# THE BREAKDOWN POINT EXAMPLES AND COUNTEREXAMPLES

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 REVSTAT Statistical Journal Volume 5, Number 1, March 2007, 1 17 THE BREAKDOWN POINT EXAMPLES AND COUNTEREXAMPLES Authors: P.L. Davies University of Duisburg-Essen, Germany, and Technical University Eindhoven, Netherlands U. Gather University of Dortmund, Germany Abstract: The breakdown point plays an important though at times controversial role in statistics. In situations in which it has proved most successful there is a group of transformations which act on the sample space and which give rise to an equivariance structure. For equivariant functionals, that is those functionals which respect the group structure, a non-trivial upper bound for the breakdown point was derived in Davies and Gather (2005). The present paper briefly repeats the main results of Davies and Gather (2005) but is mainly concerned with giving additional insight into the concept of breakdown point. In particular, we discuss the attainability of the bound and the dependence of the breakdown point on the sample or distribution and on the metrics used in its definition. Key-Words: equivariance; breakdown point; robust statistics. AMS Subject Classification: Primary: 62G07; Secondary: 65D10, 62G20.

2 2 P.L. Davies and U. Gather

3 The Breakdown Point Examples and Counterexamples 3 1. INTRODUCTION The breakdown point is one of the most popular measures of robustness of a statistical procedure. Originally introduced for location functionals (Hampel, 1968, 1971) the concept has been generalized to scale, regression and with more or less success to other situations. In Huber s functional analytic approach to robustness breakdown is related to the boundedness of a functional and the breakdown point is defined in terms of the sizes of neighbourhoods on the space of distributions. A simple and intuitive definition of the breakdown point but one restricted to finite samples, the finite sample breakdown point, was introduced by Donoho (1982) and Donoho and Huber (1983). Successful applications of the concept of breakdown point have been to the location, scale and regression models in R k and to models which are intimately related to these (see for example Ellis and Morgenthaler, 1992, Davies and Gather, 1993, Hubert, 1997, Terbeck and Davies, 1998, He and Fung, 2000, Müller and Uhlig, 2001). The reason for this is that such models have a rich equivariance structure deriving from the translation or affine group operating on R k. By restricting the class of statistical functionals to those with the appropriate equivariance structure one can prove the existence of non-trivial highest breakdown points (Davies and Gather, 2005), which in many cases can be achieved, at least locally (Huber, 1981, Davies, 1993). It is the aim of this paper to provide some additional insight into the definition of the breakdown point, to point out the limits of the concept and to give some results on the attainment of the upper bound. We proceed as follows: Chapter 2 summarizes the definitions and theorems of Davies and Gather (2005). Chapter 3 shows via examples that the breakdown point is a local concept. Chapter 4 is devoted to the attainability of the bound and Chapter 5 to the choice of metrics. Chapter 6 contains some concluding remarks. 2. DEFINITIONS AND BOUNDS FOR THE BREAKDOWN POINT Let T be a functional defined on some subfamily P T of the family P of all distributions on a sample space (X, B(X)) which takes its values in some metric space (Θ, D) with (2.1) sup D(θ 1, θ 2 ) =. θ 1,θ 2 Θ

4 4 P.L. Davies and U. Gather The finite sample breakdown point of T at a sample x n = (x 1,..., x n ), x i X, i = 1,..., n, is defined as (2.2) fsbp(t, x n, D) = 1 {k n min {1,..., n}: sup D ( T(P n ), T(Q n,k ) ) } = Q n,k where P n = n i=1 δ x i /n and Q n,k is the empirical distribution of a replacement sample with at least n k points from the original sample x n. Example 2.1. If T is the median functional T med defined on P T = P with Θ = R, and D(θ 1, θ 2 ) = θ 1 θ 2, then n+1 (2.3) fsbp(t med, x, D) = / n. 2 with A distributional definition of the breakdown point requires a metric d on P sup d(p, Q) = 1. P,Q P The breakdown point of a functional T at a distribution P P T w.r.t. d and D is then defined by { (2.4) ǫ (T, P, d, D) = inf ǫ > 0: sup D ( T(P), T(Q) ) } = d(p,q)<ǫ where D ( T(P), T(Q) ) := if Q / P T. Example 2.2. Let P and D be as in Example 2.1 and d be the Kolmogorov-metric d k (P, Q) = sup F P (x) F Q (x). For the expectation functional T E x { } T E (P) = E(P) := x dp(x), P T = P P : E(P) exists we have ǫ (T E, P, d, D) = 0 for any P P T, in contrast to the median for which ǫ (T med, P, d, D) = 1/2. As already pointed out in the introduction the derivation of a non-trivial upper bound for the breakdown point requires a group structure. Assume that G is a group of measurable transformations of the sample space X onto itself. Then G induces a group of transformations of P onto itself via P g (B)=P(g 1 (B)) for all sets B B(X). Let H g = {h g : g G} be the group of transformations h g : Θ Θ which describes the equivariance structure of the problem. A functional T : P T Θ is called equivariant with respect to G if and only if P T is closed under G and (2.5) T(P g ) = h g ( T(P) ) for all g G, P P T.

5 The Breakdown Point Examples and Counterexamples 5 Let { (2.6) G 1 := g G: lim inf D( θ, h g n(θ) ) } = n θ Θ and define } (2.7) (Q) := sup {Q(B): B B(X), g B = ι B for some g G 1 where ι is the unit element of G. We cite the main result from Davies and Gather (2005): Theorem 2.1. Suppose that the metrics d and D satisfy the properties given above and additionally ) (2.8) d (αp +(1 α)q 1, αp +(1 α)q 2 1 α, P, Q 1, Q 2 P, 0 < α < 1, (2.9) G 1. Then for all G-equivariant functionals T : P T Θ, for all P P T and for all x n we have respectively ( ) 1 (P) a) ǫ (T, P, d, D), 2 n n (Pn ) + 1 /n. b) fsbp(t, x n, D) 2 Proof: a) cf. Davies and Gather (2005). b) The proof is similar to a) but it is not given in Davies and Gather (2005). We present it here as it illustrates the simplicity of the idea of the finite sample breakdown point. The basic idea of all such proofs may be found in Huber (1981) although it was clearly known to Hampel (1975) who stated the breakdown point of what is now known as the LMS estimator (see Rousseeuw, 1984). Donoho and Huber (1983) give the first calculations for the finite sample breakdown point both for multivariate location and for a high breakdown linear regression estimator based on the multivariate location estimator of Donoho (1982). The corresponding calculations for the LMS estimator may be found in Rousseeuw (1984). Firstly we note that there are exactly n (P n ) points in x n for which g(x i ) = x i for some g G 1. We assume without loss of generality that these are the sample points x 1,..., x n (Pn). If (P n ) = 0 there are no such points and some obvious alterations to the following proof are required. To ease the notation we write n n (Pn ) + 1 l(n) =. 2 We consider the sample y n,k given by ) y n,k (x = 1,..., x n (Pn),..., x n l(n), g m (x n l(n)+1 ),..., g m (x n ) for some m 1 and some g G 1. We denote its empirical distribution by Q n,k.

