Note on Qualitative Robustness of Multivariate Sample Mean and Median

Abstract

It is known that the robustness properties of estimators depend on the choice of a metric in the space of distributions. We introduce a version of Hampel's qualitative robustness that takes into account the √𝑛-asymptotic normality of estimators in 𝑅^𝑘, and examine such robustness of two standard location estimators in ℝ^𝑘. For this purpose, we use certain combination of the Kantorovich and Zolotarev metrics rather than the usual Prokhorov type metric. This choice of the metric is explained by an intention to expose a (theoretical) situation where the robustness properties of sample mean and 𝐿₁-sample median are in reverse to the usual ones. Using the mentioned probability metrics we show the qualitative robustness of the sample multivariate mean and prove the inequality which provides a quantitative measure of robustness. On the other hand, we show that 𝐿₁-sample median could not be “qualitatively robust” with respect to the same distance between the distributions.

1. Introduction

The following Hampel’s definition (originally given for the one-dimensional case) of qualitative robustness [1, 2] deals with 𝜋-balls in the space of distributions rather than with standard “contamination neighborhoods” (see for the latter, e.g., [3, 4]).

The sequence 𝑄_𝑛, 𝑛≥1, of estimators is qualitatively robust at the distribution ℒ if for every 𝜖>0 there exists 𝛿>0 such that 𝜋(ℒ,ℒ)<𝛿 entails sup_𝑛≥1𝜋𝑄_𝑛𝑋₁,…,𝑋_𝑛,𝑄_𝑛𝑋₁,…,𝑋_𝑛<𝜖. (1)

Here, and throughout, 𝜋 denotes the Prokhorov metric, and 𝑋₁,𝑋₂,…; 𝑋₁,𝑋₂,…, are i.i.d. random vectors distributed, respectively, as ℒ and ℒ.

For a metric 𝜇 on the space of distributions and random vectors 𝑋,𝑌 we will write 𝜇(𝑋,𝑌) (as in (1)) having in mind the 𝜇-distance between the distributions of 𝑋 and 𝑌.

By all means, the use of the Prokhorov metric is only an option. For instance, in [5] other probability metrics in the definition of qualitative robustness were used. (See also [6–9] for using different probability metrics or pseudo-metrics related to the estimation of robustness).

As noted in [1, 2] in ℝ¹, sample means are not qualitatively robust at any ℒ, while sample medians are qualitatively robust at any ℒ having a unique median. (See also [10] for lack of qualitative robustness of sample means in certain Banach spaces).

Moreover, in [11] it was shown that for symmetric distributions the median is, in certain sense, the “most robust” estimator of a center location when using the pseudo-metric corresponding to neighborhoods of contamination type. (See more with this respect in [8]).

At the same time, it is known from the literature that under different circumstances, in particular, using distinct probability metrics (as, e.g., in (1) or in other definitions) the robustness properties of estimators can change considerably (see, for instance, the discussion in [12]).

The first aim of the present paper is to consider a modified version of Hampel’s definition of qualitative robustness taking into account the √𝑛-asymptotic normality of the sequence of estimators. The formal definition is given in the next section but, basically, we replace (1) with the following condition: sup_𝑛≥1√𝑛𝑙𝑄_𝑛𝑋₁,…,𝑋_𝑛,𝑄_𝑛𝑋₁,…,𝑋_𝑛+𝛾≤𝜀, (2) where 𝛾 is some constant, and 𝑙 is a probability metric (different from the Prokhorov metric in our case).

The second goal of the paper is to present an example of two probability metrics 𝑙 and 𝜇 (on the space of distributions in ℝ^𝑘) for which the following holds: (i)when 𝑄_𝑛=𝑇_𝑛,𝑛≥1, are multivariate sample means, the left-hand side of inequality (2) (with some 𝛾) is bounded by const ⋅𝜇(ℒ,ℒ); (ii)when 𝑄_𝑛=𝑀_𝑛, 𝑛≥1, are sample medians, we give an example of symmetric smooth distributions ℒ and ℒ_𝛼,  𝛼>0, in ℝ, such that 𝜇(ℒ,ℒ_𝛼)→0 as 𝛼→0, while there is a positive constant 𝑟 such that the left-hand side of inequality (2) (with 𝛾=0) is greater than 𝑟 for all sufficiently small 𝛼>0. Therefore, sample medians are not qualitative robust (in our sense) with respect to these metrics.

The metrics 𝑙 and 𝜇 are the following (the complete definitions are given in Section 2). 𝑙 is the Kantorovich metric (see, e.g., [13]) and 𝜇 is certain combination of 𝑙 and of the Zolotarev metric of order 2 (see [14]).

We should stress that this choice is not determined by any advantages for statistical applications. Moreover, the closeness of distributions in the Kantorovich metric implies the closeness in the Prokhorov metric, but also reduces the probability of “large-valued outliers” (but not the rounding errors). Therefore, our selection of metric is not quite consistent with the standard approach to qualitative robustness where the Prokhorov metric (or its invariant versions) is used.

Nevertheless, our choice allows to unveil the possible unusual robustness properties of sample means and medians and to assess the certain quantitative robustness of the multivariate sample mean (with respect to the considered metrics!). The obtained “robustness inequality" does not work in the “gross error model” but (jointly with inequalities (23)) it could be useful for quantitative assessment of the robustness of sample means, under perturbation of data of “rounding” type.

