Global robustness of location and dispersion estimates

Statistics & Probability Letters 44 (1999) 63 – 72

Global robustness of location and dispersion estimates Jose R. Berrenderoa , Ruben H. Zamar b; ∗;1 a Departamento

b Department

de MatemÃaticas, Universidad Europea de Madrid, Spain of Statistics, University of British Columbia, Room 333, 6356 Agricultural Road, Vancouver, BC, Canada V6T 1Z2 Received May 1998; received in revised form October 1998

Abstract We analyze the global robustness of location and dispersion estimates using the concept of relative explosion rate. The merits of several dispersion estimates are compared when the dispersion parameter itself is of main interest and also when they are auxiliary estimates needed to de ne scale equivariant location M-estimates. We have also compared location M-estimates and found that the choice of score function (its shape) is of secondary importance in comparison with the choice of the tuning constant and the auxiliary dispersion estimate. Finally, we use the explosion rate to assess the combined eect of the tuning constant and auxiliary dispersion estimate on the global robustness properties of location c 1999 Elsevier Science B.V. All rights reserved M-estimates. MSC: 62F35 Keywords: Breakdown point; Explosion rate; Bias robustness; Maxbias curve; M-estimates; Location; Scale

1. Introduction The maxbias function, BT (), of an estimate T measures its maximum asymptotic bias when the data includes up to a fraction  of contamination. Therefore, the maxbias provides an approximate upper bound for the nite sample estimation bias. The maxbias of a location estimate was rst introduced by Huber (1964, 1981). Other works where the maxbias of several scale and regression estimates is derived include Martin et al. (1989), Martin and Zamar (1993b), Rousseeuw and Croux (1993, 1994), Croux et al. (1994), Berrendero and Zamar (1995) and Croux et al. (1996). It is known that typical maxbias functions are continuous and increasing from zero to in nity and the point at which BT () diverges is called the breakdown point (BP) of T . An estimate whose maxbias grows slowly near zero is called locally robust and an estimate whose maxbias is relatively small for large fractions of contaminations is called globally robust. Hampel (1974) studied the ∗ 1

Corresponding author. Tel.: +1-604-822-3410; fax: +1-604-822-6960; e-mail: [email protected]. Research partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

c 1999 Elsevier Science B.V. All rights reserved 0167-7152/99/$ – see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 9 8 ) 0 0 2 9 2 - 2

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rate of growth of the maxbias when the fraction of contamination is small and introduced the concept of gross-error sensitivity (GES) to quantify this behavior GES(T ) = lim BT ()= = BT0 (0): →0

On the other hand, Berrendero et al. (1998) studied the rate of growth of the maxbias when the fraction of contamination is large and introduced the concept of relative explosion rate (RER) to quantify this behavior. Given two estimates T1 and T2 with the same BP, ∗ , their RER is given by r(T1 ; T2 ) = lim∗ →

BT1 () ; BT2 ()

provided that the limit exists. Using the RER we will study the global robustness properties of location M-estimates Tn de ned by Huber (1964) as the solution to n X

[(Xi − Tn )=Dn ] = 0;

(1)