6 6 P.L. Davies and U. Gather The sample y n,k contains at least n l(n) points of the original sample x n. The transformed sample g m (y n,k ) is equal to ) (x 1,..., x n (Pn), g m (x n (Pn)+1),..., g m (x n l(n) ), x n l(n)+1,..., x n. It contains at least n (P n )+l(n) points of the original sample x n and as n (P n ) + l(n) n l(n) it contains at least n l(n) points of x n. By the equivariance of T we have ( T(Q g m n,k ) = h g m T(Q n,k ) ) from which it follows ( ( D h g m T(Q n,k ) ) ), T(Q n,k ) D ( T(P n ), T(Q n,k )) + D ( T(P n ), T(Q g m n,k ) ). From lim inf D(θ, h g n(θ)) = for all g G 1 we have n θ ( lim D ( h g m T(Q m n,k ) ) ), T(Q n,k ) = and hence D ( T(P n ), T(Q n,k )) and D ( T(P n ), T(Q g m n,k ) ) cannot both remain bounded. We conclude that for any k n n (P n)+1 2 sup Q n,k D ( T(P n ), T(Q n,k ) ) = from which the claim of the theorem follows. For examples of Theorem 2.1 we refer to Davies and Gather (2005). 3. THE BREAKDOWN POINT IS A LOCAL CONCEPT As seen above the median T med has a finite sample breakdown point of (n+1)/2 at every real sample x n and this is the highest possible value for translation equivariant location functionals. If we consider scale functionals then the situation is somewhat different. The statistical folklore is that the highest possible finite sample breakdown point for any affine equivariant scale functional is n/2 /n and that this is attained by the median absolute deviation functional T MAD. Some authors (Croux and Rousseeuw, 1992, Davies, 1993) are aware that this is not correct as is shown by the following sample (3.1) x 11 = ( 1.0, 1.8, 1.3, 1.3, 1.9, 1.1, 1.3, 1.6, 1.7, 1.3, 1.3 ).

7 The Breakdown Point Examples and Counterexamples 7 The fsbp of T MAD at this sample is 1/11. This can be seen by replacing the data point 1.0 by 1.3 so that for the altered data set T MAD = 0 which is conventionally defined as breakdown. If a sample has no repeated observations then T MAD has a finite sample breakdown point of n/2 /n and this is indeed the highest possible finite sample breakdown point for a scale functional. The difference between the maximal finite sample breakdown points for location and scale functionals is explained by Theorem 2.1. For the sample (3.1) we have (P n ) = 5/11 and the theorem gives fsbp ( T MAD, x 11, D ) 3/11. For a sample x 11 without ties we have (P n ) = 1/n and the theorem yields fsbp ( T MAD, x 11, D ) n 2 /n = 5/11. From the above it follows that T MAD may or may not attain the upper bound. We study this in more detail in the next chapter. 4. ATTAINING THE BOUND 4.1. Location functionals From Theorem 2.1 above it is clear that the maximum breakdown point for translation equivariant location functionals is 1/2. This bound is sharp as is shown by the location equivariant L 1 -functional (4.1) T(P) = argmin µ ( x µ x ) dp(x). In general the L 1 -functional is not regarded as a satisfactory location functional as it is not affine equivariant in dimensions higher than one. For an affinely equivariant location functional the set G 1 of (2.6) is now the set of pure non-zero translations and it follows that (P) = 0 for any distribution P. Theorem 2.1 gives an upper bound of 1/2 which is clearly attainable in one dimension. It is not however clear whether this bound is attainable in higher dimensions. Work has been done in this direction but it is not conclusive (Rousseeuw and Leroy, 1987, Niinimaa, Oja and Tableman, 1990, Lopuhaä and Rousseeuw, 1991, Gordaliza, 1991, Lopuhaä, 1992, Donoho and Gasko, 1992, Davies and Gather, 2005, Chapter 5 and the Discussion of Rousseeuw in Davies and Gather, 2005). We first point out that the bound 1/2 is not globally sharp. Take a discrete measure in R 2 with point mass 1/3 on the points x 1 = (0, 1), x 2 = (0, 1), x 3 = ( 3, 0). The points form a regular simplex. For symmetry reasons every

8 8 P.L. Davies and U. Gather affinely equivariant location functional must yield the value (1/ 3, 0). On replacing ( 3, 0) by (η 3, 0) it is clear that each affinely equivariant location functional must result in (η/ 3, 0). On letting η it follows that the breakdown point of every affinely equivariant location functional cannot exceed 1/3. In k dimensions one can prove in a similar manner that 1/(k+1) is the maximal breakdown point for points on a regular simplex with k+1 sides. In spite of the above example we now show that there are probability distributions at which the finite sample replacement breakdown point is 1/2 even if this cannot be obtained globally. We consider a sample x n = (x 1,..., x n ) of size n in R k and form the empirical measure P n given by P n = 1/n n i=1 δ x i. To obtain our goal we define an appropriate affinely equivariant location functional T at Pn A for all affine transformations A and also at all measures of the form Pn A. Here Pn is any empirical measure obtained from x n by altering the values of at most (n 1)/2 of the x i. The new sample will be denoted by x n= (x 1,..., x n). We have to show that the values of T(Pn A ) can be defined in such a way that (4.2) (4.3) and (4.4) This is done in Appendix A. T(Pn A ) = A ( T(P n ) ), T(Pn A ) = A ( T(Pn) ) sup T(P n ) T(Pn) <. Pn We note that the Sample conditions 1 and 2 in Appendix A are satisfied for an i.i.d. Gaussian sample of size n if n is sufficiently large. We indicate how this may be shown in Appendix B Scatter functionals At the sample (3.1) above the median absolute deviation T MAD has a finite sample breakdown point of 1/11 compared with the upper bound of 3/11 given by Theorem 2.1. We consider a modification of T MAD as defined in Davies and Gather (2005) which attains the upper bound. For a probability measure P the interval I(P, λ) is defined by [ ] I(P, λ) = med(p) λ, med(p)+λ. We write { } (P, λ) = max P({x}): x I(P, λ). The new scale functional TMAD is defined by { TMAD(P) = min λ: P ( I(P, λ) ) ( 1+ (P, λ) ) } /2.