2. Basic Definitions

Let (𝐻,|⋅|) be a separable Hilbert space with the norm |⋅| generated by an inner product ⟨⋅,⋅⟩, and let ℬ(𝐻) be the Borel 𝜎-algebra of subsets of 𝐻. Let also 𝑋;𝑋₁,…,𝑋_𝑛,… and 𝑋;𝑋₁,…,𝑋_𝑛,… be two sequences of i.i.d. random vectors in 𝐻 with their respective distributions denoted by ℒ and ℒ.

Under the assumption 𝐸|𝑋|<∞,𝐸𝑋<∞, (3) the means 𝐸𝑋 and 𝐸𝑋 are defined as the corresponding Bochner integrals.

The definition of the median of 𝑋 is less standard (consult, e.g., [15] for different definitions in 𝐻=ℝ^𝑘). We will use the definition of median given in general setting in [16] (sometimes called 𝕃₁-median): 𝑀∶=argmin_𝑥∈𝐻𝐸(|𝑋−𝑥|−|𝑥|),𝑀∶=argmin_𝑥∈𝐻𝐸𝑋−𝑥−|𝑥|. (4)(5)

In [16] it was shown that 𝑀 exists, and it is unique unless ℒ is concentrated on a one-dimensional subspace of 𝐻. In the last case the set of minimizers in (4) is {𝑡𝑥₀∶𝑡∈[𝑎,𝑏]} for some 𝑥₀∈𝐻, and one can set 𝑀∶=((𝑎+𝑏)/2)𝑥₀.

In what follows we denote (𝑛=1,2,…): 𝑆_𝑛∶=𝑋₁+⋯+𝑋_𝑛,𝑆_𝑛∶=𝑋₁+⋯+𝑋_𝑛,𝑇_𝑛=𝑆_𝑛𝑛,𝑇_𝑛𝑆∶=^𝑛𝑛samplemeans. (6)

On the other hand, let (𝑛=1,2,…), 𝑀_𝑛=𝑀_𝑛𝑋₁,…,𝑋_𝑛,𝑀_𝑛=𝑀_𝑛𝑋₁,…,𝑋_𝑛 (7) be sample medians defined by (4) and (5) replacing ℒ and ℒ  by the corresponding empirical distributions ℒ_𝑛 and ℒ_𝑛 (obtained from 𝑋₁,…,𝑋_𝑛 and 𝑋₁,…,𝑋_𝑛, resp.). Robustness (in terms of 𝜖-contamination neighborhoods) and asymptotic normality of 𝑀_𝑛, 𝑛≥1, were proved, for instance, in [17] (see also [18]). The qualitative robustness properties of sample means {𝑇_𝑛} and sample medians {𝑀_𝑛} when the Prokhorov metric is used were discussed in Introduction.

Let us first see what happens with qualitative robustness of {𝑇_𝑛} and {𝑀_𝑛} if in the above definition we replace 𝜋 with the Kantorovich metric: 𝑙𝑋,𝑋≡𝑙ℒ,ℒ∶=sup_{𝜙∈𝐿𝑖𝑝}𝐸𝜙(𝑋)−𝐸𝜙𝑋, (8) where 𝜙(𝑥)𝑦≤.𝐿𝑖𝑝∶=𝜙∶𝐻↦ℝ∶𝜙isboundedand−𝜙𝑥−𝑦;𝑥,𝑦∈𝐻 (9)

Under condition (3)𝑙(𝑋,𝑋)<∞, and it is well known (see, e.g., [13]) that 𝑙(𝑌_𝑛,𝑌)→0 if and only if 𝜋(𝑌_𝑛,𝑌)→0 and 𝐸|𝑌_𝑛|→𝐸|𝑌|.

For 𝐻=ℝ, 𝑙𝑋,𝑋=^∞_−∞𝐹_𝑋(𝑥)−𝐹_𝑋(𝑥)𝑑𝑥. (10)

Now, applying the regularity properties of𝑙 (see, e.g., [13]): 𝑙𝑆_𝑛𝑛,𝑆_𝑛𝑛≤1𝑛^𝑛_𝑖=1𝑙𝑋_𝑖,𝑋_𝑖=𝑙ℒ,ℒ, (11) we see that the sequence of sample means {𝑇_𝑛,𝑛≥1} is “qualitatively robust” using 𝑙 instead of the Prokhorov metric.

It seems straightforward (using the approach similar to one given in [10, 19] and results in [16]) to show that the sequence of sample medians {𝑀_𝑛,𝑛≥1} is also “qualitatively robust” with respect to 𝑙. However, this is out of the scope of this paper.

3. 𝑙−𝜇−√𝑛-Robustness and Main Results

Now and in what follows we suppose that 𝐻=ℝ^𝑘 with the Euclidean norm |⋅|. The results presented in this section are an extension to the multidimensional case of the similar findings for 𝐻=ℝ, published in the hardly accessible proceedings [20]. Moreover, we improve the results of [20] even in the one-dimensional case.

In order to simplify calculations in the proof of Theorem 5 below we will use in the definitions of Kantorovich’s and Zolotarev’s metrics (see below) the following norm ‖𝑥‖∶=∑^𝑘_𝑖=1|𝑥_𝑖| in the space ℝ^𝑘.

Thus, in what follows the Kantorovich metric 𝑙 is defined by relationships (8), (9), where in (9) in place of the norm |⋅| the norm ||⋅|| is used.

Let 𝜇 be some fixed simple probability metric on the set of all probability distributions on (ℝ^𝑘,ℬ(ℝ^𝑘)).

We will consider a sequence 𝑄_𝑛, 𝑛≥1, of estimators of some parameter 𝜃∈ℝ^𝑘 of the distribution ℒ (𝜃 of the distribution ℒ, resp.).

