Case 2: If u ∈ [ F ( a$_{j}$$_{−}$$_{1}$ ) , F ( a$_{j}$ − )), similar arguments as in Case 1 yields Case 3: If u ∈ [0 , F ( a$_{1}$ )), obviously, P ( U ≤ u ) = ( u − F { a$_{j}$$_{−}$$_{1}$ } − . . . − F { a$_{1}$ } ) + ( F { a$_{1}$ } + . . . + F { a$_{j}$$_{−}$$_{1}$ } ) = u. P ( U ≤ u ) = u. P ( U ≤ u ) = ( u − F { a$_{J}$ } − . . . − F { a$_{1}$ } ) + ( F { a$_{1}$ } + . . . + F { a$_{J}$ } ) = u. Case 4: If u ∈ [ F ( a$_{J}$ ) , 1], obviously, Thus, it follows that U is uniformly distributed on [0 , 1]. □ Lemma A.2 Under Assumptions 3.1 and 3.3, F = { x ↦→ 1 { q ( x, θ ) ≤ u } : u ∈ [0 , 1] , θ ∈ Θ } is a Donsker class of functions. Proof of Lemma A.2: From Lemma A.1, simply letting Y = q ( X, θ ), we have that 1 { q ( X, θ ) ≤ u } = 1 { U ≤ F$_{θ}$ ( u ) } for all u ∈ [0 , 1], i.e., we can exploit the quantile transfor- mation and express each indicator function 1 { q ( X, θ ) ≤ u } through U via the time transfor- mation F$_{θ}$ . In light of this quantile transformation, it suffices to study the class of functions F cdf = { ¯ u ↦→ 1 { ¯ u ≤ F$_{θ}$ ( u ) } : u ∈ [0 , 1] , θ ∈ Θ } . Let N$_{[ ]}$ (¯ ϵ, F $_{cdf}$, L $_{2}$( P )) be the bracketing number of the class F cdf with respect to the un- derlying probability P , which by definition is the minimal number of ¯ ϵ -brackets under L $_{2}$( P ) metric that cover F $_{cdf}$. Henceforth, we define the underlying probability P as the probability measure of U . By Theorem 2.5.6 in van der Vaart and Wellner (1996), the Donsker property is implied by To show that such entropy result hold, we follow similar steps as the proof of Lemma 1 in Akritas and van Keilegom (2001) and of Lemma A.4 in Frazier et al. (2018). ∫ ∞ 0 √ log N$_{[ ]}$ (¯ ϵ, F $_{cdf}$, L $_{2}$( P )) d ¯ ϵ < ∞ . Let ¯ ϵ > 0 be an arbitrarily small constant, and consider partitions { Θ$_{l}$ } L l =1 of Θ. Given that Θ is compact, under Assumption 3.3, there exists a finite constant K ≤ diam (Θ / ¯ ϵ ) k such that diam (Θ$_{l}$) ≤ ¯ ϵ for every l = 1 , . . . , K . Fix l ∈ { 1 , . . . , K } and pick up some θ$_{l}$ ∈ Θ$_{l}$. Then, for any fixed u ∈ [0 , 1] and any θ ∈ Θ$_{l}$, it follows from the Lipschitz condition in Assumption 3.3 that F − l ( u ) ≤ F$_{θ}$ ( u ) ≤ F + l ( u ) , (A.4) 33