i=1

where Dn is an auxiliary robust dispersion estimate and the function is non-decreasing and odd. A tuning constant A is normally used (i.e. (x) = 0 (x=A)) to tune the estimate eciency. If 0 is linear near zero and bounded, larger values of A yield estimates closer to the sample mean and hence more ecient. On the other hand, smaller values of A give estimates which are closer to the sample median. Can we achieve arbitrarily high eciency by raising the tuning constant A without suering a loss of global robustness? Unfortunately, no. When A is large the maxbias increases very fast. On the other hand, it is well known that if the dispersion Dn has BP = 12 and the score function 0 is bounded, the corresponding location estimate has also BP = 12 . Therefore, such location estimate may have very large (but nite) bias for fractions of contamination smaller than 21 . For example, the maxbias curve can be very large for values of ¿0:25. These estimates would have, from a practical point of view, BP smaller than 0.25 (despite their formal 0.5 BP). Therefore, the BP is not a very informative summary measure to compare the global robustness properties of dierent estimates. This can be remedied by computing the BP together with the RER. It is well known (see, for instance, Hampel et al., 1986) that the eciency and GES of location M-estimates are not aected by the eciency and GES of the auxiliary dispersion estimate Dn , see (1). Therefore, Dn can be chosen on the basis of global robustness considerations alone. We will show that the RER can be used to precisely quantify the impact of Dn on the global robustness properties of the ecient location estimate. Based on these considerations we will be able to recommend an optimal choice for Dn , namely the SHORTH, to be de ned in Section 3. There are cases, however, when the estimation of dispersion is of main interest and the estimate eciency must also be taken into account. Several dispersion estimates which are robust and ecient have been proposed in the literature (see Section 3). We will show that they can be nicely compared using the concept of RER. Based on these considerations we will be able to recommend the use of a globally robust and ecient Dn , namely the dispersion -estimate, to be de ned in Section 3. The rest of the paper is organized as follows. Section 2 gives some de nitions and notations used later on. Section 3 is devoted to the study of the global robustness of dispersion estimates. The global robustness of location M-estimates is studied in Section 4. The known dispersion case is brie y studied in Section 5. Section 6 gives some conclusions and nal remarks. All the proofs and technical details are contained in Appendix A.

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65

2. Some deÿnitions and notations We will consider a parametric location and dispersion model, F0 [(x − )=], where F0 is a speci ed distribution, e.g. N(0,1). To allow deviations from this model we assume that the observations X1 ; : : : ; Xn are i.i.d. with common distribution F in the contamination neighborhood F = {F: F(x) = (1 − )F0 [(x − )=] + H (x)}; where 0 ¡  ¡ 12 , and H is an arbitrary and unknown distribution. For technical reasons, we will consider the more general de nition of location M-estimates ( ) n X [(Xi − t)=Dn ] ¡ 0 ; Tn = inf t :

(2)

i=1

which reduces to (1) when is continuous. The maxbias of Tn is given by BT () = sup |T (F) − |=; F∈F

where T (F) is the asymptotic value of Tn . Equivariance considerations allow us to assume, without loss of generality, that  = 0 and  = 1. Therefore, BT () = supF∈F |T (F)|. The breakdown point of Tn is given by ∗ = sup{ : BT () ¡ ∞}. Let Tn be a location estimate with maxbias BT () and breakdown point 12 , and let B0 () be the maxbias curve of the median. The explosion rate (ER) of Tn is de ned as r(T ) = lim

→1=2

BT () : B0 ()

Therefore, the ER is equal to the RER between Tn and the median, with the median taken as a benchmark. In view of the maxbias properties of the median, see Huber (1964), r(T )¿1 for any location equivariant estimate. Let D(F) be the asymptotic value of the dispersion estimate Dn under F and let D() = sup D(F)=

(3)

F∈V

be the corresponding maximum (explosion) bias. By equivariance considerations we can assume, without loss of generality, that =1. We will only consider dispersion estimates with BP= 12 , that is ∗ =sup{: D() ¡ ∞}= 1 2 . Given two dispersion estimates D1 and D2 their RER is given by D1 () : →1=2 D2 ()

r(D1 ; D2 ) = lim

3. Explosion rates of dispersion estimates The most commonly used auxiliary dispersion estimate is the rather inecient median absolute deviations with respect to the median MADM de ned as Mn = c med i {|xi − med j xj |};

where c = 1=F0−1 (0:75):

Notice that the MADM itself uses a location estimate, namely the median, to center the data. Another possibility studied by Rousseeuw and Leroy (1987) is to use the length of the shortest half, SHORTH, Hn = med i {|xi − Tˆ | : i = 1; : : : ; n};

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where the data are centered by the value Tˆ that minimizes m(t) = med i {|xi − t|: i = 1; : : : ; n}. Martin and Zamar (1993 b) found that the SHORTH minimizes maxbias (3) for a large class of dispersion estimates. Moreover, Grubel (1988) showed that the SHORTH has the same (low) eciency as the MADM. There are several proposals for de ning robust and ecient dispersion estimates. Yohai and Zamar (1988) proposed the dispersion -estimate (F) = inf t (F; t), where   1 2 X −t 2  (F; t) = S (F; t)EF  b S(F; t) and EF0 (X ) = b. For each t; S(F; t) is a scale M-estimate of X − t, that is, it satis es 1 2

EF [(X − t)=S(F; t)] =

for each t;

= 12 . Finally, (Fn ), where

 and  are two appropriate score functions. The corresponding nite sample where EF0 (X ) Fn is the empirical distribution function of the sample. Rousseeuw and Croux version is n = (1994) studied the properties of -estimates when the location is known. Two other ecient alternatives to the MADM have been recently proposed by Rousseeuw and Croux (1993) Sn = k med i {med j |Xi − Xj |} and Qn = d {|Xi − Xj |: i ¡ j}(p) ; where k and d are two scaling constants which depend on F0 and the quantile p is chosen to get BP = 12 . The estimate Sn is related to nested scale regression estimators (see Croux et al., 1996). The estimate Qn is related to generalized S-estimators (see Croux et al., 1994). The ER of an arbitrary dispersion estimate, Dn is de ned as the RER of Dn and the SHORTH, Hn , that is r(Dn ) = lim [D()=H ()]: →1=2

The next theorem gives the ER for Mn ; n ; Sn and Qn . Theorem 1. Assume that F0 satisÿes A1. (a) F0 is strictly increasing √ and has an even density f0 ; and (b) limx→∞ f0 (x)=[1 − F0 (x)] ¿ 0. Then, √ r(Mn ) = 2; r(n ) = 1= 2b and r(Sn ) = 2k=c. Furthermore; if F0 is the standard normal distribution, r(Qn ) = 2d=c. Table 1 gives the ER, under the Gaussian location-dispersion model, for Hn ; Sn ; Mn and Qn as de ned before, as well as the -estimate n with a 0 –1 -function (i.e. (x) = I{|x|¿c1 }) and Huber -function (i.e. (x) = min{x2 =c22 ; 1}) with c1 = 0:674 and c2 = 2:015, to achieve BP = 12 and 82.27% eciency (to match Qn eciency). Table 1 Eciency, gross-error sensitivity and explosion rate for several robust dispersion estimates Estimate

EFF (%)

GES

ER

Hn n Sn Mn Qn

36.74 82.27 58.23 36.74 82.27

1.17 1.92 1.63 1.17 2.07

1.00 1.48 1.60 2.00 2.12

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67

If the dispersion parameter is of main interest only the more ecient estimates, n and Qn , are worth comparing. Of these two, n should be preferred because of its lower ER and GES. Notice that while the SHORTH and MADM have the same eciency and GES, the SHORTH is 2 times better regarding explosion rates (independently from F0 ). We will see in the next section that these results are also relevant regarding the choice of auxiliary dispersion estimates. 4. Explosion rates of location M-estimates: unknown dispersion In this section we assume that  is a nuisance dispersion parameter which needs to be estimated to de ne a scale equivariant location M-estimate (see (1)). We wish to quantify the eect (on the global robustness properties of the nal location M-estimates) of dierent choices of tuning constants and auxiliary dispersion estimates. The following theorem gives lower and upper bounds for the ER of a location M-estimate with score function A and dispersion estimate Dn . Theorem 2. Let Tn be the location estimate deÿned by (2) with score function A and dispersion estimate Dn with BP = 12 . Assume that A1 from Theorem 1 holds and A satisÿes A2. A (x) = 0 (x=A) is odd; strictly increasing for |x| ¡ A; bounded with A (x) = 1 for x¿A; and continuous (except; perhaps; at a ÿnite number of points). Then, (a) max{1; cr(Dn )A}6r(Tn )61+cr(Dn )A; where c=1=F0−1 (0:75). (b) When the condition A (x)=1 for x¿A is removed and A is strictly increasing, r(Tn ) = ∞. From part (a), the ER bounds are linear functions of the tuning constant A. Their common slope is given by the product cr(Dn ) which depends on F0 and the dispersion estimate Dn . The dierent behavior of r(Tn ) in parts (a) and (b) of Theorem 2 also deserves further comments. M-estimates of location with monotone score functions can be viewed as the minimizers of a convex loss function. Ultimately constant score functions are related to ultimately linear loss functions, and the corresponding ER bounds increase linearly with the constant A. On the other hand, strictly increasing score functions are associated with slowly but steadily increasing loss functions. In this case, the ER = ∞. Therefore, we have established that globally robust location M-estimates must have ultimately linear loss functions. Table 2 gives the ER bounds for location M-estimates with Huber score function 0 (x)