Definition 1. We say that a sequence 𝑄_𝑛, 𝑛≥1, is (𝑙−𝜇−√𝑛)-robust at ℒ if there is some fixed vector 𝛾=𝛾(ℒ,ℒ)∈ℝ^𝑘 such that for every 𝜖>0 there exists 𝛿>0 such that 𝜇(ℒ,ℒ)≤𝛿 entails sup_𝑛≥1√𝑛𝑙𝑄_𝑛𝑋₁,…,𝑋_𝑛,𝑄_𝑛𝑋₁,…,𝑋_𝑛+𝛾≤𝜖. (12)

Remark 2. Taking into account that 𝑙(𝑎𝑋+𝑏,𝑎𝑌+𝑏)=𝑎𝑙(𝑋,𝑌), 𝑎≥0,𝑏∈ℝ^𝑘, we see that (12) can be related with the √𝑛-asymptotic normality of estimators 𝑄_𝑛, 𝑛≥1. The “scaling parameter” 𝛾 (in (12)) is necessary to ensure the equality of means of the corresponding limit normal distributions (when they exist). Only in case 𝐸𝑋=𝐸𝑋 in (12) 𝛾=0.

Zolotarev’s probability metric of order 2 𝜁₂ is defined as follows (see, e.g., [14, 21]): 𝜁₂𝑋,𝑋≡𝜁₂ℒ,ℒ∶=sup_𝜙∈𝒟2𝐸𝜙(𝑋)−𝐸𝜙𝑋, (13) where 𝒟₂∶=𝜙∶ℝ^𝑘↦ℝ∶𝐷𝜙(𝑥)−𝐷𝜙𝑦≤𝑥−𝑦;𝑥,𝑦∈ℝ^𝑘, (14) and 𝐷𝜙=(𝐷₁𝜙,…,𝐷_𝑘𝜙) is the gradient of 𝜙.

Remark 3. The distance 𝜁₂(𝑋,𝑋) can take infinite value. Particularly, from (13), (14) we see that 𝜁₂(𝑋,𝑋)=∞ if 𝐸𝑋≠𝐸𝑋. On the other hand, if there exist second moments, 𝐸𝑋=𝐸𝑋 and 𝐸𝑋²<∞, 𝐸𝑋²<∞, then 𝜁₂(𝑋,𝑋)<∞ (see, e.g., [14]). The function 𝜙(𝑥)∶=(1/2)|𝑥|²∈𝒟₂ in (14). Therefore, by (13), 𝐸𝑋²−𝐸𝑋²≤𝜁₂𝑋,𝑋. (15) Thus, if 𝐸𝑋²<∞ and 𝜁₂(𝑋,𝑋)<∞ then 𝐸𝑋²<∞. Consequently, if 𝐸𝑋=𝐸𝑋 and 𝐸|𝑋|²<∞, then 𝜁₂(𝑋,𝑋)<∞ if and only if 𝐸|𝑋|²<∞.

We now define the metric 𝜇 to work with: 𝜇𝑋,𝑋∶=max2𝑙𝑋,𝑋,inf_{𝑏∈ℝ𝑘}𝜁₂𝑋,𝑋+𝑏. (16)

3.1. (𝑙−𝜇−√𝑛)-Robustness of Sample Means

To prove the inequality in Theorem 5 below we need to impose the following restriction on the distribution ℒ.

Assumption 4. (i) 𝐸|𝑋|^𝑘+2<∞, and the covariance matrix ℳ of 𝑋 is positive definite.
The distribution ℒ of 𝑋 has a density 𝑓 such that for some 𝑠≥1:
(ii) the density 𝑓_𝑠 of 𝑆_𝑠∶=𝑋₁+⋯+𝑋_𝑠 is bounded and differentiable;
(iii) the gradient 𝐷𝑓_𝑠 is bounded and |𝐷𝑓_𝑠| belongs to 𝕃₁(ℝ^𝑘);
(iv) for some 𝛼>0_{|𝑥|>𝛼𝑛}𝐷𝑓_𝑠𝑑𝑥=𝑂𝑛^−1/2,as𝑛→∞. (17)

Note that in view of Remark 3 under the condition (i) 𝜇(𝑋,𝑋)<∞ if and only if 𝐸𝑋²<∞, (18) (see (15)).

Theorem 5. Under Assumption 4 and supposing (18) it holds sup_𝑛≥1√𝑛𝑙𝑇_𝑛,𝑇_𝑛+𝛾≤𝑐𝜇ℒ,ℒ, (19) where 𝛾=𝐸𝑋−𝐸𝑋, 𝑐=max(10𝑠−1)^1/2,5.4𝑑𝑘,𝑑=sup_𝑛≥𝑠max_{1≤𝑖≤𝑘}𝐷_𝑖𝑔_{𝑛𝕃1(ℝ𝑘)}<∞, (20)(21) and 𝑔_𝑛 is the density of (1/√𝑛)(𝑋₁+⋯+𝑋_𝑛), 𝑛≥1.

