Note on Qualitative Robustness of Multivariate Sample Mean and Median

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, and examine such robustness of two standard location estimators in R. 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) situationwhere the robustness properties of samplemean and L1-samplemedian 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 quantitativemeasure of robustness. On the other hand, we show that L1-samplemedian could not be “qualitatively robust” with respect to the same distance between the distributions.


Introduction
The following Hampel's definition (originally given for the one-dimensional case) of qualitative robustness [1,2] deals with -balls in the space of distributions rather than with standard "contamination neighborhoods" (see for the latter, e.g., [3,4]).
For a metric  on the space of distributions and random vectors ,  we will write (, ) (as in (1)) having in mind the -distance between the distributions of  and .
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][7][8][9] for using different probability metrics or pseudo-metrics related to the estimation of robustness).
As noted in [1,2] in R 1 , 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 √-asymptotic normality of the sequence of where  is some constant, and  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  and  (on the space of distributions in R  ) for which the following holds: (i) when   =   ,  ≥ 1, are multivariate sample means, the left-hand side of inequality (2) (with some ) is bounded by const ⋅(L, L); (ii) when   =   ,  ≥ 1, are sample medians, we give an example of symmetric smooth distributions L and L ,  > 0, in R, such that (L, L ) → 0 as  → 0, while there is a positive constant  such that the lefthand side of inequality (2) (with  = 0) is greater than  for all sufficiently small  > 0. Therefore, sample medians are not qualitative robust (in our sense) with respect to these metrics.
The metrics  and  are the following (the complete definitions are given in Section 2). is the Kantorovich metric (see, e.g., [13]) and  is certain combination of  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.
Under the assumption the means  and  X are defined as the corresponding Bochner integrals.
The definition of the median of  is less standard (consult, e.g., [15] for different definitions in  = R  ).We will use the definition of median given in general setting in [16] (sometimes called L 1 -median): In [16] it was shown that  exists, and it is unique unless L is concentrated on a one-dimensional subspace of .In the last case the set of minimizers in ( 4) is { 0 :  ∈ [, ]} for some  0 ∈ , and one can set  := (( + )/2) 0 .
For  = R, Now, applying the regularity properties of  (see, e.g., [13]): we see that the sequence of sample means {  ,  ≥ 1} is "qualitatively robust" using  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 {  ,  ≥ 1} is also "qualitatively robust" with respect to .However, this is out of the scope of this paper.

𝑙−𝜇−√𝑛-Robustness and Main Results
Now and in what follows we suppose that  = R  with the Euclidean norm |⋅|.The results presented in this section are an extension to the multidimensional case of the similar findings for  = R, published in the hardly accessible proceedings [20].Moreover, we improve the results of [20] even in the onedimensional 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 ‖‖ := ∑  =1 |  | in the space R  .Thus, in what follows the Kantorovich metric  is defined by relationships ( 8), (9), where in (9) in place of the norm | ⋅ | the norm || ⋅ || is used.
Let  be some fixed simple probability metric on the set of all probability distributions on (R  , B(R  )).
The "scaling parameter"  (in (12)) is necessary to ensure the equality of means of the corresponding limit normal distributions (when they exist).Only in case  =  X in (12)  = 0.
We now define the metric  to work with: (ii) the density   of   :=  1 + ⋅ ⋅ ⋅ +   is bounded and differentiable; (iii) the gradient   is bounded and (iv) for some  > 0 Note that in view of Remark 3 under the condition (i) (, X) < ∞ if and only if (see (15)).

Remark 6. (i)
The constant  in ( 20), ( 21) is entirely determined by the distribution L of .For various particular densities of  the constant  in ( 21) can be bounded by means of computer calculations.For this one can use the fact (true under wide conditions) that the sequence     ,  ≥ , converges in L 1 -norm to the corresponding partial derivative of the limit normal density with covariance matrix M (and zero mean since ||    || L 1 (R  ) is invariant under translations).For example, let  = 2 and  = (  ,   ), where   and   are independent random variables;   has the gamma density with  = 2.1 and arbitrary   > 0, while   has the gamma density with  = 3 and arbitrary   .Simple computer calculations show that in (21)  < max{0.7065 , 0.5414  }, and since we can take  = 1, we obtain in (20) that  < max {3, 10.8 max {0.7065  , 0.5414  }} . ( For instance,  < 7.6302 for   =   = 1.(For these values of   ,   we can take  = 3 in ( 20) and obtain  < 6.3829.)(ii) Since under the above assumption (L, L) < ∞ entails (18), inequality (19) ensures ( −  − √)-robustness of the sequence of sample means   ,  ≥ 1.
(iii) For  = 1, in [20] an example is given showing that in general the sequence of sample means   ,  ≥ 1 is not ( −  − √)-robust (even if (18) holds and  =  X).It is also almost evident that the sample means   ,  ≥ 1, are not ( −  − √)-robust, for example, if  is the total variation metric  (or, if  = max(, )).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 := || T − ||.Then from (19) it follows that (The simple proof is similar to the one given in [20]).

About (𝑙−𝜇−√𝑛)-Robustness of Sample Medians.
Let again  be the metric defined in (16).We show that the sequence of sample medians   ,  ≥ 1, in general, is not ( −  − √)-robust even when  and X have strictly positive, bounded, smooth densities symmetric with respect to the origin, and the sequences of sample medians   ,  ≥ 1, M ,  ≥ 1, are √-asymptotically normal.We consider a modified version of the corresponding example from [20].
We have obtained that the left-hand side of ( 12) is positive for all small enough  > 0. On the other hand, (, X ) → 0 as  → 0. It follows from the inequality (valued when  = ; see [21, page 376]) and from (10).
Remark 9.The densities as in (24) represent the following somewhat strange type of "contamination." Since max    () =   (0) sample points from   tend to concentrate around the origin.But  X (0) → 0 as  → 0, and therefore sample points from  X frequently in some extent are separated from 0.
If  = M then under certain conditions the closeness in μ guarantees the closeness of normal densities which are limiting for {√  ,  ≥ 1} and for {√ M ,  ≥ 1}, respectively.To attempt proving ( − μ − √)-robustness of   ,  ≥ 1 (as in (12)) one can show Hampel's qualitative robustness of   ,  ≥ 1 with respect to the metric  and then use the property √(  , M ) = (√  , √ M ).A not clear point of this plan is finding conditions under which |√(  − )| → 0 as  → ∞.
Example 10.Let us give another (very simple) example of the sequence of estimators which is ( −  − √)-robust with  :=  (on the class of distributions described below).For  = R we consider the class  of all random variables  with bounded supports Supp() = [0,   ], having density   such that   () ≥  > 0,  ∈ [0,   ].We suppose that   ≤  * < ∞,  ∈ , and  is the same for all  ∈ .Assume that parameter  =   is unknown, and the sequence of estimators   := max 1≤≤   is used to estimate it.
By elementary calculations we bound the right-hand side of (31) by From this relationship it follows that for , X ∈ , θ →  as  (, X) → 0.

𝑏∈R 𝑘 𝜁 2 (
, X + )} .(16) 3.1.(−−√)-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) || +2 < ∞, and the covariance matrix M of  is positive definite.The distribution L of  has a density  such that for some  ≥ 1: