content_image/296666.jpg content_image/296665.jpg content_image/296667.jpg content_image/296668.jpg content_image/296670.jpg content_image/296669.jpg content_image/296671.jpg
## 5.1 M -estimators with an arbitrary rate of convergence 

For the purposes of explaining the inconsistency of the nonparametric bootstrap, the unusual rate of convergence of ˆ θ$_{n}$ is a bit of a “red herring”. Accordingly, suppose ˆ θ$_{n}$ satisfies (1) and that, for some increasing function r$_{n}$ of n (such as 3 √$_{n}$ or √$_{n}$$_{),}$ r$_{n}$ ( ˆ θ$_{n}$ − θ 0 ) has a non-degenerate limiting distribution. Then, in perfect analogy with Section 2, the shape of that limiting distribution can usually be anticipated with the help of the representation 

where 

and 

<img src='content_image/296666.jpg'>

<img src='content_image/296665.jpg'>

<img src='content_image/296667.jpg'>

Specifically, assuming the functions G$_{n}$ and Q$_{n}$ satisfy convergence properties of the form (5) and (4), respectively, it stands to reason that 

<img src='content_image/296668.jpg'>

In particular, for example, if r$_{n}$ = √$_{n}$ and G ( s ) = s ′ ˙ G with ˙ G some zero-mean Gaussian vector, then S becomes the usual normal distribution of √$_{n}$$_{-consistent (parametric) M-estimators.}$ 

The bootstrap analogue of ˆ θ$_{n}$ satisfies 

where 

and 

<img src='content_image/296670.jpg'>

<img src='content_image/296669.jpg'>

<img src='content_image/296671.jpg'>

