The quarter median

We introduce and discuss a multivariate version of the classical median that is based on an equipartition property with respect to quarter spaces. These arise as pairwise intersections of the half-spaces associated with the coordinate hyperplanes of an orthogonal basis. We obtain results on existence, equivariance, and asymptotic normality.

For a data set x 1 , . . . , xn of real numbers a (sample) median is any (distributional) median associated with the empirical distribution Pn :" n´1 ř n k"1 δx k , where δx denotes unit mass at x. With x p1q ď . . . ď x pnq as the values obtained by arranging x 1 , . . . , xn in increasing order, is the unique sample median if n is odd, and we get " x pn{2q , x pn{2`1q ‰ as the set of sample medians if n is even.
Historically, the first attempt to generalize this to dimensions d ą 1 was to use coordinates: If x 1 , . . . , xn is a subset of R d for some d ě 2 then a coordinatewise or marginal median of the empirical distribution Pn :" n´1 ř n k"1 δx k is any vector whose ith coordinate is a (one-dimensional) median of the respective ith coordinates of the data vectors. This is the sample version; for a probability measure P on the Borel subsets B d of R d the push-forwards P πi of P under the coordinate projections π i , i " 1, . . . , d, would be used. Formally, the marginal medians of P are elements of the ddimensional interval " MMed´pP q, MMed`pP q ‰ where MMed˘pP q have Med˘pP πi q as their ith component, i " 1, . . . , d, and MMedmpP q :"`MMed´pP q`MMed`pP q˘{2 ( 3) is the midpoint marginal median. It is well known that such component-wise extensions of the one-dimensional median are not equivariant with respect to orthogonal transformations and thus depend on the coordinate system chosen for the representation of the data. One way to repair this is by 'averaging out' (symmetrization), see Grübel (1996).
Other generalizations of the one-dimensional median to higher dimensions start with some characterizing property of the classical version and then use a multidimensional analogue. For example, it is known that in the one-dimensional case any median θ minimizes the function x Þ Ñ ş |y´x| P pdyq, which leads to the spatial or L 1 -median of the data set, defined as a minimizer of x Þ Ñ ř n k"1 }x´x k }, where }¨} denotes Euclidean distance. As this distance is invariant under orthogonal transformation, the spatial median does not depend on the chosen coordinates.
Closer to the above equipartition property are the various concepts related to data depth, where for each x P R d and each affine hyperplane H with x P H the number of data points to one side of H is the basic ingredient: A centerpoint median is defined as any such x with the property that the minimum number is at least a fraction n{pd`1q of the data, and Tukey's median is a point that maximizes these data depth values; see e.g. Donoho and Gasko (1992). This whole area has attracted the attention of many researchers, often with emphasis on robustness. The literature on multivariate medians up to about 1990 is covered in the review of Small (1990), which includes an interesting discussion of the history of the concept. For more recent reviews of this subject we refer to the treatises of Oja (2013) and Dhar and Chaudhuri (2011).
In the present paper, following an idea of the late Dietrich Morgenstern, we again start with the marginal median as in the work of Grübel (1996), but then use a variant of the equipartition property instead of symmetrization. In dimension d " 2, for example, a coordinate system specifies four quarter spaces via the pairwise intersections of the half-spaces underlying the marginal median, and a quarter median of P may be defined as any vector θ P R 2 such that for some coordinate system centered at θ all four of the respective quarter spaces have probability at least 1{4. Figure 1 shows a simple example that Professor Morgenstern would have liked: The data are the locations of the cities in Germany with a population of at least 100,000 (in 2017). We have n " 76 such cities in total, so we can aim for a partition with 19 cities in each open quarter space. One resulting quarter median is nearby the exit Freudenberg of the A45 (Sauerlandlinie), and with the North-South axis tilted by about 39 degrees counterclockwise; see the lines in Figure 1. The data (longitude/latitude) for drawing the German border are extracted from the file gadm36´DEU´0´sp.rds available at https://gadm.org/data.html.
Investigation of this concept leads to a number of interesting questions, starting with the obvious ones concerning existence and uniqueness. Next, what are the equivariance properties of the quarter median? Further, in contrast to the spatial median a quarter median comes with a system of associated hyperplanes, which may be regarded as a part of the estimator. In the two-dimensional case these are just two orthogonal lines and can be parametrized by a counterclockwise rotation of the horizontal axis, thus providing a connection to a real interval. In higher dimensions, however, specification of a topological notion of nearness of systems of coordinate hyperplanes is less straightforward, but it is needed in connection with consistency of the estimators. For asymptotic normality even a local linearization is required. Finally, there is the algorithmic issue: How do we find a quarter median for a given data set?
Section 2 contains our main results, on existence, equivariance, and large sample behavior. Elliptical distributions are treated in some detail and we compare the quarter median with other estimators for several specific families. We also consider algorithmic aspects together with the problem of measurable selection from the respective solution sets. Proofs are collected in Section 4.

