content_image/317912.jpg content_image/317908.jpg content_image/317913.jpg content_image/317914.jpg content_image/317915.jpg content_image/317919.jpg
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 

where 

is a zero-mean random process, while 

<img src='content_image/317912.jpg'>

<img src='content_image/317908.jpg'>

<img src='content_image/317913.jpg'>

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 

and 

<img src='content_image/317914.jpg'>

<img src='content_image/317915.jpg'>

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 

<img src='content_image/317919.jpg'>