Remark 6. (i) The constant 𝑐 in (20), (21) is entirely determined by the distribution ℒ of 𝑋. For various particular densities of 𝑋 the constant 𝑑 in (21) can be bounded by means of computer calculations. For this one can use the fact (true under wide conditions) that the sequence 𝐷_𝑖𝑔_𝑛, 𝑛≥𝑠, converges in 𝕃₁-norm to the corresponding partial derivative of the limit normal density with covariance matrix ℳ (and zero mean since ||𝐷_𝑖𝑔_𝑛||_𝕃1(ℝ^𝑘₎ is invariant under translations).
For example, let 𝑘=2 and 𝑋=(𝑋^′,𝑋^′′), where 𝑋^′ and 𝑋^′′ are independent random variables; 𝑋^′ has the gamma density with 𝛼=2.1 and arbitrary 𝜆^′>0, while 𝑋^′′ has the gamma density with 𝛼=3 and arbitrary 𝜆^′′. Simple computer calculations show that in (21) 𝑑<max{0.7065𝜆^′,0.5414𝜆^′′}, and since we can take 𝑠=1, we obtain in (20) that 𝑐<max3,10.8max0.7065𝜆^′,0.5414𝜆^′′. (22) For instance, 𝑐<7.6302 for 𝜆^′=𝜆^′′=1. (For these values of 𝜆^′, 𝜆^′′ we can take 𝑠=3 in (20) and obtain 𝑐<6.3829.)
(ii) Since under the above assumption 𝜇(ℒ,ℒ)<∞ entails (18), inequality (19) ensures (𝑙−𝜇−√𝑛)-robustness of the sequence of sample means 𝑇_𝑛, 𝑛≥1.
(iii) For 𝑘=1, in [20] an example is given showing that in general the sequence of sample means 𝑇_𝑛, 𝑛≥1 is not (𝑙−𝑙−√𝑛)-robust (even if (18) holds and 𝐸𝑋=𝐸𝑋). It is also almost evident that the sample means 𝑇_𝑛, 𝑛≥1, are not (𝑙−𝜇−√𝑛)-robust, for example, if 𝜇 is the total variation metric 𝑉 (or, if 𝜇=max(𝑙,𝑉)). The appearance of Zolotarev’s metric on the right-hand side of (19) is related to closeness of corresponding limit normal distributions.

Corollary 7. Suppose for a moment that 𝜃=𝜃 and that one evaluates the quality of estimators by mean of absolute errors: 𝛿_𝑛∶=𝐸||𝑇_𝑛−𝜃||, 𝛿_𝑛∶=𝐸||𝑇_𝑛−𝜃||. Then from (19) it follows that 𝛿_𝑛−𝛿_𝑛≤𝑛^−1/2const𝜇ℒ,ℒ,𝑛≥1. (23)

(The simple proof is similar to the one given in [20]).

3.2. About (𝑙−𝜇−√𝑛)-Robustness of Sample Medians

Let again 𝜇 be the metric defined in (16). We show that the sequence of sample medians 𝑀_𝑛, 𝑛≥1, in general, is not (𝑙−𝜇−√𝑛)-robust even when 𝑋 and 𝑋 have strictly positive, bounded, smooth densities symmetric with respect to the origin, and the sequences of sample medians 𝑀_𝑛, 𝑛≥1, 𝑀_𝑛, 𝑛≥1, are √𝑛-asymptotically normal. We consider a modified version of the corresponding example from [20].

Example 8. Let 𝐻=ℝ, 𝑋∼𝑁(0,1), and for 𝜖∈(0,1) let 𝑋_𝜖 be a random variable with the density: 𝑓𝑋_𝜖(𝑥)1∶=𝑐(𝜖)𝑥𝜖+²𝜖³+𝑥²𝑒^−𝑥2/2,𝑥∈ℝ, (24) where 𝑐(𝜖) is a normalizing constant. By symmetry of the density, we get 𝑀=𝑀=𝐸𝑋=𝐸𝑋=0, (25) and also it is clear that 𝑓_𝑋,𝑓_𝑋∈ℂ^∞(ℝ), and sup_{()𝜖∈0,1,𝑥∈ℝ}𝑓𝑋_𝜖(𝑥)<∞. (26)

First of all let us show that for this example the left-hand side of (12) is infinite for any 𝛾≠0. It is well known (see, e.g., [16]) that 𝑀_𝑛→𝑀=0, 𝑀_𝑛→𝑀=0 as 𝑛→∞ with probability 1. Also, from the results of [16] we can obtain that |𝑀_𝑛|≤(2/𝑛)∑^𝑛_𝑖=1|𝑋_𝑖|. Using this inequality it is easy to show that the sequence |𝑀_𝑛|, 𝑛≥1, is uniformly integrable. Therefore, 𝐸|𝑀_𝑛|→0 (and also 𝐸|𝑀_𝑛|→0) as 𝑛→∞, and for this 𝑙(𝑀_𝑛,𝑀_𝑛+𝛾)→𝑙(0,𝛾)=𝛾.

Let now 𝛾=0 in (12). By the well-known asymptotic normality (see [22, page 307]), 𝜋√𝑛𝑀_𝑛,𝜂→0,𝜋√𝑛𝑀_𝑛,𝜂_𝜖→0, (27) where 1𝜂∼𝑁0,2𝑓_𝑋(0)=𝑁0,𝜋2,𝜂_𝜖1∼𝑁0,𝑋2𝑓_𝜖(0)𝑐(𝜖)=𝑁0,𝜖. (28)

From (27) and (28) it follows that there is 𝑟>0 such that for all small enough 𝜖 and all large enough 𝑛 we get 𝜋(√𝑛𝑀_𝑛,√𝑛𝑀_𝑛)≥𝑟. Consequently, for all such 𝜖 and 𝑛, √𝑛𝑙𝑀_𝑛,𝑀_𝑛=𝑙√𝑛𝑀_𝑛,√𝑛𝑀_𝑛≥𝑟²>0, (29) since 𝑙≥𝜋² (see [13, page 86]).