Results
We first assemble some formal definitions and then give our results in separate subsections.

Preliminaries and basic definitions
We regard vectors x P R d as column vectors and write xx, yy for the inner product of x, y P R d . The transpose of the vector x is x t , so that xx, yy " x t y. Similarly, A t denotes the transpose of a matrix A. We write }¨} for the Euclidean norm on R d , S d´1 is the unit sphere in R d , Opdq is the group of orthogonal dˆd-matrices, and SOpdq denotes the group of rotations, consisting of those elements of Opdq that have determinant 1.
If pΩ, A, P q is a probability space, if pΩ 1 , A 1 q is a measurable space, and if the mapping S : Ω Ñ Ω 1 is pA, A 1 q-measurable then we write P S for the push-forward of P under S, i.e. P S pA 1 q " P pS´1pA 1 qq for all A 1 P A 1 . If pΩ, Aq " pR d , B d q and b P S d´1 or U P Opdq then P b and P U respectively refer to the push-forwards of P under the mappings x Þ Ñ xb, xy and x Þ Ñ U x.
For a P R d and b P S d´1 we formally define the associated half-spaces by H´pa; bq :" x P R d : xb, x´ay ď 0 ( , H`pa; bq :" x P R d : xb, x´ay ě 0 ( . Let e 1 , . . . , e d be the vectors of the canonical basis of R d . Then a marginal median for a probability measure P on pR d , B d q is any vector θ P R d with the property that P`H`pθ; e i q˘ě 1{2, P`H´pθ; e i q˘ě 1{2, 1 ď i ď d.
For two orthogonal unit vectors b, b 1 P S d´1 and a P R d the quarter spaces are the intersections of these half-spaces, V˘˘pa; b, b 1 q :" H˘pa; bq X H˘pa; b 1 q, with the obvious notational convention for the four possible combinations of plus and minus signs. We then define a quarter median of a probability measure P on pR d , B d q as any vector θ P R d with the property that, for some orthonormal basis B " tb 1 , .
for all combinations of plus and minus signs. If (5) and (6) hold then we call the pair pθ, U q P R dˆO pdq, where U has rows b t 1 , . . . , b t d , a solution of the quarter median problem for the probability measure P . Again, the data versions of these notions refer to the empirical distribution Pn.
Then (6) holds with b 1 " e 1 , b 2 " e 2 and θ " p1, 1{2q t , but (5) is not true. Further, with this choice of b 1 , b 2 it is easily verified that the unique marginal median θ " p1{2, 1{2q t is a quarter median. This example also shows that the quarter median may not be unique. Further, if hyperplanes have probability zero, then the inequalities in (5) and (6) can be replaced by equalities, and with pθ, U q as solution of (6) we then speak of a solution of the equipartition problem for P .

