# Note on Qualitative Robustness of Multivariate Sample Mean and Median

Authors: Evgueni Gordienko; Andrey Novikov; J. Ruiz de Chávez
Journal: Journal of Probability and Statistics
 (2013)

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## 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 then-asymptotic normality of estimators in Rk, and examine such
robustness of two standard location estimators in ℝk. 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.


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## Body

## 1. Introduction

The following Hampel’s definition (originally given for the one-dimensional case) ofqualitative 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 sequenceQn, 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
(1)supn≥1π(Qn(X1,…,Xn),Qn(X~1,…,X~n))<ϵ.Here, and throughout,π denotes the Prokhorov metric, and X1,X2,…; X~1,X~2,…, are i.i.d. random vectors distributed, respectively, as ℒ and ℒ~.For a metricμ 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 then-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:
(2)supn≥1nl(Qn(X1,…,Xn),Qn(X~1,…,X~n)+γ)≤ε,
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 metricsl and μ (on the space of distributions in ℝk) for which the following holds: (i)
whenQn=Tn,n≥1, are multivariate sample means, the left-hand side of inequality (2) (with some γ) is bounded by const ·μ(ℒ,ℒ~);(ii)
whenQn=Mn, 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 metricsl 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,…,Xn,… and X~;X~1,…,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(3)E|X|<∞,E|X~|<∞,
the meansEX and EX~ are defined as the corresponding Bochner integrals.The definition of themedian of X is less standard (consult, e.g., [15] for different definitions in H=ℝk). We will use the definition of median given in general setting in [16] (sometimes called 𝕃1-median):
(4)M:=argminx∈HE(|X-x|-|x|),(5)M~:=argminx∈HE(|X~-x|-|x|).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,…):
(6)Sn:=X1+⋯+Xn,S~n:=X~1+⋯+X~n,Tn=Snn,T~n:=S~nn(samplemeans).On the other hand, let (n=1,2,…),
(7)Mn=Mn(X1,…,Xn),M~n=M~n(X~1,…,X~n)
be sample medians defined by (4) and (5) replacing ℒ and ℒ~  by the corresponding empirical distributions ℒn and ℒ~n (obtained from X1,…,Xn and X~1,…,X~n, resp.). Robustness (in terms of ϵ-contamination neighborhoods) and asymptotic normality of Mn, n≥1, were proved, for instance, in [17] (see also [18]). The qualitative robustness properties of sample means {Tn} and sample medians {Mn} when the Prokhorov metric is used were discussed in Introduction.Let us first see what happens with qualitative robustness of{Tn} and {Mn} if in the above definition we replace π with the Kantorovich metric:
(8)l(X,X~)≡l(ℒ,ℒ~):=supϕ∈Lip|Eϕ(X)-Eϕ(X~)|,
where
(9)Lip:={ϕ:H↦ℝ:ϕisboundedand|ϕ(x)-ϕ(y)|≤|x-y|;x,y∈H}.Under condition (3)l(X,X~)<∞, and it is well known (see, e.g., [13]) that l(Yn,Y)→0 if and only if π(Yn,Y)→0 and E|Yn|→E|Y|.ForH=ℝ,
(10)l(X,X~)=∫-∞∞‍|FX(x)-FX~(x)|dx.Now, applying the regularity properties ofl (see, e.g., [13]):
(11)l(Snn,S~nn)≤1n∑i=1n‍l(Xi,X~i)=l(ℒ,ℒ~),
we see that the sequence of sample means {Tn,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 {Mn,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 thatH=ℝk 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 Theorem5 below we will use in the definitions of Kantorovich’s and Zolotarev’s metrics (see below) the following norm ∥x∥:=∑i=1k‍|xi| in the space ℝk.Thus, in what follows the Kantorovich metricl is defined by relationships (8), (9), where in (9) in place of the norm |·| the norm ||·|| is used.Letμ be some fixed simple probability metric on the set of all probability distributions on (ℝk,ℬ(ℝk)).We will consider a sequenceQn, n≥1, of estimators of some parameter θ∈ℝk of the distribution ℒ (θ~ of the distribution ℒ~, resp.).Definition 1.
We say that a sequenceQn, n≥1, is (l-μ-n)-robust at ℒ if there is some fixed vector γ=γ(ℒ,ℒ~)∈ℝk such that for every ϵ>0 there exists δ>0 such that μ(ℒ,ℒ~)≤δ entails
(12)supn≥1nl(Qn(X1,…,Xn),Qn(X~1,…,X~n)+γ)≤ϵ.Remark 2.
Taking into account thatl(aX+b,aY+b)=al(X,Y), a≥0,b∈ℝk, we see that (12) can be related with the n-asymptotic normality of estimators Qn, 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 2ζ2 is defined as follows (see, e.g., [14, 21]):
(13)ζ2(X,X~)≡ζ2(ℒ,ℒ~):=supϕ∈𝒟2|Eϕ(X)-Eϕ(X~)|,
where
(14)𝒟2:={ϕ:ℝk↦ℝ:∥Dϕ(x)-Dϕ(y)∥≤∥x-y∥;cx,y∈ℝk},
and Dϕ=(D1ϕ,…,Dkϕ) is the gradient of ϕ.Remark 3.
The distanceζ2(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<∞, EX~2<∞, then ζ2(X,X~)<∞ (see, e.g., [14]). The function ϕ(x):=(1/2)|x|2∈𝒟2 in (14). Therefore, by (13),
(15)|EX2-EX~2|≤ζ2(X,X~).
Thus, if EX2<∞ and ζ2(X,X~)<∞ then EX~2<∞. Consequently, if EX=EX~ and E|X|2<∞, then ζ2(X,X~)<∞ if and only if E|X~|2<∞.We now define the metricμ to work with:
(16)μ(X,X~):=max{2l(X,X~),infb∈ℝkζ2(X,X~+b)}.