9 The Breakdown Point Examples and Counterexamples 9 We shall show (4.5) fsbp ( TMAD, x n, D ) n n (Pn ) + 1 /n =. 2 We consider a replacement sample x n with (n n 1 + n 2 = m < n (Pn ) + 1 ) /2 points replaced and with empirical distribution P n. We show firstly that TMAD (P n) does not explode. Let λ be such that the interval [ med(p n) λ, med(p n)+λ ] contains the original sample x n. As the median does not explode we see that λ remains bounded over all replacement samples. Clearly if TMAD (P n) is to explode x n must contain points outside of this interval. We denote the number of such points by n 1. We use n 2 points to increase the size of the largest atom of x n in the interval. This is clearly done by placing these points at the largest atom of x n. The size of the largest atom of x n in the interval is therefore at most (P n ) + n 2 /n. It follows that TMAD (P n) λ if the interval contains at least ( ) n + n (P n ) + n 2 /2 observations. This will be the case if n n 1 ( ) n + n (P n ) + n 2 /2 which reduces to n1 + n 2 /2 n ( 1 (P n ) ) /2 which holds as n 1 + n 2 /2 n 1 + n 2 < n ( 1 (P n ) + 1 ) / 2. It remains to show that T MAD (P n) does not implode to zero. For this to happen we would have to be able to construct a replacement sample for which the interval I(P, λ) is arbitrarily small but for which P ( I(P, λ) ) ( 1+ (P, λ) ) /2. In order for the interval to be arbitrarily small it must contain either no points of the original sample x n or just one atom. In the latter case we denote the size of the atom by 1 (P n ). Suppose we replace n 1 +n 2 points and that the n 2 points form the largest atom in the interval I(P, λ). We see that if n 2 n 1 (P n ) then which implies n 1 + n 2 + n 1 (P n ) (n+n 2 )/2 2n 1 + 2n 2 2n 1 + n 2 + n 1 (P n ) n > n n (P n ) which contradicts n 1 + n 2 < n ( 1 (P n )+1 ) /2. If the n 2 replacement points do not compose the largest atom then this must be of size at least 1 (P n ) which implies n 1 + n 2 + n 1 (P n ) ( n + n 1 (P n ) ) /2 and hence 2n 1 + 2n 2 n n 1 (P n ) n n (P n ) which again contradicts n 1 + n 2 < n ( 1 (P n ) + 1 ) /2. TMAD (P n) cannot implode, and thus (4.5) is shown. We conclude that

10 10 P.L. Davies and U. Gather 5. THE CHOICE OF THE METRICS d AND D 5.1. The metric d Considering the parts a) and b) of Theorem 2.1 we note that there is in fact a direct connection between the two results. We consider the total variation metric d tv defined by d tv (P, Q) = sup B B(X) P(B) Q(B). If B(X) shatters every finite set of points in X then d tv (P n, P n) = k/n where P n denotes the empirical measure deriving from (x 1,..., x n ) and Pn that deriving from (x 1,..., x n) with the two samples differing in exactly k points. Suppose now that ǫ (T, P n, d tv, D) = ( 1 (P n ) ) /2. If k < n ( 1 (P n ) ) /2 then breakdown in the sense of finite sample breakdown point cannot occur and we see that (5.1) fsbp ( T, x n, D ) n n (Pn ) /n. 2 Unfortunately the inequality of Theorem 2.1 b) seems not to be provable in the same manner. We point out that the breakdown point is not necessarily the same for all metrics d. A simple counterexample is provided by the scale problem in R. If we use the Kolmogorov metric then the breakdown point of T MAD at an atomless distribution is 1/4 (Huber, 1981, page 110). However if we use the Kuiper metric d 1 ku defined in (5.3) below then the breakdown point is 1/2 in spite of the fact that both metrics satisfy the conditions of the theorem. More generally if d and d are two metrics satisfying sup d(p, Q) = 1 and (2.8) and such that d d then P,Q P (5.2) ǫ (T, P, d, D) ǫ (T, P, d, D) ( 1 (P) ) /2. In particular if ǫ (T, P, d, D) = ( 1 (P) ) /2 then ǫ (T, P, d, D) = ( 1 (P) ) /2. A class of ordered metrics is provided by the generalized Kuiper metrics d m ku defined by { (5.3) d m ku (P, Q) = sup m ( P(Ik ) Q(I k ) ) } : I 1,..., I m disjoint intervals. We have k=1 (5.4) d 1 ku... dm ku.

11 The Breakdown Point Examples and Counterexamples 11 For further remarks on the choice of d we refer to Davies and Gather (2005), Rejoinder and for a related but different generalization of the Kuiper metric of use in the context of the modality of densities we refer to Davies and Kovac (2004) The metric D As we have seen in the case of d above there seems to be no canonical choice: different choices of d can lead to different breakdown points. A similar problem exists with respect to the metric D on Θ. In the discussion of Tyler in Davies and Gather (2005) it was also pointed out that it might be difficult to achieve (2.1) when Θ is a compact space. This problem is discussed in the Rejoinder of Davies and Gather (2005), Chapter 6, and solved in Davies and Gather (2006) with applications to directional data. We now indicate a possibility of making D dependent on d. The idea is that two parameter values θ 1 and θ 2 are far apart with respect to D if and only if the corresponding distributions are far apart with respect to d. We illustrate the idea using the location problem in R. Suppose we have data with empirical distribution P n and two values of the location parameter θ 1 and θ 2. We transform the data using the translations θ 1 and θ 2 which gives rise to two further distributions P n ( θ 1 ) and P n ( θ 2 ). If these two distributions are clearly distinguishable then d ( P n ( θ 1 ), P n ( θ 2 ) ) will be almost one. An opposed case is provided by an autoregressive process of order one. The parameter space is Θ = ( 1, 1) and this may be metricized in such a manner that D(θ 1, θ 2 ) tends to infinity for fixed θ 1 as θ 2 tends to the boundary. However values of θ close to, on, or even beyond the boundary, may not be empirically distinguishable from values of θ in the parameter space. A sample of size n = 100 generated with θ 1 = 0.95 is not easily distinguishable from a series generated with θ 2 = even though D(θ 1, θ 2 ) is large. We now give a choice of D in terms of d and such that (2.1) is satisfied. We set } G(θ 1, θ 2 ) = {g G: h g (θ 1 ) = θ 2 and then define D by (5.5) D(θ 1, θ 2 ) = D P (θ 1, θ 2 ) = inf g G(θ 1,θ 2 ) log ( 1 d(p g, P) ). The interpretation is that we associate P with the parameter value θ 1 and P g with the parameter value θ 2. The requirement (2.1) will only hold if d(p g, P) may be arbitrarily close to one so that the distributions associated with θ 1 and θ 2 are as far apart as possible. It is easily checked that D defines a pseudometric

12 12 P.L. Davies and U. Gather on Θ, which is sufficient for our purposes; namely D P 0, D P is symmetric and satisfies the triangle inequality. In some situations it seems reasonable to require that d and D be invariant with respect to the groups G and H G respectively. If d is G-invariant, i.e. d(p, Q) = d(p g, Q g ), for all P, Q P, g G, then D, defined by (5.5), inherits the invariance, i.e. D(θ 1, θ 2 ) = D ( h g (θ 1 ), h g (θ 2 ) ), for all θ 1, θ 2 Θ, g G. The G-invariance of d can often be met. 6. FINAL REMARKS We conclude with a small graphic showing the connections between all ingredients which are necessary for a meaningful breakdown point concept. (X, B(X), P) T Θ d D G H G Figure 1: Connections. We point out that each object in this graphic has an important influence on the breakdown point and its upper bound: ǫ (T, P, d, D) depends on P as shown in Chapter 3, and it depends on the metrics d and D as discussed in Chapter 5. It is the equivariance structure w.r.t. the group G which allows to prove an upper bound for ǫ (T, P, d, D) and it is the condition G 1 which provides the main step in the proof. In particular, the choice of the group G determines (P), thereby the upper bound, as well as its attainability. For many P, T and G the attainability of the bound remains an open problem.