Existence and uniqueness
The existence of a median and thus of a marginal median follow immediately from the monotonicity of distribution functions. For the quarter median counting the number of variables and the number of constraints may give a first impression. For simplicity, we temporarily assume that hyperplanes have probability zero, so that we have an equipartition problem. If d " 2, there are then three unknowns, two for the location parameter and one for the angle of rotation, see also Remark 1 (b) below, and one of the four constraints is redundant, so that the number of unknowns is the same as the number of equations. For d ą 2 we first note that the conditions in (5) and (6) are not independent. Indeed, for a given basis tb 1 , . . . , b d u of R d it is enough to have probability 1{2 for H`pθ, b i q, i " 1, . . . , d, and probability 1{4 for V``pθ; b i , b j q for 1 ď i ă j ď d, so that d`dpd´1q{2 conditions remain. On the other hand, we have dpd´1q{2 parameters for the (special) orthogonal group, and there are d components of the location vector, hence the number of unknowns is again the same as the number of independent constraints. This, of course, is just a heuristic argument, as the respective conditions are nonlinear.
Using a general tool from algebraic topology, the homotopy invariance of the Brouwer degree of continuous mappings, Makeev (2007) solved the equipartition problem under the additional condition that P is absolutely continuous. A smoothing argument leads to the extension needed here.
Theorem 1 A solution pθ, U q of the quarter median problem exists for every probability measure P on pR d , B d q.
This result raises an obvious question: Is there a stronger equipartition in dimension d ą 2, such as an 'octomedian' if d " 3, with each of the octants receiving the same probability? This is certainly possible for specific distributions, such as the multivariate standard normal. However, if d " 3 then we have three unknowns for the location vector, and three for the rotation, which can be parametrized by the Euler angles, and the condition that the octants all have probability 1{8 would similarly lead to seven constraints, which is one too many. Obviously, the difference would be even greater for dimension d ą 3 if all 2 d intersections of coordinate half-spaces were to have the same probability. According to (Makeev 2007, p.554) this dimension counting argument already implies that a solution does not exist for a generic distribution.
The notion of uniqueness requires some attention. We say that a distribution P has a unique quarter median if θ 1 " θ 2 for any two solutions pθ 1 , U 1 q and pθ 2 , U 2 q of the quarter median problem. Obviously, the full pair pθ, U q will never be unique: Any linear hyperplane, i.e. a subspace H of R d of dimension d´1, may be specified as the orthogonal complement of the one-dimensional space (line) generated by a unit vector b P S d´1 via Hpbq :" tx P R d : b t x " 0u, but obviously´b would lead to the same hyperplane. This simple observation already implies that we may restrict U to be an element of SOpdq when searching for a quarter median. Further, the conditions (5) and (6) are invariant under permutations of the rows of U . Conversely, given a set of coordinate hyperplanes, a corresponding basis is a set with elements that are unique up to a factor´1. Putting this together we write U " V for U, V P Opdq if U and V lead to the same set of linear hyperplanes; we then say that the solution of the quarter median problem is unique if for any two solutions pθ 1 , U 1 q and pθ 2 , U 2 q we have θ 1 " θ 2 and U 1 " U 2 .
The following formal approach will turn out to be useful. Let G 1 be the subgroup of Opdq that consists of the diagonal matrices with diagonal entries from t´1,`1u, i.e. the subgroup that represents the compositions of reflections at the coordinate axes, and let G 2 be the subgroup of permutation matrices, which corresponds to coordinate permutations. We write G for the subgroup generated by G 1 and G 2 . This is the system of all dˆd-matrices A with entries a ij P t´1, 0, 1u and such that for some permutation π of t1, . . . , du we have a ij " 0 if and only if j " πpiq. The above transition from equality to equivalence can be then regarded as a transition from the group Opdq to the factor group Hpdq :" Opdq{G, with the equivalence classes rU s :" tV P Opdq : V " U u, U P Opdq, as elements. If we think of a solution of the quarter median problem as a pair pθ, Hq P R dˆH pdq, then uniqueness means that two solutions are equal.
Remark 1 (a) The θ-part of a solution pθ, Hq for the quarter median problem may be unique while the H-part is not. For example, if P has a density that depends on x P R d only through }x} then 0 is the unique quarter median, but all H P Hpdq lead to a solution p0, Hq.
(b) In dimension d " 2 there is an (almost) canonical choice of an element U P Opdq from an equivalence class H P Hpdq corresponding to a set of hyperplanes, here just two orthogonal lines L 1 , L 2 meeting at the point 0 P R 2 . With a suitable half-open interval I of length π{2, such as I " r´π{4, π{4q, there is exactly one α P I such that the lines are represented by which is the counterclockwise rotation of the canonical coordinate lines by the angle α, in the sense that L i " tx P R 2 : b t i x " 0u, where b t 1 , b t 2 are the rows of U . The corresponding topology would then be that of the quotient space R{I.
(c) While we work with the space R d of column vectors throughout the notions of quarter median and quarter median problem make sense for an arbitrary finitedimensional Euclidean space pE, x¨,¨yq. If pθ, U q solves the quarter median problem for a distribution P on pR d , B d q, then tb 2 , . . . , b d u is an orthonormal basis for the linear space E " Hpb 1 q, which has dimension d´1. Here b t i denotes the ith row of U , i " 1, . . . , d; equivalently, b i is the ith column of U t . If d ě 3, this can be used to obtain a projectivity or consistency property that relates pθ, U q to a solution pθ E , U E q of the quarter median problem for the push-forward of P under the orthogonal projection π E : R d Ñ E. In fact, let S E d´1 " ty P E : }y} " 1u be the unit sphere in E. For c P E and orthogonal unit vectors d, d 1 P S E d´1 the associated half-spaces and quarter spaces are given by H É pc; dq :" y P E : xd, y´cy ď 0 ( , H È pc; dq :" y P E : xd, y´cy ě 0 ( , Using that a vector x P R d can uniquely be written as x "

