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(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 . 

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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 

<img src='content_image/300129.jpg'>

be the natural bootstrap analogue of ˆ θ$_{n}$. Then the nonparametric bootstrap approximation to the distribution of ˆ θ$_{n}$ is given by 

<img src='content_image/300130.jpg'>

where P ∗ denotes a probability computed under the bootstrap distribution conditional on the data. 