13 The Breakdown Point Examples and Counterexamples 13 APPENDIX A We consider the constraints imposed upon us when defining T(P n). We start with the internal constraints which apply to each P n without reference to the other measures. Case 1: P A 1 n P A 2 n for any two different affine transformations A 1 and A 2. This is seen to reduce to Pn A Pn for any affine transformation A which is not the identity. If this is the case then there are no restrictions on the choice of T(Pn). Having chosen it we extend the definition of T to all the measures P A n by T(P A n ) = A ( T(P n) ). Case 2: Pn A =Pn for some affine transformation A which is not the identity. If this is the case then A is unique and there exists a permutation π of {1,..., n} such that A(x i ) = x π(i). This implies that for each i we can form cycles ( ) x i, A(x i ),..., A mi 1 (x i ) (A.1) with A m i (x i ) = x i. From this we see that for some sufficiently large m A m (x i ) = x i for all i. On writing A(x) = α(x) + a we see that if the x i, i = 1,..., n, span R k then α m = I where I denotes the identity transformation on R k. This implies that α must be an orthogonal transformation and that m 1 j=0 α j (a) = 0. It follows that if we set T(Pn) = µ, we must have A(µ) = µ for any affine transformation for which Pn A = Pn. The choice of µ is arbitrary subject only to these constraints. Having chosen such a µ the values of T(Pn B ) are defined to be B(µ) for all other affine transformations B. The above argument shows the internal consistency relationships which must be placed on T so that T(Pn A ) = A ( T(P n ) ) for any affine transformation A. We now consider what one may call the external restrictions. Case 3: Suppose that Pn is such that there does not exist a P n and an affine transformation A such that Pn A = P n. In this case the choice of T(Pn) is only restricted by the considerations of Case 2 above if that case applies and otherwise not at all. Case 4: Suppose that Pn is such that there exists a P n and an affine transformation A such that Pn = P n A. In this case we require T(Pn) = A ( T(P n ) ).

14 14 P.L. Davies and U. Gather We now place the following conditions on the sample x n : Sample condition 1: There do not exist two distinct subsets of x n each of size at least k+2 and an affine transformation A which transforms one subset into the other. Sample condition 2: If A(x n ) B(x n ) (n+1)/2 2k for two affine transformations A and B then A = B. Sample condition 3: k < (n 1)/2. We now construct a functional T which satisfies (4.2), (4.3) and (4.4). If the sample conditions hold then for any affine transformation A I we have Pn A Pn where Pn derives from a subset x n which differs from x n by at least one and at most (n 1)/2 points. This follows on noting that at most k+1 of the A(x i ) belong to x n by Sample condition 1. Because of this we can define T(P n ) without reference to the values of T(Pn). We set T(P n ) = 1 n x i. n i=1 If P n satisfies the conditions of Case 3 above we set T(P n) = 1 n n x π(i) where the x π(i) are those n (n + 1)/2 points of the sample x n which also belong to the sample x n. Finally we consider Case 4 above. We show that the sample assumptions and the condition Pn = P n A uniquely determine the affine transformation A. To see this we suppose that there exists a second affine transformation B and a distribution P n i=1 such that P n = P B n. Let x π(1),..., x π(n ) denote those points of x n not contained in the sample x n. Because of Sample condition 1 this set contains at least (n+1)/2 k 2 points of the form A(x i ). Similarly it also contains at least (n + 1)/2 k 2 points of the form B(x i ). The intersection of these two sets is of size at least (n+1)/2 2k and we may conclude from Sample condition 2 that A = B. The representation is therefore unique. Let x π(1),..., x π(m) be those points of x n which belong to the sample x n and for which A(x π(1) ),..., A(x π(m) ) belong to the sample x n. It is clear that m 1. We define and by equivariance T(P n ) = 1 m T(P n) = 1 m m i=1 x π(i) m A ( ) x π(i). i=1

15 The Breakdown Point Examples and Counterexamples 15 It follows that T(P n) is well defined and in both cases the sums involved come from the sample x n. The functional T is extended to all P n B and P n B by affine equivariance. In all cases the definition of T(P n) is as the mean of a subset of x n. From this it is clear that (4.4) is satisfied. APPENDIX B We now show that Sample conditions 1 and 2 hold for independent random samples X 1,..., X n with probability one. Let A=A+a and B=B+b with A and B nonsingular matrices and a and b points in R k. We suppose that A B. On taking differences we see that there exist variables X i1,..., X ik+1 and X j1,..., X jk+1 such that A(X il X ik+1 ) = B(X jl X jk+1 ), j = 1,..., k. This implies that B 1 A and B 1 (b a) are functions of the chosen sample points (B.1) B 1 A = C ( X i1,..., X ik+1, X j1,..., X jk+1 ), B 1 (b a) = c ( X i1,..., X ik+1, X j1,..., X jk+1 ). For n sufficiently large there exist four further sample points X i, i = 1,...,4 which are not contained in { X i1,..., X ik+1, X j1,..., X jk+1 } and for which This implies A(X 1 ) + a = B(X 2 ) + b, A(X 3 ) + a = B(X 4 ) + b. (B.2) B 1 A(X 3 X 1 ) = X 4 X 2. However as the X i, i = 1,...,4, are independent of X i1,..., X ik+1, X j1,..., X jk+1 it follows from (B.1) that (B.2) holds with probability zero. From this we conclude that A = B. Similarly we can show that a = b and hence A = B. ACKNOWLEDGMENTS Research supported in part by Sonderforschungsbereich 475, University of Dortmund.

16 16 P.L. Davies and U. Gather REFERENCES [1] Croux, C. and Rousseeuw, P.J. (1992). A class of high-breakdown estimators based on subranges, Commun. Statist. Theory Meth., 21, [2] Davies, P.L. (1993). Aspects of robust linear regression, Ann. Statist., 21, [3] Davies, P.L. and Gather, U. (1993). The identification of multiple outliers (with discussion and rejoinder), J. Amer. Statist. Assoc., 88, [4] Davies, P.L. and Gather, U. (2005). Breakdown and groups (with discussion and rejoinder), Ann. Statist., 33, [5] Davies, P.L. and Gather, U. (2006). Addendum to the discussion of breakdown and groups, Ann. Statist., 34, [6] Davies, P.L. and Kovac, A. (2004). Densities, spectral densities and modality, Ann. Statist., 32, [7] Donoho, D.L. (1982). Breakdown properties of multivariate location estimators, Ph. D. qualifying paper, Dept. Statistics, Harvard University. [8] Donoho, D.L. and Gasko, M. (1992). Breakdown properties of location estimates based on halfspace depth and projected outlyingness, Ann. Statist., 20, [9] Donoho, D.L. and Huber, P.J. (1983). The notion of breakdown point. In A Festschrift for Erich Lehmann (P.J. Bickel, K. Doksum and J.L. Hodges, Jr., Eds.), Wadsworth, Belmont, CA, [10] Ellis, S.P. and Morgenthaler, S. (1992). Leverage and breakdown in L 1 regression, J. Amer. Statist. Assoc., 87, [11] Gordaliza, A. (1991). On the breakdown point of multivariate location estimators based on trimming procedures, Statist. Probab. Letters, 11, [12] Hampel, F.R. (1968). Contributions to the theory of robust estimation, Ph. D. thesis, Dept. Statistics, Univ. California, Berkeley. [13] Hampel, F.R. (1971). A general qualitative definition of robustness, Ann. Math. Statist., 42, [14] Hampel, F.R. (1975). Beyond location parameters: robust concepts and methods (with discussion), Proceedings of the 40th Session of the ISI, Vol. 46, Book 1, [15] He, X. and Fung, W.K. (2000). High breakdown estimation for multiple populations with applications to discriminant analysis, J. Multivariate Anal., 72, [16] Huber, P.J. (1981). Robust Statistics, Wiley, New York. [17] Hubert, M. (1997). The breakdown value of the L 1 estimator in contingency tables, Statist. Probab. Letters, 33, [18] Lopuhaä, H.P. (1992). Highly efficient estimators of multivariate location with high breakdown point, Ann. Statist., 20,

