As is well documented, however, this distributional approximation is inconsistent under cube root asymptotics (e.g., Abrevaya and Huang, 2005; L´ eger and MacGibbon, 2006). For the purpose of giving a heuristic, yet constructive, explanation of the inconsistency of the nonparametric bootstrap, it is helpful to recall that for ˆ θ$_{n}$ defined in (1), a proof of (2) can be based on the representation 3 √$_{n}$$_{(}$ ˆ θ$_{n}$ − θ 0 ) = argmax s ∈ R d { Q$_{n}$ ( s ) + G$_{n}$ ( s ) } , (3) where is a zero-mean random process, while is a non-random function that is correctly centered in the sense that argmax$_{s}$$_{∈}$$_{R}$ d Q$_{n}$ ( s ) = 0 . In cases where m is non-smooth but M is smooth, Q$_{n}$ and G$_{n}$ are usually asymptotically quadratic and asymptotically Gaussian, respectively, in the sense that G$_{n}$ ( s ) = n 2 / $^{3}$[ M$_{n}$ ( θ 0 + s /n 1 / $^{3}$) − M$_{n}$ ( θ 0 ) − M ( θ 0 + s /n 1 / $^{3}$) + M ( θ 0 )] Q$_{n}$ ( s ) = n 2 / $^{3}$[ M ( θ 0 + s /n 1 / $^{3}$) − M ( θ 0 )] Q$_{n}$ ( s ) → Q ( s ) (4) G$_{n}$ ( s ) ⇝ G ( s ) . (5) and Under regularity conditions ensuring among other things that the convergence in (4) and (5) is suitably uniform in s, the proof of (2) can then be completed by applying a continuous mapping- type theorem for the argmax functional to the representation in (3). Similarly to (3), the bootstrap analogue of ˆ θ$_{n}$ admits a representation of the form 3 √$_{n}$$_{(}$ ˆ θ ∗ n − ˆ θ$_{n}$ ) = argmax s ∈ R d { Q ∗ $_{n}$( s ) + G ∗ $_{n}$( s ) } , (6) 4