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

To prove the inequality in Theorem5 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 distributionℒ of X has a density f such that for some s≥1:
(ii) the densityfs of Ss:=X1+⋯+Xs is bounded and differentiable;
(iii) the gradientDfs is bounded and |Dfs| belongs to 𝕃1(ℝk);
(iv) for someα>0(17)∫|x|>αn‍|Dfs|dx=O(n-1/2),asn→∞.Note that in view of Remark3 under the condition (i) μ(X,X~)<∞ if and only if
(18)E|X~|2<∞,
(see (15)).Theorem 5.
Under Assumption4 and supposing (18) it holds
(19)supn≥1nl(Tn,T~n+γ)≤cμ(ℒ,ℒ~),
where γ=EX-EX~,
(20)c=max{(10s-1)1/2,5.4dk},(21)d=supn≥smax1≤i≤k∥Dign∥𝕃1(ℝk)<∞,
and gn is the density of (1/n)(X1+⋯+Xn), n≥1.Remark 6.
(i) The constantc 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 Dign, n≥s, converges in 𝕃1-norm to the corresponding partial derivative of the limit normal density with covariance matrix ℳ (and zero mean since ||Dign||𝕃1(ℝk) is invariant under translations).
For example, letk=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
(22)c<max{3,10.8max{0.7065λ′,0.5414λ′′}}.
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 assumptionμ(ℒ,ℒ~)<∞ entails (18), inequality (19) ensures (l-μ-n)-robustness of the sequence of sample meansTn, n≥1.
(iii) Fork=1, in [20] an example is given showing that in general the sequence of sample means Tn, n≥1 is not (l-l-n)-robust (even if (18) holds and EX=EX~). It is also almost evident that the sample means Tn, 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 thatθ=θ~ and that one evaluates the quality of estimators by mean of absolute errors: δn:=E||Tn-θ||, δ~n:=E||T~n-θ||. Then from (19) it follows that
(23)|δn-δ~n|≤n-1/2constμ(ℒ,ℒ~),n≥1.(The simple proof is similar to the one given in [20]).