We have obtained that the left-hand side of (12) is positive for all small enough 𝜖>0. On the other hand, 𝜇(𝑋,𝑋_𝜖)→0 as 𝜖→0. It follows from the inequality 𝜁₂()≤1𝑋,𝑌2^∞_−∞𝑥²𝑓_𝑋(𝑥)−𝑓_𝑌(𝑥)𝑑𝑥 (30) (valued when 𝐸𝑋=𝐸𝑌; see [21, page 376]) and from (10).

Remark 9. The densities as in (24) represent the following somewhat strange type of “contamination.” Since max_𝑥𝑓_𝑋(𝑥)=𝑓_𝑋(0) sample points from 𝑓_𝑋 tend to concentrate around the origin. But 𝑓𝑋_𝜖(0)→0 as 𝜖→0, and therefore sample points from 𝑓𝑋_𝜖 frequently in some extent are separated from 0.

A natural question is “how to choose the metric 𝜇 to ensure (𝑙−𝜇−√𝑛)-robustness of the sequence of sample medians 𝑀_𝑛, 𝑛≥1?” Our conjecture is (for 𝑘=1, e.g.) to try 𝜇(𝑋,𝑋)=max{𝑙(𝑋,𝑋),esssup_𝑥∈ℝ|𝑓_𝑋(𝑥)−𝑓_𝑋(𝑥)|} (supposing the existence of densities).

If 𝑀=𝑀 then under certain conditions the closeness in 𝜇 guarantees the closeness of normal densities which are limiting for {√𝑛𝑀_𝑛,𝑛≥1} and for {√𝑛𝑀_𝑛,𝑛≥1}, respectively. To attempt proving (𝑙−𝜇−√𝑛)-robustness of 𝑀_𝑛, 𝑛≥1 (as in (12)) one can show Hampel’s qualitative robustness of 𝑀_𝑛, 𝑛≥1 with respect to the metric 𝑙 and then use the property √𝑛𝑙(𝑀_𝑛,𝑀_𝑛)=𝑙(√𝑛𝑀_𝑛,√𝑛𝑀_𝑛). A not clear point of this plan is finding conditions under which 𝐸|√𝑛(𝑀_𝑛−𝑀)|→0 as 𝑛→∞.

Example 10. Let us give another (very simple) example of the sequence of estimators which is (𝑙−𝜇−√𝑛)-robust with 𝜇∶=𝑙 (on the class of distributions described below). For 𝐻=ℝ we consider the class 𝑊 of all random variables 𝑋 with bounded supports Supp(𝑋)=[0,𝜃_𝑋], having density 𝑓_𝑋 such that 𝑓_𝑋(𝑥)≥𝛽>0, 𝑥∈[0,𝜃_𝑋]. We suppose that 𝜃_𝑋≤𝜃^∗<∞, 𝑋∈𝑊, and 𝛽 is the same for all 𝑋∈𝑊. Assume that parameter 𝜃=𝜃_𝑋 is unknown, and the sequence of estimators 𝑄_𝑛∶=max_{1≤𝑖≤𝑛}𝑋_𝑖 is used to estimate it.
Denoting 𝜃=𝜃_𝑋, 𝜃=𝜃_𝑋 and choosing in (12) 𝛾=𝜃−𝜃, we get 𝑙𝑄_𝑛,𝑄_𝑛+𝛾=𝑙𝜃−max_{1≤𝑖≤𝑛}𝑋_𝑖,𝜃−max_{1≤𝑖≤𝑛}𝑋_𝑖≤𝐸𝜃−max_{1≤𝑖≤𝑛}𝑋_𝑖−𝜃−max_{1≤𝑖≤𝑛}𝑋_𝑖≤𝐸𝜃−max_{1≤𝑖≤𝑛}𝑋_𝑖+𝐸𝜃−max_{1≤𝑖≤𝑛}𝑋_𝑖, (31) because of the metric 𝑙 is minimal for the compound metric 𝐸|𝑋−𝑌| (see, e.g., [13]).
By elementary calculations we bound the right-hand side of (31) by 2𝛽().𝑛+1 (32)
Now for each 𝑛 fixed, by induction we obtain (see (10)) 𝑙max_{1≤𝑖≤𝑛}𝑋_𝑖,max_{1≤𝑖≤𝑛}𝑋_𝑖=^∞₀𝐹^𝑛_𝑋(𝑥)−𝐹^𝑛_𝑋(𝑥)𝑑𝑥≤𝑛^∞₀𝐹_𝑋(𝑥)−𝐹_𝑋(𝑥)𝑑𝑥=𝑛𝑙𝑋,𝑋. (33)
On the other hand, let, for example, 𝜃≤𝜃. Then 𝑙𝑋,𝑋=^𝜃₀𝐹_𝑋(𝑥)−𝐹_𝑋(𝑥𝜃)𝑑𝑥+_𝜃1−𝐹_𝑋(𝑥)𝑑𝑥. (34) But for 𝑥∈[𝜃,𝜃], 1−𝐹_𝑋(𝑥)=_𝑥𝜃𝑓_𝑋𝑦𝑑𝑦≥𝛽𝜃−𝑥. (35) From this relationship it follows that for𝑋,𝑋∈𝑊,𝜃→𝜃as𝑙𝑋,𝑋→0. (36) Finally, applying (32), we can select 𝑁 in such a way that sup_𝑛≥𝑁√𝑛𝑙𝑄_𝑛,𝑄_𝑛≤𝜖+𝛾2. (37) For 𝑛<𝑁 we can use the inequality 𝑙(𝑋,𝑌+𝑏)≤𝑙(𝑋,𝑌)+|𝑏| and (33), (36). Thus, choosing small enough 𝛿>0 we can ensure inequality (12) for all 𝑋,𝑋∈𝑊 with 𝜇(𝑋,𝑋)≡𝑙(𝑋,𝑋)≤𝛿. In this way we proved (𝑙−𝑙−√𝑛)-robustness of 𝑄_𝑛 on the set 𝑊.