= min{max{−1; x}; 1}

and several choices of dispersion estimates Dn and tuning constants A. The corresponding Gaussian eciencies and GESs are also displayed. Since all the estimates on the same row of Table 2 have the same eciency, GES and BP, the only summary measure able to distinguish them is the ER. Given our bounds for the ER, we can only reach a

Table 2 Lower and upper bounds for the explosion rates of Huber’s location M-estimates for several eciencies (tuning constants) and auxiliary dispersion estimates A

EFF (%)

GES

Hn

n

Sn

Mn

Qn

0.98 1.34 2.01

90 95 99

1.46 1.63 2.10

1.45 –2.45 1.99 –2.99 2.98–3.98

2.15 –3.15 2.94 –3.94 4.41–5.41

2.34 –3.34 3.20 – 4.20 4.80 –5.80

2.91–3.91 3.97– 4.97 5.96 – 6.96

3.08– 4.08 4.21–5.21 6.32–7.32

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conclusion when the ER upper bound for an estimate is smaller than the ER lower bound for another. In this regard, the SHORTH is the clear winner on the last row (99% eciency). In the 95% eciency case (second row) the SHORTH clearly wins except against the  where there is a small range overlap. For 90% eciency ( rst row) the comparison of SHORTH,  and Sn does not allow for any conclusions. We can only conclude that the SHORTH is preferred to the MADM and Qn . An important point regarding the ER bounds displayed in Table 2 is that they do not depend on the particular shape of 0 , and therefore hold for any score function with the same truncation point. 5. Explosion rates of location M-estimates: known dispersion To assess (by comparison) the impact of estimating the unknown dispersion parameter, we consider the rather unrealistic case when the dispersion parameter is known. In this case we have the following surprising result: Theorem 3. Let B0 () be the maxbias of the median and let BT () be the maxbias of the location M-estimate with score function A . Assume that A1(a) from Theorem 1 and A2 from Theorem 2 hold. Then, BT ()6B0 () + A

for 0 ¡  ¡ 12 :

Therefore, when the dispersion is known, the maxbias behavior of location M-estimates cannot be very dierent from that of the median. An immediate consequence of Theorem 3 is that, when the dispersion is known, r(Tn ) = 1. The dierence between the known and unknown dispersion cases is now apparent: constant versus linearly increasing explosion rates. 6. Conclusions We have shown that the RER and ER are eective analytic tools to compare location and dispersion estimates. When comparing dispersion estimates we must distinguish two cases: (i) dispersion is of main interest; and (ii) dispersion is a nuisance parameter. In the rst case, we have shown that a particular  estimate is better than other estimates which have been recently de ned to achieve high breakdown point and eciency. In the second case, we have shown that the SHORTH is a very attractive choice, twice better than the popular MADM, in terms of explosion rate. An interesting nding, regarding the de nition of location M-estimates, is that the choice of the shape of the score function is of secondary importance in comparison with the other choices that one must face: the tuning constant and the auxiliary dispersion estimate. The only important feature regarding the shape of the score function is that it must be constant beyond a certain point. Appendix A. Proofs The following lemma gives the maxbias of dispersion -estimates. Compare with the identical result obtained by Rousseeuw and Croux (1994) in the case of known location. Lemma A.1. Assume that A1(a) from Theorem 1 holds. Assume that the score functions  and  are; even; bounded; non-decreasing for positive arguments; continuous except in at most a ÿnite number of points; and

