U-statistics of growing order and sub-Gaussian mean estimators with sharp constants

Let $X_{1},\ldots,X_{N}$ be i.i.d. random variables with distribution $P$ having mean $\mu$ and finite variance $\sigma^{2}$. At the core of this paper is the following question: given $1\leq t\leq t_{\max}(N)$, construct an estimator $\widetilde{\mu}_{N}=\widetilde{\mu}_{N}(X_{1},\ldots,X_{N})$ such that

for some absolute positive constant $L$. Estimators that satisfy this deviation property are called sub-Gaussian. For example, the sample mean $\bar{X}_{N}=\frac{1}{N}\sum_{j=1}^{N}X_{j}$ is sub-Gaussian for $t_{\max}\asymp q(N,P)$ where $q(N,P)\to\infty$ as $N\to\infty$ and the constant $L$ equals $2$: this immediately follows from the fact that convergence of the distribution functions is uniform in the central limit theorem. However, $q(N,P)$ can grow arbitrarily slow in general, and it grows as $\log^{1/2}(N)$ if $\mathbb{E}|X|^{2+\varepsilon}<\infty$ for some $\varepsilon>0$ in view of the Berry-Esseen theorem (for instance, see the book by Petrov,, 1975). At the same time, the so-called median of means (MOM) estimator, originally introduced by Nemirovski and Yudin, (1983); Alon et al., (1996); Jerrum et al., (1986) and studied recently in relation to the problem at hand satisfies inequality (1) with $t_{\max}$ of order $N$ and $L=24e$ (Lerasle and Oliveira,, 2011), although the latter can be improved. A large body of existing work used the MOM estimator as a core subroutine to relax underlying assumptions for a variety of statistical problems, in particular the methods based on the empirical risk minimization; we refer the reader to an excellent survey paper by Lugosi and Mendelson, (2019) for a detailed overview of the recent advances.

The exact value of constant $L$ in inequality (1) is less important in problems where only the minimax rates are of interest, but it becomes crucial in terms of practical value and sample efficiency of the algorithms. The benchmark here is the situation when observations are normally distributed: Catoni, (2012) showed that no estimator can outperform the sample mean in this situation. The latter satisfies the relation

where $\Phi^{-1}(\cdot)$ denotes the quantile function of the standard normal law. As $\Phi^{-1}(1-e^{-t/2})=(1+o(1))\sqrt{t}$ as $t\to\infty$, the best guarantee of the form (1) one can hope for is attained for $L=2$. It is therefore natural to ask whether there exist sharp sub-Gaussian estimators of the mean, that is, estimators satisfying (1) with $L=2(1+o(1))$ where $o(1)$ is a sequence that converges to $0$ as $N\to\infty$, under minimal assumptions on the underlying distribution. This question has previously been posed by Devroye et al., (2016) as an open problem, and several results appeared since then that give partial answers. We proceed with a brief review of the state of the art.

Recall that we are given an i.i.d. sample $X_{1},\ldots,X_{N}$ from distribution $P$ with mean $\mu$ and variance $\sigma^{2}$. The median of means estimator of $\mu$ is constructed as follows: let $G_{1}\cup\ldots\cup G_{k}\subseteq[N]$ be an arbitrary (possibly random but independent from the data) collection of $k\leq N/2$ disjoint subsets (“blocks”) of cardinality $\lfloor N/k\rfloor$ each, $\bar{X}_{j}:=\frac{1}{|G_{j}|}\sum_{i\in G_{j}}X_{i}$ and

