I'm looking for references (textbook if possible) that treat the strong consistency of bootstrapped maximum likelihood (MLH) or more generally M-estimators. By strong consistency I mean that the difference between the cdf of the original statistic and the bootstrap counterpart goes a.s. to zero uniformly (see below).

So far I have been following the textbook:

Shao, Tu: The Jacknife and Bootstrap. 1995.

that provides a really nice account of the bootstrap and theorems of this type. For example, see Theorem 3.1 (page 80) in Shao & Tu, denote by $\bar{X}_n$ the sample mean of $X_1,\ldots,X_n$ (i.i.d.), and by $h\colon \mathbb{R}\to\mathbb{R}$ a function such that $h'(\mu) \neq 0$ and $h'$ is continuous at $\mu$, where $\mu = \mathbb{E}[X]$ and $\mathbb{E}[X^2] < \infty$. For a statistic $$ T_n = h(\bar{X}_n) $$ with bootstrap counterpart $T^*_n$ we then have $$\Vert F_{T_n^*} - F_{T_n} \Vert_{\infty} = \sup_{x\in\mathbb{R}} \vert F_{T_n^*}(x) - F_{T_n}(x) \vert \xrightarrow[n\to\infty]{a.s.} 0. $$ Here $F_{T_n^*}$ and $F_{T_n}$ are the respective distribution functions and the convergence a.s. is necessary since $F_{T_n^*}$ is random.

This settles method of moment type estimators (in fact in Shao & Tu the general multivariate version is considered), but I'm wondering about implicitly defined estimators like MLH (or M-estimators).

My question is therefore, if $\mathcal{M} = \{f_{\theta};\theta\in\Theta\}$ is a parametric statistical model and $\widehat{\theta}_n$ is the MLH estimator based on $X_1,\ldots,X_n$ (i.i.d. with density $f_{\theta_0}$) what can be said about the limit $$\Vert F_{\widehat{\theta}_n^*} - F_{\widehat{\theta}_n} \Vert_{\infty}.$$

I'm looking for references that either give conditions and a proof, or, preferably, a textbook that outlines the results in a digestible way.


1 Answer 1


It depends on what you count as 'digestible', but I like the version in van der Vaart's Asymptotic Statistics, which develops consistency of the bootstrap from the delta method in both the finite-dimensional and infinite-dimensional case.


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