Appendix

A. The Proofs

To start with the proof of Theorem 5 of the previous section, first of all we note that in (19) 𝑙𝑇_𝑛,𝑇_𝑛𝑆+𝛾=𝑙^𝑛𝑛,𝑆_𝑛𝑛𝑋𝑆+𝐸𝑋−𝐸=𝑙^𝑛−𝑛𝐸𝑋𝑛,𝑆_𝑛−𝑛𝐸𝑋𝑛, (A.1) because of 𝑙(𝑋+𝑏,𝑌+𝑏)=𝑙(𝑋,𝑌), 𝑏∈ℝ^𝑘. Also it is easy to see that 𝜁₂(𝑋+𝑏,𝑌+𝑏)=𝜁₂(𝑋,𝑌).

On the other hand, by definition of 𝜇 in (16) inf_{𝑏∈ℝ𝑘}𝜁₂𝑋,𝑋+𝑏=𝜁₂𝑋,𝑋+𝐸𝑋−𝐸𝑋=𝜁₂𝑋−𝐸𝑋,𝑋−𝐸𝑋 (A.2) (see Remark 3). Also 𝑙𝑋𝑋𝑋𝑋𝑋,𝑋−𝐸𝑋,𝑋−𝐸=𝑙𝑋,𝑋+𝐸𝑋−𝐸≤𝑙𝑋,+𝐸𝑋−≤2𝑙𝑋, (A.3) because the metric 𝑙 is minimal for the metric 𝐸|⋅|.

The above arguments show that to establish inequality (19) it suffices to prove the version of (19) with 𝛾=0, 𝐸𝑋=𝐸𝑋=0, and the metric 𝜇^′∶=max{𝑙,𝜁₂} on the right-hand side of (19).

The proof is based on the two following lemmas.

Let 𝑔∶ℝ^𝑘↦ℝ be a differentiable function. We will write 𝐷𝑔∶=(𝐷₁𝑔,…,𝐷_𝑘𝑔), where 𝐷_𝑖𝑔(𝑥)∶=(𝜕/𝜕𝑥_𝑖)𝑔(𝑥), 𝑖=1,2,…,𝑘.

Lemma A.1. Let 𝑋,𝑌, and 𝜉 be random vectors in ℝ^𝑘 such that (a)𝜉 is independent of 𝑋 and 𝑌; (b)𝐸|𝑋|²<∞, 𝐸|𝑌|²<∞; 𝐸𝑋=𝐸𝑌; (c)𝜉 has a bounded differentiable density 𝑓_𝜉 (with respect to the Lebesgue measure) such that 𝐷_𝑖𝑓_𝜉∈𝕃₁(ℝ^𝑘), 𝑖=1,2,…,𝑘.
Then 𝑙(𝑋+𝜉,𝑌+𝜉)≤𝑘max_{1≤𝑖≤𝑘}𝐷_𝑖𝑓_{𝜉𝕃1(ℝ𝑘)}𝜁₂(𝑋,𝑌). (A.4)

Proof. From the definition (8), (9) of the metric 𝑙 (with the norm ||⋅|| instead of |⋅| !) it follows (the proof is simple) that in (8) the class of functions Lip (given in (9)) can be replaced by the class of all bounded differentiable functions 𝑔∶ℝ^𝑘↦ℝ such that sup_{𝑥∈ℝ𝑘}𝜕𝑔(𝑥)𝜕𝑥_𝑖≤1,𝑖=1,2,…,𝑘. (A.5)
Let us fix any such function 𝑔 and arbitrary 𝑖,𝑗∈{1,2,…,𝑘}. Then (by the Fubini theorem), ()()=𝐸𝑔𝑋+𝜉−𝐸𝑔𝑌+𝜉_ℝ𝑘𝑔(𝑥)𝑑𝑥_ℝ𝑘𝑓_𝜉(𝑥−𝑡)𝑑𝐹_𝑋(𝑡)−𝑑𝐹_𝑌(𝑡×_ℝ𝑘)𝑑𝐹_𝑋(𝑡)−𝑑𝐹_𝑌(𝑡)_ℝ𝑘𝑔(𝑥)𝑓_𝜉(𝑥−𝑡)𝑑𝑥. (A.6) For each 𝑥∈ℝ fixed let ℎ(𝑡)∶=_ℝ𝑘𝑔(𝑥)𝑓_𝜉()=𝑥−𝑡𝑑𝑥_ℝ𝑘𝑔𝑦+𝑡𝑓_𝜉𝑦𝑑𝑦,𝑡∈ℝ^𝑘. (A.7) Because of boundedness of 𝜕𝑔/𝜕𝑡_𝑖 and integrability of 𝐷_𝑖𝑓_𝜉 we can differentiate in (A.7) under the integral sign (see [23, Appendix A]). Thus, 𝜕ℎ𝜕𝑡_jℎ(𝑡)=_ℝ𝑘𝜕𝑔𝜕𝑡_𝑗𝑔𝑦+𝑡𝑓_𝜉𝑦𝜕𝑑𝑦,₂ℎ𝜕𝑡_𝑖𝜕𝑡_𝑗ℎ(𝑡)=−_ℝ𝑘𝜕𝑔𝜕𝑥_𝑗(𝑥)𝜕𝑓_𝜉𝜕𝑡_𝑖(𝑥−𝑡)𝑑𝑥. (A.8) In view of (A.5) we obtain 𝜕²ℎ𝜕𝑡_𝑖𝜕𝑡_𝑗ℎ(𝑡)≤_ℝ𝑘𝜕𝑓_𝜉𝜕𝑡_𝑖(𝑥−𝑡)𝑑𝑥=𝐷_𝑖𝑓_{𝜉𝕃1(ℝ𝑘)}. (A.9)
For every 𝑥,𝑦∈ℝ^𝑘 we have 𝐷ℎ(𝑥)−𝐷ℎ𝑦=^𝑘_𝑗=1𝐷_𝑗ℎ(𝑥)−𝐷_𝑗ℎ𝑦,𝐷_𝑗ℎ(𝑥)−𝐷_𝑗ℎ𝑦≤^𝑘_𝑖=1sup_{𝑥∈ℝ𝑘}𝜕²ℎ𝜕𝑥_𝑖𝜕𝑥_𝑗(𝑥)𝑥_𝑖−𝑦_𝑖. (A.10)(A.11) Comparing (A.6)–(A.11) and taking into account the definition of 𝜁₂ in (13), (14), we obtain inequality (A.4).