17 The Breakdown Point Examples and Counterexamples 17 [19] Lopuhaä, H.P. and Rousseeuw, P.J. (1991). Breakdown points of affine equivariant estimators of multivariate location and covariance matrices, Ann. Statist., 19, [20] Müller, C.H. and Uhlig, S. (2001). Estimation of variance components with high breakdown point and high efficiency, Biometrika, 88, [21] Niinimaa, A.; Oja, H. and Tableman, M. (1990). The finite-sample breakdown point of the Oja bivariate median and of the corresponding half-samples version, Statist. Probab. Letters, 10, [22] Rousseeuw, P.J. (1984). Least median of squares regression, Journal of the American Statistical Association, 79, [23] Rousseeuw, P.J. and Leroy, A.M. (1987). Robust Regression and Outlier Detection, Wiley, New York. [24] Terbeck, W. and Davies, P.L. (1998). Interactions and outliers in the two-way analysis of variance, Ann. Statist., 26,

### ON MULTIVARIATE MONOTONIC MEASURES OF LOCATION WITH HIGH BREAKDOWN POINT

Sankhyā : The Indian Journal of Statistics 999, Volume 6, Series A, Pt. 3, pp. 362-380 ON MULTIVARIATE MONOTONIC MEASURES OF LOCATION WITH HIGH BREAKDOWN POINT By SUJIT K. GHOSH North Carolina State University,

### 1 Robust statistics; Examples and Introduction

Robust Statistics Laurie Davies 1 and Ursula Gather 2 1 Department of Mathematics, University of Essen, 45117 Essen, Germany, laurie.davies@uni-essen.de 2 Department of Statistics, University of Dortmund,

### General Foundations for Studying Masking and Swamping Robustness of Outlier Identifiers

General Foundations for Studying Masking and Swamping Robustness of Outlier Identifiers Robert Serfling 1 and Shanshan Wang 2 University of Texas at Dallas This paper is dedicated to the memory of Kesar

### Maximum likelihood in log-linear models

Graphical Models, Lecture 4, Michaelmas Term 2010 October 22, 2010 Generating class Dependence graph of log-linear model Conformal graphical models Factor graphs Let A denote an arbitrary set of subsets

### High Breakdown Analogs of the Trimmed Mean

High Breakdown Analogs of the Trimmed Mean David J. Olive Southern Illinois University April 11, 2004 Abstract Two high breakdown estimators that are asymptotically equivalent to a sequence of trimmed

### Robust estimation of principal components from depth-based multivariate rank covariance matrix

Robust estimation of principal components from depth-based multivariate rank covariance matrix Subho Majumdar Snigdhansu Chatterjee University of Minnesota, School of Statistics Table of contents Summary

### Statistics 612: L p spaces, metrics on spaces of probabilites, and connections to estimation

Statistics 62: L p spaces, metrics on spaces of probabilites, and connections to estimation Moulinath Banerjee December 6, 2006 L p spaces and Hilbert spaces We first formally define L p spaces. Consider

### Symmetrised M-estimators of multivariate scatter

Journal of Multivariate Analysis 98 (007) 1611 169 www.elsevier.com/locate/jmva Symmetrised M-estimators of multivariate scatter Seija Sirkiä a,, Sara Taskinen a, Hannu Oja b a Department of Mathematics

### Lehrstuhl für Statistik und Ökonometrie. Diskussionspapier 88 / 2012

Lehrstuhl für Statistik und Ökonometrie Diskussionspapier 88 / 2012 Robustness Properties of Quasi-Linear Means with Application to the Laspeyres and Paasche Indices Ingo Klein Vlad Ardelean Lange Gasse

### REGRESSION ANALYSIS AND ANALYSIS OF VARIANCE

REGRESSION ANALYSIS AND ANALYSIS OF VARIANCE P. L. Davies Eindhoven, February 2007 Reading List Daniel, C. (1976) Applications of Statistics to Industrial Experimentation, Wiley. Tukey, J. W. (1977) Exploratory

### On robust and efficient estimation of the center of. Symmetry.

On robust and efficient estimation of the center of symmetry Howard D. Bondell Department of Statistics, North Carolina State University Raleigh, NC 27695-8203, U.S.A (email: bondell@stat.ncsu.edu) Abstract

### GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. 2π) n

GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. A. ZVAVITCH Abstract. In this paper we give a solution for the Gaussian version of the Busemann-Petty problem with additional

### ASYMPTOTICS OF REWEIGHTED ESTIMATORS OF MULTIVARIATE LOCATION AND SCATTER. By Hendrik P. Lopuhaä Delft University of Technology

The Annals of Statistics 1999, Vol. 27, No. 5, 1638 1665 ASYMPTOTICS OF REWEIGHTED ESTIMATORS OF MULTIVARIATE LOCATION AND SCATTER By Hendrik P. Lopuhaä Delft University of Technology We investigate the

### Linear Algebra Massoud Malek

CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product

### Supplementary Notes for W. Rudin: Principles of Mathematical Analysis

Supplementary Notes for W. Rudin: Principles of Mathematical Analysis SIGURDUR HELGASON In 8.00B it is customary to cover Chapters 7 in Rudin s book. Experience shows that this requires careful planning

### A Mathematical Programming Approach for Improving the Robustness of LAD Regression

A Mathematical Programming Approach for Improving the Robustness of LAD Regression Avi Giloni Sy Syms School of Business Room 428 BH Yeshiva University 500 W 185 St New York, NY 10033 E-mail: agiloni@ymail.yu.edu

### Robust Estimation of Cronbach s Alpha

Robust Estimation of Cronbach s Alpha A. Christmann University of Dortmund, Fachbereich Statistik, 44421 Dortmund, Germany. S. Van Aelst Ghent University (UGENT), Department of Applied Mathematics and

### Math 5520 Homework 2 Solutions

Math 552 Homework 2 Solutions March, 26. Consider the function fx) = 2x ) 3 if x, 3x ) 2 if < x 2. Determine for which k there holds f H k, 2). Find D α f for α k. Solution. We show that k = 2. The formulas

### Communications in Statistics Simulation and Computation. Impact of contamination on training and test error rates in statistical clustering

Impact of contamination on training and test error rates in statistical clustering Journal: Communications in Statistics - Simulation and Computation Manuscript ID: LSSP-00-0.R Manuscript Type: Review

### The Skorokhod reflection problem for functions with discontinuities (contractive case)

The Skorokhod reflection problem for functions with discontinuities (contractive case) TAKIS KONSTANTOPOULOS Univ. of Texas at Austin Revised March 1999 Abstract Basic properties of the Skorokhod reflection

