Approximating symmetrized estimators of scatter via balanced incomplete U-statistics

Abstract

We derive limiting distributions of symmetrized estimators of scatter. Instead of considering all $n(n-1)/2$ pairs of the n observations, we only use nd suitably chosen pairs, where $d \ge 1$ is substantially smaller than n. It turns out that the resulting estimators are asymptotically equivalent to the original one whenever $d = d(n) \rightarrow \infty$ at arbitrarily slow speed. We also investigate the asymptotic properties for arbitrary fixed d. These considerations and numerical examples indicate that for practical purposes, moderate fixed values of d between 10 and 20 yield already estimators which are computationally feasible and rather close to the original ones.

A auxiliary results

1.1 A.1 A particular coupling of random permutations

Preparations. For an integer $n \ge 1$, let ${\mathcal {S}}_n$ be the set of all permutations of $\langle n\rangle := \{1,2,\ldots ,n\}$. A cycle in ${\mathcal {S}}_n$ is a permutation $\sigma \in {\mathcal {S}}_n$ such that for $m \ge 1$ pairwise different points $a_1,\ldots ,a_m \in \langle n\rangle$,

$$\begin{aligned} a_1 \ \mapsto \ a_2 \mapsto \ \cdots \ \mapsto \ a_m \ \mapsto \ a_1, \end{aligned}$$

while $\sigma (i) = i$ for $i \in \langle n\rangle {\setminus } \{a_1,\ldots ,a_m\}$. (In the case of $m = 1$, $\sigma (i) = i$ for all $i \in \langle n\rangle$.) We write

$$\begin{aligned} \sigma \ = \ (a_1,\ldots ,a_m)_{\textrm{c}} \end{aligned}$$

for this mapping and note that it has m equivalent representations

$$\begin{aligned} \sigma \ = \ (a_1,\ldots ,a_m)_{\textrm{c}} \ = \ (a_2,\ldots ,a_m,a_1)_{\textrm{c}} \ = \ \cdots \ = \ (a_m,a_1,\ldots ,a_{m-1})_{\textrm{c}}. \end{aligned}$$

Any permutation $\sigma \in {\mathcal {S}}_n$ can be written as

$$\begin{aligned} \sigma \ = \ (a_{11},\ldots ,a_{1m(1)})_{\textrm{c}} \circ \cdots \circ (a_{k1},\ldots ,a_{km(k)})_{\textrm{c}}, \end{aligned}$$

where the sets $\{a_{j1},\ldots ,a_{jm(j)}\}$, $1 \le j \le k$, form a partition of $\langle n\rangle$. Note that the cycles $(a_{j1},\ldots ,a_{jm(j)})_{\textrm{c}}$, $1\le j\le m$, commute. This representation of $\sigma$ as a combination of cycles is unique if we require, for instance, that

$$\begin{aligned} a_{jm(j)} \ = \ \min \{a_{j1},\ldots ,a_{jm(j)}\} \quad \text {for} \ 1 \le j \le k \end{aligned}$$

and

$$\begin{aligned} a_{1m(1)}< \cdots < a_{km(k)}. \end{aligned}$$

In what follows, let ${\mathcal {S}}_n^*$ be the set of all permutations $\sigma \in {\mathcal {S}}_n$ consisting of just one cycle, i.e.,

$$\begin{aligned} \sigma \ = \ (a_1,a_2,\ldots ,a_n)_{\textrm{c}} \end{aligned}$$

with pairwise different numbers $a_1, a_2, \ldots , a_n \in \langle n\rangle$.

The coupling. The standardized cycle representation of $\sigma \in {\mathcal {S}}_n$ gives rise to a particular mapping ${\mathcal {S}}_n \ni \pi \mapsto (\sigma ,\sigma ^*) \in {\mathcal {S}}_n \times {\mathcal {S}}_n^*$ such that $\pi \mapsto \sigma$ is bijective. For fixed $\pi \in {\mathcal {S}}_n$ and any index $i \in \langle n\rangle$, let

$$\begin{aligned} M_i \,= \ \langle n\rangle \setminus \{\pi (s): 1 \le s < i\}, \end{aligned}$$

i.e., $\langle n\rangle = M_1 \supset M_2 \supset \cdots \supset M_n = \{\pi (n)\}$, and $\# M(i) = n+1-i$. Let $1 \le t_1< t_2< \cdots < t_k = n$ be those indices i such that $\pi (i) = \min (M_i)$. Then,

$$\begin{aligned} \sigma \,= \ \left( \pi (1), \ldots , \pi (t_1) \right) _{\textrm{c}} \circ \left( \pi (t_1+1),\ldots ,\pi (t_2) \right) _{\textrm{c}} \circ \cdots \circ \left( \pi (t_{k-1}+1), \ldots , \pi (t_k) \right) _{\textrm{c}} \end{aligned}$$