It is known (e.g. Lerasle and Oliveira,, 2011; Devroye et al.,, 2016) that $\widehat{\mu}_{\mathrm{MOM}}$ satisfies inequality (1) for $t=k$ and $L=8e^{2}$. This value of $L$ appears to be overly pessimistic however: it follows from Theorem 5 in (Minsker,, 2019) that if $k\to\infty$ sufficiently slow so that the bias of $\widehat{\mu}_{\mathrm{MOM}}$ is of order $o(N^{-1/2})$, then

as $k,N/k\to\infty$. In particular, if $\mathbb{E}|X|^{2+\delta}<\infty$ for some $0<\delta\leq 1$, then $k=o\left(N^{\delta/(1+\delta)}\right)$ suffices for the asymptotic unbiasedness and asymptotic normality to hold. Asymptotic relation (2) suggests that the best value of the constant $L$ in the deviation inequality (1) for the estimator $\widehat{\mu}_{\mathrm{MOM}}$ is $\pi+o(1)$. We will demonstrate that this is indeed the case. Denote

Clearly, $g(m)\to 0$ as $m\to\infty$ for distributions with finite variance. Feller, (1968) proved that $\sup_{t\in\mathbb{R}}\left|\Phi_{m}(t)-\Phi(t)\right|\leq 6g(m)$ where $\Phi_{m}$ and $\Phi$ are the distribution functions of $\frac{\sum_{j=1}^{m}X_{j}-\mu}{\sigma\sqrt{m}}$ and the standard normal law respectively. It is well known that $g(m)\leq C\mathbb{E}\left|\frac{X_{1}-\mu}{\sigma}\right|^{q}m^{-(q-2)/2}$ whenever $\mathbb{E}|X_{1}-\mu|^{q}<\infty$ for some $q\in(2,3]$. The next result can be viewed as a non-asymptotic analogue of relation (2).

One may ask whether the median of means estimator admits a more sample-efficient modification, one that would satisfy inequality (1) with a constant $L$ smaller than $\pi$. A natural idea is to require that the estimator is invariant with respect to permutations of the data or, equivalently, is a function of order statistics only. Such an extension of the MOM estimator was proposed by Minsker, (2019), however no provable improvements for the performance over the standard MOM estimator were established rigorously. The question of such improvements, especially the guarantees expressed in the form (1), is addressed next. Let us recall the proposed construction. Assume that $2\leq m<N$ and, given $J\subseteq[N]$ of cardinality $|J|=m$, set $\bar{X}_{J}:=\frac{1}{m}\sum_{j\in J}X_{j}$. Define $\mathcal{A}_{N}^{(m)}=\left\{J\subset[N]:\ |J|=m\right\}$ and

where $\left\{\bar{X}_{J},\ J\in\mathcal{A}_{N}^{(m)}\right\}$ denotes the set of sample averages computed over all possible subsets of $[N]$ of cardinality $m$; in particular, unlike the standard median-of-means estimator, $\widehat{\mu}_{N}$ is uniquely defined. Note that for $m=2$, $\widehat{\mu}_{N}$ coincides with the well known Hodges-Lehmann estimator of location (Hodges and Lehmann,, 1963). When $m$ is a fixed integer greater than $2$, $\widehat{\mu}_{N}$ is known as the generalized Hodges-Lehmann estimator. Its asymptotic properties are well-understood and can be deduced from results by Serfling, (1984), among other works. For example, its breakdown point is $1-(1/2)^{1/m}$ and, in case of normally distributed data, the asymptotic distribution of $\sqrt{N}(\widehat{\mu}_{N}-\mu)$ is centered normal with variance $\Delta_{m}^{2}=m\sigma^{2}\arctan\left(\frac{1}{\sqrt{m^{2}-1}}\right)$. In particular, $\Delta_{m}^{2}=\sigma^{2}(1+o(1))$ as $m\to\infty$. When the underlying distribution is not symmetric however, $\widehat{\mu}_{N}$ is biased for the mean, and the properties of this estimator in the regime $m\to\infty$ have not been investigated in the robust statistics literature (to the best of our knowledge). Only very recently, DiCiccio and Romano, (2022) proved that whenever $m\to\infty$, $m=o(\sqrt{N})$ and the sample is normally distributed, $\sqrt{N}(\widehat{\mu}_{N}-\mu)\to N(0,\sigma^{2})$. We will extend this result in several directions: first, by allowing a much wider class of underlying distributions, second, by including the case when $\sqrt{N}\ll m\ll N$ which is interesting as $\mathrm{bias}\left(\widehat{\mu}_{N}\right)$ is $o\left(N^{-1/2}\right)$ in this regime, and finally by presenting sharp sub-Gaussian deviation inequalities for $\widehat{\mu}_{N}$ that hold for heavy-tailed data.