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69

satisfy (0) = (0) = 0 and (∞) = (∞) = 1. Assume further that s → s2 EF0 (X=s) is non-decreasing for s¿0. Then,  1=2    1 X (1 − )EF0  + ; ()  = BS () b BS () where BS () is the (explosion) maxbias of the scale M-estimate with score function . Proof. Let F ∞ = (1 − )F0 + ∞ be the distribution obtained by placing a point mass contamination at in nity. Then, ()  = sup (F)¿(F ∞ ) = inf (F ∞ ; t): t

F∈F

(A.1)

Furthermore, inf (F ∞ ; t) = (F ∞ ; 0) = sup (F; 0): t

F∈F

(A.2)

The rst equality in (A.2) follows since, under the assumptions of the lemma, S(F∞ ; t)¿S(F∞ ; 0), for each t, and using that s → s2 EF0 (X=s) is non-decreasing. The second equality follows since, by Theorem 4 of Martin and Zamar (1993 b), S(F; 0)6S(F ∞ ; 0), for each F ∈ F , and using again the monotonicity of s → s2 EF0 (X=s). Now, it is straightforward that  sup (F; 0)¿ sup inf (F; t) = ():

F∈F

F∈F

t

(A.3)

Putting together Eqs. (A.1) – (A.3), it follows that  1=2    1 X (1 − )EF0  + : ()  = (F ∞ ; 0) = BS () b BS () The following lemma is needed to prove the main results. We use it to compare H () = sup H (F) and F M () = sup M (F) with the maxbias function of the median, B0 (). F

Lemma A.2. Assume that A1 from Theorem 1 holds. Then, (a) lim→1=2 [H ()=B0 ()] = c and (b) lim→1=2 [M ()=B0 ()] = 2c, where c = 1=F0−1 (0:75). Proof. (a) From Huber (1981), p. 74, and Martin and Zamar (1993b), Lemma A.1, we have that F −1 [(3 − 2)=(4(1 − ))] H () : = c lim 0 −1 →1=2 B0 () →1=2 F0 [1=(2(1 − ))] lim

De ne G1 () = F0−1 [(3 − 2)=(4(1 − ))] and G2 () = F0−1 [1=(2(1 − ))]. By the Mean Value Theorem and given that f0 is unimodal, it follows that 06

1 − 2 1 − F0 [G1 ()] G1 () − G2 () 6 = G2 () 4(1 − )G2 ()f0 [G1 ()] G2 ()f0 [G1 ()]

and applying A1(b), the last expression goes to zero as  goes to 12 . Therefore, lim→b G1 ()=G2 () = 1. This shows part (a). (b) Let a = 1=c. Following Martin and Zamar (1993b), Lemma A.2, M () satis es   1 − 2 X − B0 () ; where a (x) = I {|x|¿a}: = (A.4) EF0 a 2(1 − ) M ()

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After some calculations, it follows from (A.4) that F0 [B0 () + M ()=c] − F0 [B0 () − M ()=c] =

1 : 2(1 − )

(A.5)

Now de ne G1 () = 2cB0 () and G2 () = c[B0 () + H ()=c]: We will show that G1 () ¡ M () ¡ G2 ():

(A.6)

To prove the rst inequality, we replace G1 () by M () in (A.5): F0 [3B0 ()] − 1 + F0 [B0 ()] = F0 [3B0 ()] − 1 + 1=[2(1 − )] ¡ 1=[2(1 − )]: Since F0 [B0 () + ax] − F0 [B0 () − ax] is strictly increasing in x, it follows that G1 () ¡ M (). On the other hand, replacing G2 () with M () in Eq. (A.5) we obtain      3 − 2 3 − 2 1 3 − 2 − 1− ¿2 −1= F0 2B0 () + F0−1 4(1 − ) 4(1 − ) 4(1 − ) 2(1 − ) and, therefore, M () ¡ G2 (). From (A.6) and applying part (a) of this lemma, 2c = lim

→1=2

G1 () M () G2 () 6 lim 6 lim = 2c: B0 () →1=2 B0 () →1=2 B0 ()