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

Let againμ be the metric defined in (16). We show that the sequence of sample medians Mn, 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 Mn, n≥1, M~n, n≥1, are n-asymptotically normal. We consider a modified version of the corresponding example from [20].Example 8.
LetH=ℝ, X~N(0,1), and for ϵ∈(0,1) let X~ϵ be a random variable with the density:
(24)fX~ϵ(x):=1c(ϵ)(ϵ+x2ϵ3+x2)e-x2/2,x∈ℝ,
where c(ϵ) is a normalizing constant. By symmetry of the density, we get
(25)M=M~=EX=EX~=0,
and also it is clear that fX,fX~∈ℂ∞(ℝ), and
(26)supϵ∈(0,1),x∈ℝfX~ϵ(x)<∞.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 Mn→M=0, M~n→M~=0 as n→∞ with probability 1. Also, from the results of [16] we can obtain that |Mn|≤(2/n)∑i=1n‍|Xi|. Using this inequality it is easy to show that the sequence |Mn|, n≥1, is uniformly integrable. Therefore, E|Mn|→0 (and also E|M~n|→0) as n→∞, and for this l(Mn,M~n+γ)→l(0,γ)=γ.Let nowγ=0 in (12). By the well-known asymptotic normality (see [22, page 307]),
(27)π(nMn,η)→0,π(nM~n,ηϵ)→0,
where
(28)η~N(0,12fX(0))=N(0,π2),ηϵ~N(0,12fX~ϵ(0))=N(0,c(ϵ)ϵ).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(nMn,nM~n)≥r. Consequently, for all such ϵ and n,
(29)nl(Mn,M~n)=l(nMn,nM~n)≥r2>0,
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
(30)ζ2(X,Y)≤12∫-∞∞‍x2|fX(x)-fY(x)|dx
(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 maxxfX(x)=fX(0) sample points from fX 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 metricμ~ to ensure (l-μ~-n)-robustness of the sequence of sample medians Mn, n≥1?” Our conjecture is (for k=1, e.g.) to try μ~(X,X~)=max{l(X,X~),esssupx∈ℝ|fX(x)-fX~(x)|} (supposing the existence of densities).IfM=M~ then under certain conditions the closeness in μ~ guarantees the closeness of normal densities which are limiting for {nMn,n≥1} and for {nM~n,n≥1}, respectively. To attempt proving (l-μ~-n)-robustness of Mn, n≥1 (as in (12)) one can show Hampel’s qualitative robustness of Mn, n≥1 with respect to the metric l and then use the property nl(Mn,M~n)=l(nMn,nM~n). A not clear point of this plan is finding conditions under which E|n(Mn-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,θX], having density fX such that fX(x)≥β>0, x∈[0,θX]. We suppose that θX≤θ*<∞, X∈W, and β is the same for all X∈W. Assume that parameter θ=θX is unknown, and the sequence of estimators Qn:=max1≤i≤nXi is used to estimate it.
Denotingθ=θX, θ~=θ~X and choosing in (12) γ=θ-θ~, we get
(31)l(Qn,Q~n+γ)=l(θ-max1≤i≤nXi,θ~-max1≤i≤nX~i)≤E|(θ-max1≤i≤nXi)-(θ~-max1≤i≤nX~i)|≤E(θ-max1≤i≤nXi)+E(θ~-max1≤i≤nX~i),
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
(32)2β(n+1).
Now for eachn fixed, by induction we obtain (see (10))
(33)l(max1≤i≤nXi,max1≤i≤nX~i)=∫0∞‍|FXn(x)-FX~n(x)|dx≤n∫0∞‍|FX(x)-FX~(x)|dx=nl(X,X~).
On the other hand, let, for example,θ≤θ~. Then
(34)l(X,X~)=∫0θ‍|FX(x)-FX~(x)|dx+∫θθ~‍(1-FX~(x))dx.
But for x∈[θ,θ~],
(35)1-FX~(x)=∫xθ~‍fX~(y)dy≥β(θ~-x).
From this relationship it follows that
(36)forX,X~∈W,θ~→θasl(X,X~)→0.
Finally, applying (32), we can select N in such a way that
(37)supn≥Nnl(Qn,Q~n+γ)≤ϵ2.
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 Qn on the set W.


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

To prove the inequality in Theorem5 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 distributionℒ of X has a density f such that for some s≥1:
(ii) the densityfs of Ss:=X1+⋯+Xs is bounded and differentiable;
(iii) the gradientDfs is bounded and |Dfs| belongs to 𝕃1(ℝk);
(iv) for someα>0(17)∫|x|>αn‍|Dfs|dx=O(n-1/2),asn→∞.Note that in view of Remark3 under the condition (i) μ(X,X~)<∞ if and only if
(18)E|X~|2<∞,
(see (15)).Theorem 5.
Under Assumption4 and supposing (18) it holds
(19)supn≥1nl(Tn,T~n+γ)≤cμ(ℒ,ℒ~),
where γ=EX-EX~,
(20)c=max{(10s-1)1/2,5.4dk},(21)d=supn≥smax1≤i≤k∥Dign∥𝕃1(ℝk)<∞,
and gn is the density of (1/n)(X1+⋯+Xn), n≥1.Remark 6.
(i) The constantc 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 Dign, n≥s, converges in 𝕃1-norm to the corresponding partial derivative of the limit normal density with covariance matrix ℳ (and zero mean since ||Dign||𝕃1(ℝk) is invariant under translations).
For example, letk=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
(22)c<max{3,10.8max{0.7065λ′,0.5414λ′′}}.
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 assumptionμ(ℒ,ℒ~)<∞ entails (18), inequality (19) ensures (l-μ-n)-robustness of the sequence of sample meansTn, n≥1.
(iii) Fork=1, in [20] an example is given showing that in general the sequence of sample means Tn, n≥1 is not (l-l-n)-robust (even if (18) holds and EX=EX~). It is also almost evident that the sample means Tn, 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 thatθ=θ~ and that one evaluates the quality of estimators by mean of absolute errors: δn:=E||Tn-θ||, δ~n:=E||T~n-θ||. Then from (19) it follows that
(23)|δn-δ~n|≤n-1/2constμ(ℒ,ℒ~),n≥1.(The simple proof is similar to the one given in [20]).


