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}$$_{),}$ r$_{n}$ ( ˆ θ$_{n}$ − θ 0 ) has a non-degenerate limiting distribution. Then, in perfect analogy with Section 2, the shape of that limiting distribution can usually be anticipated with the help of the representation where Specifically, assuming the functions G$_{n}$ and Q$_{n}$ satisfy convergence properties of the form (5) and (4), respectively, it stands to reason that In particular, for example, if r$_{n}$ = √$_{n}$ and G ( s ) = s ′ ˙ G with ˙ G some zero-mean Gaussian vector, then S becomes the usual normal distribution of √$_{n}$$_{-consistent (parametric) M-estimators.}$ The bootstrap analogue of ˆ θ$_{n}$ satisfies where and 12 and r$_{n}$ ( ˆ θ$_{n}$ − θ 0 ) = argmax s ∈ R d { Q$_{n}$ ( s ) + G$_{n}$ ( s ) } , G$_{n}$ ( s ) = r 2 $_{n}$[ M$_{n}$ ( θ 0 + s /r$_{n}$ ) − M$_{n}$ ( θ 0 ) − M ( θ 0 + s /r$_{n}$ ) + M ( θ 0 )] Q$_{n}$ ( s ) = r 2 $_{n}$[ M ( θ 0 + s /r$_{n}$ ) − M ( θ 0 )] . ˆ s$_{n}$ = r$_{n}$ ( ˆ θ$_{n}$ − θ 0 ) ⇝ argmax s ∈ R d {Q ( s ) + G ( s ) } = S . r$_{n}$ ( ˆ θ ∗ n − ˆ θ$_{n}$ ) = argmax s ∈ R d { Q ∗ $_{n}$( s ) + G ∗ $_{n}$( s ) } , G ∗ $_{n}$( s ) = r 2 $_{n}$[ M ∗ $_{n}$( ˆ θ$_{n}$ + s /r$_{n}$ ) − M ∗ $_{n}$( ˆ θ$_{n}$ ) − M$_{n}$ ( ˆ θ$_{n}$ + s /r$_{n}$ ) + M$_{n}$ ( ˆ θ$_{n}$ )] Q ∗ $_{n}$( s ) = r 2 $_{n}$[ M$_{n}$ ( ˆ θ$_{n}$ + s /r$_{n}$ ) − M$_{n}$ ( ˆ θ$_{n}$ )] .