defines a permutation of $\langle n\rangle$ with standardized cycle representation. This is essentially the construction used by Feller (1945) to investigate the number of cycles of a random permuation. Moreover,

$$\begin{aligned} \sigma ^* \,= \ \left( \pi (1), \pi (2), \ldots , \pi (n) \right) _{\textrm{c}} \end{aligned}$$

defines a permutation in ${\mathcal {S}}_n^*$ such that

$$\begin{aligned} \left\{ i \in \langle n\rangle : \sigma (i) \ne \sigma ^*(i) \right\} \ = \ {\left\{ \begin{array}{ll} \emptyset &{} \text {if} \ k = 1, \\ \{t_1,\ldots ,t_k\} &{} \text {if} \ k \ge 2. \end{array}\right. } \end{aligned}$$

Suppose that $\pi$ is a random permutation with uniform distribution on ${\mathcal {S}}_n$. Then, $\sigma$ is a random permutation with uniform distribution on ${\mathcal {S}}_n$ too, because $\pi \mapsto \sigma$ is a bijection. Since the conditional distribution of $\pi (i)$, given $(\pi (s))_{1 \le s < i}$, is the uniform distribution on $M_i$, the random variables

$$\begin{aligned} Y_i \,= \ 1_{[\pi (i) = \min (M_i)]}, \quad i \in \langle n\rangle , \end{aligned}$$

are stochastically independent Bernoulli random variables with $\textrm{I}\!\textrm{P}(Y_i = 1) = (n+1-i)^{-1} = 1 - \textrm{I}\!\textrm{P}(Y_i = 0)$. Consequently,

$$\begin{aligned} \textrm{I}\!\textrm{E}\left( \# \left\{ i \in \langle n\rangle : \sigma (i) \ne \sigma ^*(i) \right\} \right) \ \le \ \sum _{i=1}^n (n+1-i)^{-1} \ = \ 1 + \sum _{j=2}^n j^{-1} \ \le \ 1 + \log (n), \end{aligned}$$

because $j^{-1} \le \int _{j-1}^j x^{-1} \, dx = \log (j) - \log (j-1)$ for $2 \le j \le n$.

1.2 A.2 Some inequalities related to Lindeberg-type conditions

In connection with Gaussian approximations and Stein’s method, see Stein (1986) or Barbour and Chen (2005), the quantity

$$\begin{aligned} L(X) \,= \ \textrm{I}\!\textrm{E}\left( X^2 \min (|X|,1) \right) \end{aligned}$$

for a square-integrable random variable X plays an important role. Elementary considerations show that

$$\begin{aligned} h(x) \ \le \ x^2 \min (|x|,1) \ \le \ \sqrt{2} \, h(x) \quad \text {with}\quad h(x) \,= \ \frac{|x|^3}{\sqrt{1 + x^2}} \end{aligned}$$

for arbitrary $x \in {\mathbb {R}}$. Moreover, $h: {\mathbb {R}}\rightarrow [0,\infty )$ is an even, convex function such that $h(2x) \le 8 h(x)$. Consequently, for arbitrary $x,y \in {\mathbb {R}}$, Jensen’s inequality implies that

$$\begin{aligned} (x + y)^2 \min (|x + y|, 1)&\le \ \sqrt{2} \, \textrm{I}\!\textrm{E}h(x + y) \\&\le \ 2^{-1/2} \left( h(2x) + h(2y) \right) \\&\le \ \sqrt{32} \, \textrm{I}\!\textrm{E}h(x) + \sqrt{32} \, \textrm{I}\!\textrm{E}h(y) \\&\le \ 6 x^2 \min (|x|, 1) + 6 y^2 \min (|y|,1) \ \le \ 6 x^2 \min (|x|,1) + 6 y^2 . \end{aligned}$$

For a symmetric matrix $A \in {\mathbb {R}}^{n\times n}$, we define its row means $\bar{A}_i:= n^{-1} \sum _{j=1}^n A_{ij}$ and its overall mean $\bar{A}:= n^{-2} \sum _{i,j=1}^n A_{ij}$. Let $\tilde{A}:= (A_{ij} - \bar{A}_i - \bar{A}_j + \bar{A})_{i,j=1}^n$. Then, elementary calculations and the previous inequalities reveal that

$$\begin{aligned} 0 \ \le \ n^{-1} \sum _{i,j=1}^n A_{ij}^2 - n^{-1} \sum _{i,j=1}^n \tilde{A}_{ij}^2 \ \le \ 2 \sum _{i=1}^n \bar{A}_i^2 \end{aligned}$$

and

$$\begin{aligned} n^{-1} \sum _{i,j=1}^n \tilde{A}_{ij}^2 \min (|\tilde{A}_{ij}|,1) \ \le \ 6 n^{-1} \sum _{i,j=1}^n A_{ij}^2 \min (|A_{ij}|,1) + 12 \sum _{i=1}^n \bar{A}_i^2. \end{aligned}$$