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

Let againμ be the metric defined in (16). We show that the sequence of sample medians Mn, 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 Mn, n≥1, M~n, n≥1, are n-asymptotically normal. We consider a modified version of the corresponding example from [20].Example 8.
LetH=ℝ, X~N(0,1), and for ϵ∈(0,1) let X~ϵ be a random variable with the density:
(24)fX~ϵ(x):=1c(ϵ)(ϵ+x2ϵ3+x2)e-x2/2,x∈ℝ,
where c(ϵ) is a normalizing constant. By symmetry of the density, we get
(25)M=M~=EX=EX~=0,
and also it is clear that fX,fX~∈ℂ∞(ℝ), and
(26)supϵ∈(0,1),x∈ℝfX~ϵ(x)<∞.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 Mn→M=0, M~n→M~=0 as n→∞ with probability 1. Also, from the results of [16] we can obtain that |Mn|≤(2/n)∑i=1n‍|Xi|. Using this inequality it is easy to show that the sequence |Mn|, n≥1, is uniformly integrable. Therefore, E|Mn|→0 (and also E|M~n|→0) as n→∞, and for this l(Mn,M~n+γ)→l(0,γ)=γ.Let nowγ=0 in (12). By the well-known asymptotic normality (see [22, page 307]),
(27)π(nMn,η)→0,π(nM~n,ηϵ)→0,
where
(28)η~N(0,12fX(0))=N(0,π2),ηϵ~N(0,12fX~ϵ(0))=N(0,c(ϵ)ϵ).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(nMn,nM~n)≥r. Consequently, for all such ϵ and n,
(29)nl(Mn,M~n)=l(nMn,nM~n)≥r2>0,
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
(30)ζ2(X,Y)≤12∫-∞∞‍x2|fX(x)-fY(x)|dx
(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 maxxfX(x)=fX(0) sample points from fX 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 metricμ~ to ensure (l-μ~-n)-robustness of the sequence of sample medians Mn, n≥1?” Our conjecture is (for k=1, e.g.) to try μ~(X,X~)=max{l(X,X~),esssupx∈ℝ|fX(x)-fX~(x)|} (supposing the existence of densities).IfM=M~ then under certain conditions the closeness in μ~ guarantees the closeness of normal densities which are limiting for {nMn,n≥1} and for {nM~n,n≥1}, respectively. To attempt proving (l-μ~-n)-robustness of Mn, n≥1 (as in (12)) one can show Hampel’s qualitative robustness of Mn, n≥1 with respect to the metric l and then use the property nl(Mn,M~n)=l(nMn,nM~n). A not clear point of this plan is finding conditions under which E|n(Mn-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,θX], having density fX such that fX(x)≥β>0, x∈[0,θX]. We suppose that θX≤θ*<∞, X∈W, and β is the same for all X∈W. Assume that parameter θ=θX is unknown, and the sequence of estimators Qn:=max1≤i≤nXi is used to estimate it.
Denotingθ=θX, θ~=θ~X and choosing in (12) γ=θ-θ~, we get
(31)l(Qn,Q~n+γ)=l(θ-max1≤i≤nXi,θ~-max1≤i≤nX~i)≤E|(θ-max1≤i≤nXi)-(θ~-max1≤i≤nX~i)|≤E(θ-max1≤i≤nXi)+E(θ~-max1≤i≤nX~i),
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
(32)2β(n+1).
Now for eachn fixed, by induction we obtain (see (10))
(33)l(max1≤i≤nXi,max1≤i≤nX~i)=∫0∞‍|FXn(x)-FX~n(x)|dx≤n∫0∞‍|FX(x)-FX~(x)|dx=nl(X,X~).
On the other hand, let, for example,θ≤θ~. Then
(34)l(X,X~)=∫0θ‍|FX(x)-FX~(x)|dx+∫θθ~‍(1-FX~(x))dx.
But for x∈[θ,θ~],
(35)1-FX~(x)=∫xθ~‍fX~(y)dy≥β(θ~-x).
From this relationship it follows that
(36)forX,X~∈W,θ~→θasl(X,X~)→0.
Finally, applying (32), we can select N in such a way that
(37)supn≥Nnl(Qn,Q~n+γ)≤ϵ2.
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 Qn on the set W.



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*Source: 208950-2013-02-17.xml*