### Robust scale estimation with extensions

Robust scale estimation with extensions Garth Tarr, Samuel Müller and Neville Weber School of Mathematics and Statistics THE UNIVERSITY OF SYDNEY Outline The robust scale estimator P n Robust covariance

### Lecture 35: December The fundamental statistical distances

36-705: Intermediate Statistics Fall 207 Lecturer: Siva Balakrishnan Lecture 35: December 4 Today we will discuss distances and metrics between distributions that are useful in statistics. I will be lose

### AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES

AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim

### On Kusuoka Representation of Law Invariant Risk Measures

MATHEMATICS OF OPERATIONS RESEARCH Vol. 38, No. 1, February 213, pp. 142 152 ISSN 364-765X (print) ISSN 1526-5471 (online) http://dx.doi.org/1.1287/moor.112.563 213 INFORMS On Kusuoka Representation of

### Exercise Solutions to Functional Analysis

Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n

### Notes, March 4, 2013, R. Dudley Maximum likelihood estimation: actual or supposed

18.466 Notes, March 4, 2013, R. Dudley Maximum likelihood estimation: actual or supposed 1. MLEs in exponential families Let f(x,θ) for x X and θ Θ be a likelihood function, that is, for present purposes,

### Robust repeated median regression in moving windows with data-adaptive width selection SFB 823. Discussion Paper. Matthias Borowski, Roland Fried

SFB 823 Robust repeated median regression in moving windows with data-adaptive width selection Discussion Paper Matthias Borowski, Roland Fried Nr. 28/2011 Robust Repeated Median regression in moving

### Fast and Robust Classifiers Adjusted for Skewness

Fast and Robust Classifiers Adjusted for Skewness Mia Hubert 1 and Stephan Van der Veeken 2 1 Department of Mathematics - LStat, Katholieke Universiteit Leuven Celestijnenlaan 200B, Leuven, Belgium, Mia.Hubert@wis.kuleuven.be

### Math 117: Topology of the Real Numbers

Math 117: Topology of the Real Numbers John Douglas Moore November 10, 2008 The goal of these notes is to highlight the most important topics presented in Chapter 3 of the text [1] and to provide a few

### MAXIMUM VARIANCE OF ORDER STATISTICS

Ann. Inst. Statist. Math. Vol. 47, No. 1, 185-193 (1995) MAXIMUM VARIANCE OF ORDER STATISTICS NICKOS PAPADATOS Department of Mathematics, Section of Statistics and Operations Research, University of Athens,

### Detecting outliers in weighted univariate survey data

Detecting outliers in weighted univariate survey data Anna Pauliina Sandqvist October 27, 21 Preliminary Version Abstract Outliers and influential observations are a frequent concern in all kind of statistics,

### Asymptotic Relative Efficiency in Estimation

Asymptotic Relative Efficiency in Estimation Robert Serfling University of Texas at Dallas October 2009 Prepared for forthcoming INTERNATIONAL ENCYCLOPEDIA OF STATISTICAL SCIENCES, to be published by Springer

### s P = f(ξ n )(x i x i 1 ). i=1

Compactness and total boundedness via nets The aim of this chapter is to define the notion of a net (generalized sequence) and to characterize compactness and total boundedness by this important topological

### Invariant coordinate selection for multivariate data analysis - the package ICS

Invariant coordinate selection for multivariate data analysis - the package ICS Klaus Nordhausen 1 Hannu Oja 1 David E. Tyler 2 1 Tampere School of Public Health University of Tampere 2 Department of Statistics

### INVARIANT COORDINATE SELECTION

INVARIANT COORDINATE SELECTION By David E. Tyler 1, Frank Critchley, Lutz Dümbgen 2, and Hannu Oja Rutgers University, Open University, University of Berne and University of Tampere SUMMARY A general method

### 3. 4. Uniformly normal families and generalisations

Summer School Normal Families in Complex Analysis Julius-Maximilians-Universität Würzburg May 22 29, 2015 3. 4. Uniformly normal families and generalisations Aimo Hinkkanen University of Illinois at Urbana

### Robust Online Scale Estimation in Time Series: A Regression-Free Approach

Robust Online Scale Estimation in Time Series: A Regression-Free Approach Sarah Gelper 1,, Karen Schettlinger, Christophe Croux 1, and Ursula Gather May 3, 7 1 Faculty of Economics and Management, Katholieke

### Nonamenable Products are not Treeable

Version of 30 July 1999 Nonamenable Products are not Treeable by Robin Pemantle and Yuval Peres Abstract. Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism

### Testing Statistical Hypotheses

E.L. Lehmann Joseph P. Romano Testing Statistical Hypotheses Third Edition 4y Springer Preface vii I Small-Sample Theory 1 1 The General Decision Problem 3 1.1 Statistical Inference and Statistical Decisions

### Accurate and Powerful Multivariate Outlier Detection

Int. Statistical Inst.: Proc. 58th World Statistical Congress, 11, Dublin (Session CPS66) p.568 Accurate and Powerful Multivariate Outlier Detection Cerioli, Andrea Università di Parma, Dipartimento di

### Finite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product

Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )

### Examples of Dual Spaces from Measure Theory

Chapter 9 Examples of Dual Spaces from Measure Theory We have seen that L (, A, µ) is a Banach space for any measure space (, A, µ). We will extend that concept in the following section to identify an

### Boolean Inner-Product Spaces and Boolean Matrices

Boolean Inner-Product Spaces and Boolean Matrices Stan Gudder Department of Mathematics, University of Denver, Denver CO 80208 Frédéric Latrémolière Department of Mathematics, University of Denver, Denver

### Generalized Multivariate Rank Type Test Statistics via Spatial U-Quantiles

Generalized Multivariate Rank Type Test Statistics via Spatial U-Quantiles Weihua Zhou 1 University of North Carolina at Charlotte and Robert Serfling 2 University of Texas at Dallas Final revision for

### HOMEWORK ASSIGNMENT 6

HOMEWORK ASSIGNMENT 6 DUE 15 MARCH, 2016 1) Suppose f, g : A R are uniformly continuous on A. Show that f + g is uniformly continuous on A. Solution First we note: In order to show that f + g is uniformly

### Sharp bounds on the VaR for sums of dependent risks

Paul Embrechts Sharp bounds on the VaR for sums of dependent risks joint work with Giovanni Puccetti (university of Firenze, Italy) and Ludger Rüschendorf (university of Freiburg, Germany) Mathematical

### Gaussian processes. Chuong B. Do (updated by Honglak Lee) November 22, 2008

Gaussian processes Chuong B Do (updated by Honglak Lee) November 22, 2008 Many of the classical machine learning algorithms that we talked about during the first half of this course fit the following pattern:

### HW1 solutions. 1. α Ef(x) β, where Ef(x) is the expected value of f(x), i.e., Ef(x) = n. i=1 p if(a i ). (The function f : R R is given.