Lemma A.2. Under the assumption of the previous section the constant 𝑑 in (21) is finite.

Remark A.3. A similar assertion was proved in [24] for ||⋅||_𝕃1-norm of second derivatives of the densities 𝑔_𝑛 (under sightly different conditions). For this reason we give only a sketch of the proof of Lemma A.2 indicating only differences in comparison to the proof of Lemma  4.1 in [24] (where the omitted details can be seen).

Proof. As before, let 𝑔_𝑛 (𝑛≥𝑠) be the density of 𝑆_𝑛/√𝑛, and 𝜙_𝑛 its characteristic function. Let also 𝑓_𝑛 denote the density of 𝑆_𝑛. There is 𝑙≥𝑠 such that |𝑡|𝜙_𝑙(𝑡)∈𝕃₁(ℝ^𝑘) and therefore |𝑡|𝜙_𝑛(𝑡)∈𝕃₁ℝ^𝑘,for𝑛≥𝑙. (A.12) Indeed, since 𝑣_𝑖𝑥_𝑖𝜕∶=𝜕𝑥_𝑖𝑓_𝑠…,𝑥_𝑖,…∈𝕃₁(ℝ) (A.13) (fixing other variables), we can integrate by parts: 𝜙_𝑠√𝑠𝑡=_ℝ𝑘−1𝑒^{𝑖⟨𝑡′,𝑥′⟩}𝑑𝑥^′_ℝ𝑓_𝑠…,𝑥_𝑖,…𝑒^{𝑖𝑡𝑖𝑥𝑖}=𝑑𝑥_ℝ𝑘−1𝑒^{𝑖⟨𝑡′,𝑥′⟩}𝑑𝑥^′1𝑖𝑡_𝑖ℝ𝑣_𝑖𝑥_𝑖𝑒^{𝑖𝑡𝑖𝑥𝑖}𝑑𝑥_𝑖, (A.14) and (A.12) follows for large enough 𝑙.
In view of (A.12) we can write down the inverse Fourier transform for 𝑛≥𝑙: 𝑔_𝑛(𝑥)=1(2𝜋)^𝑘ℝ𝑘𝜙_𝑛(𝑡)𝑒^{−𝑖⟨𝑡,𝑥⟩}𝑑𝑡, (A.15) and differentiate under the integral sign in (A.15).
The condition 𝜙_𝑛∈𝕃₁(ℝ^𝑘) (𝑛≥𝑙) and Assumption (i) in Section 2 ensure the hypothesis of in [25, Theorem 19.2, Ch. 4]. By this theorem, 𝑔_𝑛(𝑥)=^𝑘_𝑚=0𝑛^−𝑚/2𝑃_𝑚(𝑥)𝜙_0,ℳ(𝑥)1+𝑜𝑛^𝑘/2, (A.16)𝑥∈ℝ^𝑘 (as 𝑛→∞), where 𝑃_𝑚, 𝑚=0,…,𝑘, are certain polynomials and 𝜙_0,ℳ is the normal density with zero mean and the covariance matrix ℳ (see Assumption (i)).
Let 𝑞_𝑛 denote the Fourier transform of 𝐷_𝑗ℎ_𝑛, where ℎ_𝑛(𝑥)∶=𝑔_𝑛(𝑥)−^𝑘_𝑚=0𝑛^−𝑚/2𝑃_𝑚(𝑥)𝜙_0,ℳ(𝑥). (A.17) Using (A.15), (A.16), and in [25, Lemma 7.2, Ch. 2], we can obtain that for each fixed 𝑗𝑞_𝑛(𝑡)=𝑡_𝑗𝜙_𝑛(𝑡)−^𝑘_𝑚=0𝑛^−𝑚/2𝑃_𝑚(𝑡)−1exp2⟨𝑡,ℳ𝑡⟩, (A.18) with certain polynomials 𝑃_𝑚.
By arguments similar to those given in [24], it follows from (A.18) that there exist constants 𝛽>0, 𝑐<∞ such that _{√|𝑡|≤𝛽𝑛}𝑞_𝑛(𝑡𝑐)𝑑𝑡≤𝑛^𝑘/2(𝑛≥𝑙). (A.19) On the other hand, to prove that _{√|𝑡|>𝛽𝑛}𝑞_𝑛(𝑡1)=𝑂𝑛^𝑘/2,as𝑛→∞, (A.20) it is sufficient to show that _{√|𝑡|>𝛾𝑛}𝑡_𝑗𝜙_𝑛(𝑡1)𝑑𝑡=𝑂𝑛^𝑘/2,as𝑛→∞. (A.21) But the last equality follows from (A.12) and the fact that sup_{√|𝑡|>𝛾𝑛}𝜙₁𝑡√𝑛≤𝜅<1. (A.22) Expressing 𝐷_𝑗ℎ_𝑛 in terms of 𝑞_𝑛 (as an inverse Fourier transform), and using (A.16)–(A.20) we can establish that there exist a constant 𝑐 and polynomials 𝑃_𝑚(𝑥), 𝑚=0,…,𝑘 such that for each 𝑗∈{1,…,𝑘}, sup_{𝑥∈ℝ𝑘}𝐷_𝑗𝑔_𝑛(𝑥)−^𝑘_𝑚=0𝑛^−𝑚/2𝑃_𝑚(𝑥)𝜙_0,ℳ(𝑥)≤𝑐𝑛^𝑘/2, (A.23)𝑛=𝑙,𝑙+1,…