√ Proof of Theorem 1. Note that r(Mn ) = 2 is obvious from Lemma A.2 and r(n ) = 1= 2b is obvious from Lemma A.1. Next, we prove that r(S) = 2k=c. In Rousseeuw and Croux (1993), Theorem 4, it is shown that F0 [H ()=c + S()=k] − F0 [H ()=c − S()=k] =

1 : 2(1 − )

De ne G1 () = 2kB0 () and G2 () = 2kH ()=c. Following a similar proof to that of Lemma A.2(b) it is not dicult to show that G1 () ¡ S() ¡ G2 (). Now, dividing by H (), taking limits when  goes to 12 and applying Lemma A.2, we obtain that r(Sn )=2k=c. Finally, assuming that F0 is the standard normal distribution, Theorem 7 in Rousseeuw and Croux (1993) implies that   5 − 8 + 42 1=2 −1 Q() = d2 F0 : 8(1 − )2 Now, de ne G1 () = (1=c)H () = F0−1 [(3 − 2)=(4(1 − ))] and G2 () = d−1 2−1=2 Q() and repeat the lines of the proof of Lemma A.2(a). It follows that lim→1=2 G1 ()=G2 () = 1 and, therefore, r(Qn ) = 21=2 d=c applying Lemma A.2. Proof of Theorem 2. To prove (a) and (b), we will use the following expression (due to Martin and Zamar (1993a), Lemma 1) for the maxbias function of a location M-estimate Tn with score function A : [=(1 − ); D()]; BT () = g−1 A where

Z g A (t; s) = −

∞

−∞

(A.7) Z

A [(x

− t)=s] dF0 (x) = s

(·; s) is the inverse function of g and g−1 A that 0 ¡ D() ¡ ∞.

A

0

∞

(x)[f0 (sx − t) − f0 (sx + t)] d x

(A.8)

with respect to its rst argument. This expression is valid provided

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71

(a) To prove the upper bound de ne the auxiliary score function a∗ (x) = sgn(x)I {|x|¿A} and let T ∗ be the corresponding location M-estimate. Then, g a∗ (t; s) = F0 (As + t) − F0 (As − t) (see Eq. (A.8)). Applying (A.7) and letting t = BT ∗ () and s = D(), it follows that  : (A.9) F0 [AD() + BT ∗ ()] − F0 [AD() − BT ∗ ()] = 1− De ne G1 () = AD() + F0−1 [=(1 − )] and G2 () = AD() + B0 () and repeat again the proof of Lemma 2(b) to prove that G1 () ¡ B A∗ () ¡ G2 (). Dividing by B0 (), taking limits as  → 12 and applying Lemma 2, we have that r(Tn∗ ) = 1 + cr(Dn )A. (Notice that, under assumption A1, lim→1=2 F0−1 [=(1 − )]=B0 () = 1.) Now, let A be any function satisfying A2. Then A (x)¿ A∗ (x) for all x¿0 and hence g a (t; s)¿g A∗ (t; s), for all s ¿ 0 and t ¿ 0. Applying (A.7) it follows that BT ()6BT ∗ () and, therefore, r(Tn )6r(Tn∗ ) = 1 + cr(Dn )A. Next, we will show the lower bound. De ne, for 0 ¡ l ¡ A, the following auxiliary score function: l (x)

= sgn(x)[ A (l)I {|x|6l} + I {|x| ¿ l}]

