The numerical simulations reported in Section 4 pertained to the independent sampling framework. Here we report analogous simulations for the matched pairs sampling framework. The simulation design is the same as described in Section 4.1, except that dependence between pairs was induced by linking the random variables X 1 i and X 2 i with a Gaussian copula with parameter ρ = 0.25, 0.5, 0.75. In Tables B.1, B.2 and B.3 we report results analogous to those reported in Table 4.1, and in Figure B.1 we report results analogous to those reported in Figure 4.2. Qualitatively, the results for the matched pairs sampling framework are similar to those for the independent sampling framework.  Table B.1: Null rejection rates with X 1 ∼ X 2 ∼ dP(α, β) and n = 2000 matched pairs linked by a Gaussian copula with correlation parameter ρ = 0.25. Rejection rates are in bold when they exceed the corresponding rate obtained with τ n = ∞ by more than 0.1 percentage point.  Table B.2: Null rejection rates with X 1 ∼ X 2 ∼ dP(α, β) and n = 2000 matched pairs linked by a Gaussian copula with correlation parameter ρ = 0.50. Rejection rates are in bold when they exceed the corresponding rate obtained with τ n = ∞ by more than 0.1 percentage point. 6.0 5.9 5.9 5.9 5.9 5.9 5.8 5.7 3 5.  Table B.3: Null rejection rates with X 1 ∼ X 2 ∼ dP(α, β) and n = 2000 matched pairs linked by a Gaussian copula with correlation parameter ρ = 0.75. Rejection rates are in bold when they exceed the corresponding rate obtained with τ n = ∞ by more than 0.1 percentage point. Power with X 1 ∼ dP(3, 1.5) and X 2 (β) ∼ dP(2.1, β) as a function of the parameter β. Going from top to bottom in each panel, the five power curves correspond to our test with τ n = 1, 2, 3, 4, and the test of Barrett et al. (2014). Samples are n = 2000 matched pairs linked by a Gaussian copula with correlation parameter ρ.