(1990) found that M -estimators such as often exhibit cube root asymptotics: where ⇝ denotes weak convergence, Q ( s ) is a quadratic form given by and G is a non-degenerate zero-mean Gaussian process with G (0) = 0 . Whereas the matrix V 0 governing the shape of Q is finite-dimensional, the covariance kernel of G in (2) typically involves infinite-dimensional unknown quantities. As a consequence, the limiting distribution of ˆ θ$_{n}$ tends to be more difficult to approximate than conventional Gaussian distributions, implying in turn that basing inference on ˆ θ$_{n}$ is more challenging under cube root asymptotics than in the more familiar case where ˆ θ$_{n}$ $_{is (}$√$_{n}$$_{-consistent and) asymptotically normally}$ distributed. As a candidate method of approximating the distribution of ˆ θ$_{n}$, consider the standard nonpara- metric bootstrap. To describe it, let { z ∗ i : 1 ≤ i ≤ n } denote a random sample from the empirical distribution of { z$_{i}$ : 1 ≤ i ≤ n } and let be the natural bootstrap analogue of ˆ θ$_{n}$. Then the nonparametric bootstrap approximation to the distribution of ˆ θ$_{n}$ is given by where P ∗ denotes a probability computed under the bootstrap distribution conditional on the data. ˆ θ ∗ n = argmax θ ∈ Θ M ∗ $_{n}$( θ ) , M ∗ $_{n}$( θ ) = 1 n n ∑ i =1 m ( z ∗ $_{i}$, θ ) , P [ 3 √$_{n}$$_{(}$ ˆ θ$_{n}$ − θ 0 ) ≤ t ] ≈ P $^{∗}$[ 3 √$_{n}$$_{(}$ ˆ θ ∗ n − ˆ θ$_{n}$ ) ≤ t ] , ˆ θ$_{n}$ = argmax θ ∈ Θ M$_{n}$ ( θ ) , M$_{n}$ ( θ ) = 1 n n ∑ i =1 m ( z$_{i}$ , θ ) , (1) 3 √$_{n}$$_{(}$ ˆ θ$_{n}$ − θ 0 ) ⇝ argmax s ∈ R d {Q ( s ) + G ( s ) } , (2) Q ( s ) = − 1 2 s $^{′}$V 0 s , V 0 = − ∂θ 2 ∂θ∂θ $_{′}$M ( θ 0 ) , 3