Let us remark that an argument behind Theorem 2.1 combined with a version of Bernstein’s inequality for U-statistics due to Hoeffding, (1963) immediately implies that $\widehat{\mu}_{N}$ satisfies relation (4). Similar reasoning applies to other deviation guarantees for the classical median of means estimator that exist in the literature, so in this sense $\widehat{\mu}_{N}$ always performs at least as good as $\widehat{\mu}_{\mathrm{MOM}}$.

Analysis of the estimator $\widehat{\mu}_{N}$ is most naturally carried out using the language of U-statistics. The following section introduces the necessary background, while additional useful facts are summarized in section 7.1.

Let $Y_{1},\ldots,Y_{N}$ be i.i.d. random variables with distribution $P_{Y}$ and assume that $h_{m}:\mathbb{R}^{m}\mapsto\mathbb{R},\ m\geq 1$ are square-integrable with respect to $P_{Y}^{m}$ and permutation-symmetric functions, meaning that that $\mathbb{E}h_{m}^{2}(Y_{1},\ldots,Y_{m})<\infty$ and $h_{m}(x_{\pi(1)},\ldots,x_{\pi(m)})=h_{m}(x_{1},\ldots,x_{m})$ for any $x_{1},\ldots,x_{m}\in\mathbb{R}$ and any permutation $\pi:[m]\mapsto[m]$. Without loss of generality, we will also assume that $\mathbb{E}h_{m}:=\mathbb{E}h_{m}(Y_{1},\ldots,Y_{m})=0$. Recall that $\mathcal{A}_{N}^{(m)}=\left\{J\subseteq[N]:\ |J|=m\right\}$. The U-statistic with kernel $h_{m}$ is defined as

For $i\in[N]$, let

We will assume that $\mathbb{P}{\left(h_{m}^{(1)}(Y_{1})\neq 0\right)}>0$ for all $m$, meaning that the kernels $h_{m}$ are non-degenerate. The random variable

known as the Hájek projection of $U_{N,m}$, is essentially the best approximation of $U_{N,m}$ in terms of the sum of i.i.d. random variables of the form $f(Y_{1})+\ldots+f(Y_{m})$. We are interested in the sufficient conditions guaranteeing that $\frac{U_{N,m}-S_{N,m}}{\sqrt{\mathrm{Var}(S_{N,m})}}=o_{P}(1)$ as $N,m\to\infty$. Such asymptotic relation immediately implies that the limiting behavior of $U_{N,m}$ is defined by the Hájek projection $S_{N,m}$. Results of these type for U-statistics of fixed order $m$ are standard and well-known (Hoeffding,, 1948; Serfling,, 2009; Lee,, 2019). However, we are interested in the situation when $m$ is allowed to grow with $N$, possibly up to the order $m=o(N)$. U-statistics of growing order were studied for example by Frees, (1989), however existing results are not readily applicable in our framework. Very recently, such U-statistics have been investigated in relation to performance of Breiman’s random forests algorithm (e.g. see the papers by Song et al., (2019) and Peng et al., (2022)). The following theorem is essentially due to Peng et al., (2022); we give a different proof of this fact in Appendix 7.2 as we rely on parts of the argument elsewhere in the paper.

It is easy to see that asymptotic normality of $\frac{U_{N,m}}{\sqrt{\mathrm{Var}(S_{N,m})}}$ immediately follows from the previous theorem whenever its assumptions are satisfied. Next, we will apply this result to establish asymptotic normality of the estimator $\widehat{\mu}_{N}$ defined via (8).