The next step is to find for each 𝑗∈{1,…,𝑘} an upper bound for ||𝐷_𝑗𝑔_𝑛||_𝕃1(ℝ^𝑘₎ which does not depend on 𝑛≥𝑠.
For 𝑛>𝑠, 𝑔_𝑛(𝑥)=𝑛^𝑘/2𝑓_𝑛(√𝑛𝑥) and 𝑓_𝑛(𝑥)=∫_ℝ𝑘𝑓_𝑠(𝑥−𝑡)𝑓_{𝑛−𝑠(𝑡)}𝑑𝑡. Thus, using Assumptions (ii), (iii), and the corresponding theorems in [23, Appendix A], we get 𝐷_𝑗𝑓_𝑛(𝑥)=_ℝ𝑘𝐷_𝑗𝑓_𝑠(𝑥−𝑡)𝑓_𝑛−𝑠(𝑡)𝑑𝑡. (A.24) We have 𝐷_𝑗𝑔_{𝑛𝕃1(ℝ𝑘)}≤_{√|𝑥|≤2𝑛𝛼}𝐷_𝑗𝑔_𝑛𝑑𝑥+_{√|𝑥|>2𝑛𝛼}𝐷_𝑗𝑔_𝑛𝑑𝑥, (A.25) where 𝛼 is the constant from Assumption (iv). The first summands on the right-hand side of (A.25) are uniformly bounded in 𝑛≥𝑙 due to (A.23). To bound the second terms in (A.25) we write (see (A.24)) _{√|𝑥|>2𝑛𝛼}𝐷_𝑗𝑔_𝑛𝑑𝑥=𝑛^1/2_{|𝑧|>2𝛼𝑛}𝐷_𝑗𝑓_𝑛(𝑧)𝑑𝑧≤𝑛^1/2_{|𝑧|>2𝛼𝑛}𝑑𝑧_ℝ𝑘𝐷_𝑗𝑓_𝑠(𝑧−𝑡)𝑓_𝑛−𝑠(𝑡)𝑑𝑡=∶𝑛^1/2𝐼_𝑛. (A.26) Now 𝐼_𝑛=_{|𝑧|>2𝛼𝑛}𝑑𝑧_{|𝑡|≤𝛼𝑛}𝐷_𝑗𝑓_𝑠(𝑧−𝑡)𝑓_𝑛−𝑠(𝑡)+𝑑𝑡_{|𝑧|>2𝛼𝑛}𝑑𝑧_{|𝑡|>𝛼𝑛}𝐷_𝑗𝑓_𝑠(𝑧−𝑡)𝑓_𝑛−𝑠(𝑡)𝑑𝑡. (A.27) From |𝑧|>2𝛼𝑛 and |𝑡|≤𝛼𝑛 it follows that |𝑧−𝑡|>𝛼𝑛. Thus, the first terms on the right-hand side of (A.27) are less than 𝑐^′/𝑛^1/2 due to Assumption (iv). Applying the Fubini theorem and using the fact of integrability of 𝐷_𝑗𝑓_𝑠 we see that the second summand in (A.27) is bounded by const 𝑃(|∑^𝑛−𝑠_𝑖=1𝑋_𝑖|>𝛼𝑛), which is 𝑂(1/𝑛) by the Chebyshev and Rosenthal inequalities.

Exploiting Lemmas A.1 and A.2, the rest of the proof of the Theorem 5 in Section 3 is carried out exactly as the proof in one-dimensional case given in [20]. The “ideality properties” of the metric 𝜁₂, 𝜁₂𝑎^𝑛_𝑖=1𝑋_𝑖,𝑎^𝑛_𝑖=1𝑋_𝑖≤𝑎^2𝑛_𝑖=1𝜁₂𝑋_𝑖,𝑋_𝑖, (A.28) used there hold true for random vectors (see, e.g., [14]). The general version of inequality (3.15) in [20] which relates the Kantorovich and the total variation metrics is proved in [13, page 89]. Note that the proof presented in [20] uses the so-called convolution approach (see, e.g., [26]) and induction arguments. This method has been widely used to estimate rates of convergence in multidimensional Central Limit Theorems (see, e.g., [13, 14, 21, 26]).

Acknowledgments

The authors thank the National System of Investigators (SNI) of CONACYT, Mexico for the partial support of this work. The authors are grateful to the anonymous referee for his valuable suggestions on the improvement of an earlier version of this paper.