HW1 solutions Exercise 1 (Some sets of probability distributions.) Let x be a real-valued random variable with Prob(x = a i ) = p i, i = 1,..., n, where a 1 < a 2 < < a n. Of course p R n lies in the standard

### Statistical Depth Function

Machine Learning Journal Club, Gatsby Unit June 27, 2016 Outline L-statistics, order statistics, ranking. Instead of moments: median, dispersion, scale, skewness,... New visualization tools: depth contours,

### In particular, if A is a square matrix and λ is one of its eigenvalues, then we can find a non-zero column vector X with

Appendix: Matrix Estimates and the Perron-Frobenius Theorem. This Appendix will first present some well known estimates. For any m n matrix A = [a ij ] over the real or complex numbers, it will be convenient

### The minimum volume ellipsoid (MVE), introduced

Minimum volume ellipsoid Stefan Van Aelst 1 and Peter Rousseeuw 2 The minimum volume ellipsoid (MVE) estimator is based on the smallest volume ellipsoid that covers h of the n observations. It is an affine

### Working Paper Convergence rates in multivariate robust outlier identification

econstor www.econstor.eu Der Open-Access-Publikationsserver der ZBW Leibniz-Informationszentrum Wirtschaft The Open Access Publication Server of the ZBW Leibniz Information Centre for Economics Gather,

### On the limiting distributions of multivariate depth-based rank sum. statistics and related tests. By Yijun Zuo 2 and Xuming He 3

1 On the limiting distributions of multivariate depth-based rank sum statistics and related tests By Yijun Zuo 2 and Xuming He 3 Michigan State University and University of Illinois A depth-based rank

### Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices

Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices by Jack W. Silverstein* Department of Mathematics Box 8205 North Carolina State University Raleigh,

### Krzysztof Burdzy University of Washington. = X(Y (t)), t 0}

VARIATION OF ITERATED BROWNIAN MOTION Krzysztof Burdzy University of Washington 1. Introduction and main results. Suppose that X 1, X 2 and Y are independent standard Brownian motions starting from 0 and

### Regression Analysis for Data Containing Outliers and High Leverage Points

Alabama Journal of Mathematics 39 (2015) ISSN 2373-0404 Regression Analysis for Data Containing Outliers and High Leverage Points Asim Kumer Dey Department of Mathematics Lamar University Md. Amir Hossain

### Lecture Notes 1: Vector spaces

Optimization-based data analysis Fall 2017 Lecture Notes 1: Vector spaces In this chapter we review certain basic concepts of linear algebra, highlighting their application to signal processing. 1 Vector

### CLASSIFICATION EFFICIENCIES FOR ROBUST LINEAR DISCRIMINANT ANALYSIS

Statistica Sinica 8(008), 58-599 CLASSIFICATION EFFICIENCIES FOR ROBUST LINEAR DISCRIMINANT ANALYSIS Christophe Croux, Peter Filzmoser and Kristel Joossens K.U. Leuven and Vienna University of Technology

### x log x, which is strictly convex, and use Jensen s Inequality:

2. Information measures: mutual information 2.1 Divergence: main inequality Theorem 2.1 (Information Inequality). D(P Q) 0 ; D(P Q) = 0 iff P = Q Proof. Let ϕ(x) x log x, which is strictly convex, and

### Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

### Series 7, May 22, 2018 (EM Convergence)

Exercises Introduction to Machine Learning SS 2018 Series 7, May 22, 2018 (EM Convergence) Institute for Machine Learning Dept. of Computer Science, ETH Zürich Prof. Dr. Andreas Krause Web: https://las.inf.ethz.ch/teaching/introml-s18

### PACKING-DIMENSION PROFILES AND FRACTIONAL BROWNIAN MOTION

PACKING-DIMENSION PROFILES AND FRACTIONAL BROWNIAN MOTION DAVAR KHOSHNEVISAN AND YIMIN XIAO Abstract. In order to compute the packing dimension of orthogonal projections Falconer and Howroyd 997) introduced

### Complements on Simple Linear Regression

Complements on Simple Linear Regression Terry R. McConnell Syracuse University March 16, 2015 Abstract We present a simple-minded approach to a variant of simple linear regression that seeks to minimize

### In English, this means that if we travel on a straight line between any two points in C, then we never leave C.

Convex sets In this section, we will be introduced to some of the mathematical fundamentals of convex sets. In order to motivate some of the definitions, we will look at the closest point problem from

### Trace Class Operators and Lidskii s Theorem

Trace Class Operators and Lidskii s Theorem Tom Phelan Semester 2 2009 1 Introduction The purpose of this paper is to provide the reader with a self-contained derivation of the celebrated Lidskii Trace

### Applications of Information Geometry to Hypothesis Testing and Signal Detection

CMCAA 2016 Applications of Information Geometry to Hypothesis Testing and Signal Detection Yongqiang Cheng National University of Defense Technology July 2016 Outline 1. Principles of Information Geometry

### Outlier Detection via Feature Selection Algorithms in

Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin (Session CPS032) p.4638 Outlier Detection via Feature Selection Algorithms in Covariance Estimation Menjoge, Rajiv S. M.I.T.,

### DEPARTAMENTO DE MATEMÁTICA DOCUMENTO DE TRABAJO. Impartial Trimmed Means for Functional Data. Juan Antonio Cuesta-Albertos and Ricardo Fraiman

DEPARTAMENTO DE MATEMÁTICA DOCUMENTO DE TRABAJO Impartial Trimmed Means for Functional Data Juan Antonio Cuesta-Albertos and Ricardo Fraiman D.T.: N 29 Diciembre 2003 Vito Dumas 284, (B1644BID) Victoria,

### Closest Moment Estimation under General Conditions

Closest Moment Estimation under General Conditions Chirok Han and Robert de Jong January 28, 2002 Abstract This paper considers Closest Moment (CM) estimation with a general distance function, and avoids

### Identification of Multivariate Outliers: A Performance Study

AUSTRIAN JOURNAL OF STATISTICS Volume 34 (2005), Number 2, 127 138 Identification of Multivariate Outliers: A Performance Study Peter Filzmoser Vienna University of Technology, Austria Abstract: Three

### CoDa-dendrogram: A new exploratory tool. 2 Dept. Informàtica i Matemàtica Aplicada, Universitat de Girona, Spain;

CoDa-dendrogram: A new exploratory tool J.J. Egozcue 1, and V. Pawlowsky-Glahn 2 1 Dept. Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Barcelona, Spain; juan.jose.egozcue@upc.edu 2 Dept.