l

and let T be the corresponding location M-estimate. After some calculations, we have that g l (t; s) = [1 − (l)][F0 (sl + t) − F0 (sl − t)] + (l)[2F0 (t) − 1] for t ¿ 0 and s ¿ 0. Letting t = BT l () and s = D() and applying (A.7) we obtain that, for 0 ¡  ¡ 12 , =(1 − ) = [1 − (l)][F0 (lD() + BT l ()) − F0 (lD() − BT l ())] + (l)[2F0 (BT l ()) − 1]: Now, we take limits as  goes to 12 in both sides of this equation. Since the left-hand side converges to one, the right-hand side must also converge to one. This implies in particular that lim→1=2 [lD() − BT l ()] = −∞. Therefore, there exists 0 such that if 0 ¡  ¡ 12 , then BT l () ¿ lD(). By Lemma 2, it follows that r(Tnl )¿cr(Dn )l. Given that A (x)6 l (x), for x¿0 and 0¡l¡A, we have that BT ()¿BT l (), for 0 ¡  ¡ 12 and 0¡l¡A. Hence, r(Tn )¿r(Tnl )¿cr(Dn )l, for 0¡l¡A. This fact proves the lower bound and, therefore, part (a) of the theorem. Part (b) follows by noticing that, under the new assumptions, the proof of the lower bound is valid for all l¿0. Proof of Theorem 3. When the dispersion parameter is known, the maxbias curve of the R ∞location M-estimate with score function A is given, under A1 and A2, by g−1 [=(1 − )], where g (t) = − −∞ (x − t)f(x) d x = R∞ (x)[f0 (x − t) − f0 (x + t)] d x. De ne A∗ (x) = sgn(x)I {|x|¿A} and let T ∗ be the corresponding loca0 tion M-estimate. Given that A∗ (x)6 A (x), for x¿0, it follows that g A∗ (t)6g (t), for t¿0 and, therefore, BT ∗ ()¿BT (), for 0 ¡  ¡ 12 . Now, some manipulations show that g A∗ (t)=F0 (t −A)+F0 (t +A)−1 ¿ 2F0 (t − A)−1, for t ¿ 0. Letting t=BT ∗ () in this inequality and arranging terms, we have 1=[2(1−)] ¿ F0 [BT ∗ ()−A], for 0 ¡  ¡ 12 . Applying F0−1 to both sides of this inequality, we obtain B0 () ¿ BT ∗ () − A, for 0 ¡  ¡ 12 . Therefore, BT ()6BT ∗ () ¡ B0 () + A, for 0 ¡  ¡ 12 . References Berrendero, J.R., Mazzi, S., Romo, J., Zamar, R.H., 1998. On the explosion rate of maximum bias functions. Canad. J. Statist., to appear. Berrendero, J.R., Zamar, R., 1995. On the maxbias curve of residual admissible robust regression estimates. Working Paper 95 – 62. Universidad Carlos III de Madrid, Spain. Croux, C., Rousseeuw, P.J., Hossjer, O., 1994. Generalized S-estimators. J. Amer. Statist. Assoc. 89, 1271–1281. Croux, C., Rousseeuw, P.J., Van Bael, A., 1996. Positive-breakdown regression by minimizing nested scale estimators. J. Statist. Plann. Inference. 53, 197–235. Grubel, R., 1988. The length of the shorth. Ann. Statist. 16, 619–628. Hampel, F.R., 1974. The in uence curve and its role in robust estimation. J. Amer. Statist. Assoc. 69, 383–393.

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Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A., 1986. Robust Statistics. The Approach Based on In uence Functions. Wiley, New York. Huber, P.J., 1964. Robust estimation of a location parameter. Ann. Math. Statist. 35, 73–101. Huber, P.J., 1981. Robust Statistics. Wiley, New York. Martin, R.D., Yohai, V.J., Zamar, R.H., 1989. Min–max bias robust regression. Ann. Statist. 17, 1608–1630. Martin, R.D., Zamar, R.H., 1993a. Eciency-constrained bias-robust estimation of location. Ann. Statist. 21, 338–354. Martin, R.D., Zamar, R.H., 1993b. Bias robust estimation of scale. Ann. Statist. 21, 991–1017. Rousseeuw, P.J., Croux, C., 1993. Alternatives to the median absolute deviation. J. Amer. Statist. Assoc. 88, 1273–1283. Rousseeuw, P.J., Croux, C., 1994. The bias of k-step M-estimators. Statist. Probab. Lett. 20, 411–420. Rousseeuw, P.J., Leroy, A.M., 1987. Robust Regression and Outlier Detection. Wiley, New York. Yohai, V.J., Zamar, R.H., 1988. High breakdown point estimates of regression by means of the minimization of an ecient scale. J. Amer. Statist. Assoc. 83, 406–413.