### Partial cubes: structures, characterizations, and constructions

Partial cubes: structures, characterizations, and constructions Sergei Ovchinnikov San Francisco State University, Mathematics Department, 1600 Holloway Ave., San Francisco, CA 94132 Abstract Partial cubes

### PARAMETER CONVERGENCE FOR EM AND MM ALGORITHMS

Statistica Sinica 15(2005), 831-840 PARAMETER CONVERGENCE FOR EM AND MM ALGORITHMS Florin Vaida University of California at San Diego Abstract: It is well known that the likelihood sequence of the EM algorithm

### MAJORIZING MEASURES WITHOUT MEASURES. By Michel Talagrand URA 754 AU CNRS

The Annals of Probability 2001, Vol. 29, No. 1, 411 417 MAJORIZING MEASURES WITHOUT MEASURES By Michel Talagrand URA 754 AU CNRS We give a reformulation of majorizing measures that does not involve measures,

### Maximal perpendicularity in certain Abelian groups

Acta Univ. Sapientiae, Mathematica, 9, 1 (2017) 235 247 DOI: 10.1515/ausm-2017-0016 Maximal perpendicularity in certain Abelian groups Mika Mattila Department of Mathematics, Tampere University of Technology,

### Doubly Indexed Infinite Series

The Islamic University of Gaza Deanery of Higher studies Faculty of Science Department of Mathematics Doubly Indexed Infinite Series Presented By Ahed Khaleel Abu ALees Supervisor Professor Eissa D. Habil

### Goodness-of-Fit Tests for Time Series Models: A Score-Marked Empirical Process Approach

Goodness-of-Fit Tests for Time Series Models: A Score-Marked Empirical Process Approach By Shiqing Ling Department of Mathematics Hong Kong University of Science and Technology Let {y t : t = 0, ±1, ±2,

### OPERATOR PENCIL PASSING THROUGH A GIVEN OPERATOR

OPERATOR PENCIL PASSING THROUGH A GIVEN OPERATOR A. BIGGS AND H. M. KHUDAVERDIAN Abstract. Let be a linear differential operator acting on the space of densities of a given weight on a manifold M. One

### Small sample corrections for LTS and MCD

Metrika (2002) 55: 111 123 > Springer-Verlag 2002 Small sample corrections for LTS and MCD G. Pison, S. Van Aelst*, and G. Willems Department of Mathematics and Computer Science, Universitaire Instelling

### Machine Learning Lecture 7

Course Outline Machine Learning Lecture 7 Fundamentals (2 weeks) Bayes Decision Theory Probability Density Estimation Statistical Learning Theory 23.05.2016 Discriminative Approaches (5 weeks) Linear Discriminant

### Computationally Easy Outlier Detection via Projection Pursuit with Finitely Many Directions

Computationally Easy Outlier Detection via Projection Pursuit with Finitely Many Directions Robert Serfling 1 and Satyaki Mazumder 2 University of Texas at Dallas and Indian Institute of Science, Education

### arxiv: v2 [math.ca] 4 Jun 2017

EXCURSIONS ON CANTOR-LIKE SETS ROBERT DIMARTINO AND WILFREDO O. URBINA arxiv:4.70v [math.ca] 4 Jun 07 ABSTRACT. The ternary Cantor set C, constructed by George Cantor in 883, is probably the best known

### CHAPTER 1. Metric Spaces. 1. Definition and examples

CHAPTER Metric Spaces. Definition and examples Metric spaces generalize and clarify the notion of distance in the real line. The definitions will provide us with a useful tool for more general applications

### Statistics & Decisions 7, (1989) R. Oldenbourg Verlag, München /89 \$

Statistics & Decisions 7, 377-382 (1989) R. Oldenbourg Verlag, München 1989-0721-2631/89 \$3.00 + 0.00 A NOTE ON SIMULTANEOUS ESTIMATION OF PEARSON MODES L. R. Haff 1 ) and R. W. Johnson 2^ Received: Revised

### Machine Learning. VC Dimension and Model Complexity. Eric Xing , Fall 2015

Machine Learning 10-701, Fall 2015 VC Dimension and Model Complexity Eric Xing Lecture 16, November 3, 2015 Reading: Chap. 7 T.M book, and outline material Eric Xing @ CMU, 2006-2015 1 Last time: PAC and

### Shape Matching: A Metric Geometry Approach. Facundo Mémoli. CS 468, Stanford University, Fall 2008.

Shape Matching: A Metric Geometry Approach Facundo Mémoli. CS 468, Stanford University, Fall 2008. 1 General stuff: class attendance is mandatory will post possible projects soon a couple of days prior

### Lecture 2: Linear Algebra Review

EE 227A: Convex Optimization and Applications January 19 Lecture 2: Linear Algebra Review Lecturer: Mert Pilanci Reading assignment: Appendix C of BV. Sections 2-6 of the web textbook 1 2.1 Vectors 2.1.1

### Solving a linear equation in a set of integers II

ACTA ARITHMETICA LXXII.4 (1995) Solving a linear equation in a set of integers II by Imre Z. Ruzsa (Budapest) 1. Introduction. We continue the study of linear equations started in Part I of this paper.

### Independent Component (IC) Models: New Extensions of the Multinormal Model

Independent Component (IC) Models: New Extensions of the Multinormal Model Davy Paindaveine (joint with Klaus Nordhausen, Hannu Oja, and Sara Taskinen) School of Public Health, ULB, April 2008 My research

### The Arzelà-Ascoli Theorem

John Nachbar Washington University March 27, 2016 The Arzelà-Ascoli Theorem The Arzelà-Ascoli Theorem gives sufficient conditions for compactness in certain function spaces. Among other things, it helps

### Outlier detection for skewed data

Outlier detection for skewed data Mia Hubert 1 and Stephan Van der Veeken December 7, 27 Abstract Most outlier detection rules for multivariate data are based on the assumption of elliptical symmetry of

### Regression models for multivariate ordered responses via the Plackett distribution

Journal of Multivariate Analysis 99 (2008) 2472 2478 www.elsevier.com/locate/jmva Regression models for multivariate ordered responses via the Plackett distribution A. Forcina a,, V. Dardanoni b a Dipartimento

### arxiv: v1 [math.na] 25 Sep 2012

Kantorovich s Theorem on Newton s Method arxiv:1209.5704v1 [math.na] 25 Sep 2012 O. P. Ferreira B. F. Svaiter March 09, 2007 Abstract In this work we present a simplifyed proof of Kantorovich s Theorem

### COMPOSITE RELIABILITY MODELS FOR SYSTEMS WITH TWO DISTINCT KINDS OF STOCHASTIC DEPENDENCES BETWEEN THEIR COMPONENTS LIFE TIMES

COMPOSITE RELIABILITY MODELS FOR SYSTEMS WITH TWO DISTINCT KINDS OF STOCHASTIC DEPENDENCES BETWEEN THEIR COMPONENTS LIFE TIMES Jerzy Filus Department of Mathematics and Computer Science, Oakton Community

### Weighted empirical likelihood estimates and their robustness properties

Computational Statistics & Data Analysis ( ) www.elsevier.com/locate/csda Weighted empirical likelihood estimates and their robustness properties N.L. Glenn a,, Yichuan Zhao b a Department of Statistics,

### A note on a construction of J. F. Feinstein

STUDIA MATHEMATICA 169 (1) (2005) A note on a construction of J. F. Feinstein by M. J. Heath (Nottingham) Abstract. In [6] J. F. Feinstein constructed a compact plane set X such that R(X), the uniform

### Some Aspects of 2-Fuzzy 2-Normed Linear Spaces

BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 32(2) (2009), 211 221 Some Aspects of 2-Fuzzy 2-Normed Linear Spaces 1 R. M. Somasundaram

### is a Borel subset of S Θ for each c R (Bertsekas and Shreve, 1978, Proposition 7.36) This always holds in practical applications.

Stat 811 Lecture Notes The Wald Consistency Theorem Charles J. Geyer April 9, 01 1 Analyticity Assumptions Let { f θ : θ Θ } be a family of subprobability densities 1 with respect to a measure µ on a measurable