Equivariance
In the context of equivariance properties of location estimators for d-dimensional data we start with a family (often a group) S of measurable transformations S : R d Ñ R d and we regard the location parameter θ P R d as a function θ " T pP q of P . Recall that P S is the push-forward of P under the transformation S. Then T is said to be S-equivariant if T pP S q " pS˝T qpP q for all S P S.
In cases where the parameter is not unique this might better be expressed as Spθq being a parameter of this type for P S if θ is for P . For example, with S being one of the groups G 1 and G 2 that were used in connection with (7), it is easy to check that S´1`MMedmpP S q˘" MMedmpP q for all S P S, i.e. the specific marginal median introduced in (3) is equivariant with respect to reflections and coordinate permutations. It is easily seen that we may even take S " G. This specific marginal median is also equivariant with respect to separate rescalings, where the corresponding group is the set of all regular diagonal matrices. The general idea is of course that statistical inference should respect the structure of the data space. Equivariance of a location estimator with respect to shifts and orthogonal linear transformations means that the estimator interacts in this way with the full isometry group of Euclidean space.
At various stages the following function will be important, with U P Opdq and P a probability measure on pR d , B d q; Ψ pU, P q is the specific marginal median in the coordinate system associated with U , transformed back to canonical coordinates. By definition, a quarter median is always a marginal median in a specific coordinate system. Below, marginal medians in will be of particular interest.
Proposition 1 Let P be a probability distribution on pR d , B d q, and let S be the set of Euclidean motions.
(a) If θ is a quarter median for the distribution P then, for all S P S, Spθq is a quarter median for P S .
(b) If U, V P Opdq are such that U " V then Ψ pU, P q " Ψ pV, P q. (c) If pθ, U q solves the quarter median problem for P , then there is a marginal median η P MpP U q such that pθ, U q withθ " U t η is also a solution.
(d) If pθ, U q solves the quarter median problem for P and S P S, Spxq " Ax`b, then pAθ`b, UA t q solves the quarter median problem for P S . Moreover, if P b has a unique median for all b P S d´1 , Ψ`UA t , P S˘" S`Ψ pU, P q˘.
(e) Suppose that P b has a unique median for all b P S d´1 . Then, the function Ψ p¨, P q : Opdq Ñ R d is continuous.  Part (a) is the equivariance on the set level. Part (b) shows that Ψ p¨, P q may be regarded as a function on Hpdq. Part (c) is of importance when developing an algorithm for searching a solution of empirical quarter median problems: For each U P Opdq the set of potential θ-values in (11) consists of three vectors only, separately for each coordinate. It is tempting to think that this could be reduced further to the respective midpoint, but the uniform distribution on the six points p´2,´2q t , p´1, 3q t , p1,´1q t , p2, 2q t , p3, 4q t , p4, 0q t in the plane provides a counterexample: The pair`p2, 2q t , U˘with U " diagp1, 1q solves the quarter median problem for this distribution, but the midpoint of the associated rectangle r1, 2sˆr0, 2s of multivariate medians would lead to`p1.5, 1q t , U˘, which is not a solution. Part (d) notices the Euclidean motion equivariance of the quarter median for distributions P with the property that P b has a unique median for all b P S d´1 , a condition that will be used repeatedly below.
Similar to the spatial median and the orthomedian introduced by Grübel (1996), the quarter median is not affine equivariant. In particular, it is not equivariant with respect to separate rescalings of the coordinates, in contrast to the special marginal median, but for distributions P with the property that P b has a unique median for all b P S d´1 , equivariance does hold (and is used in the proofs below) for simultaneous rescalings, where the group S consists of all positive scalar multiples of the identity matrix.
Any quarter median is a marginal median in a suitably chosen coordinate system, which we may take to be generated by an orthogonal matrix with determinant 1. Grübel (1996) introduced the orthomedian of P as the integral of the function Ψ p¨, P q over SOpdq with respect to the Haar measure with total mass 1. Thus, the orthomedian may be seen as the expected value of this function with U chosen uniformly at random, and its orthogonal equivariance is then an immediate consequence of the fact that the Haar measure is invariant under the group operations. In contrast, the quarter median picks a marginal median from the range of the set-valued function via the partition constraint (6). Figure 2 shows the range of the function U Þ Ñ Ψ pU, Pnq for the empirical distribution Pn associated with the city data in Figure 1, where n " 76. The parts colored red correspond to possible quarter medians.

Asymptotics
Throughout this section we assume that P is a probability measure on pR d , B d q with the property that P a has a unique median for all a P S d , and that X 1 , X 2 , . . . are independent copies of a random vector X with distribution P . We further assume that, for each n P N, the pair pθn, Unq P R dˆO pdq with θn " θnpX 1 , . . . , Xnq, Un " UnpX 1 , . . . , Xnq, is a solution of the quarter median problem for the (random) empirical distribution Pn associated with X 1 , . . . , Xn. We also assume that θn and Un are measurable with respect to the respective Borel σ-fields on pR d q n . A discussion of the associated selection problem and an extension of the results to more general random elements is given in Section 2.6. We deal with consistency first and then consider asymptotic normality. Recall that uniqueness and convergence refer not to the U -matrices themselves, but to the associated equivalence classes rU s P Hpdq, hence we have to specify what convergence of the U -part means. On Opdq we use the topology induced by R dˆd where a sequence of matrices converges if all entries converge individually. It is well known that this makes Opdq a compact and separable Hausdorff space. We use the quotient topology on Hpdq, which is the finest topology with the property that the mapping SOpdq Ñ Hpdq, U Þ Ñ rU s, is continuous. As we have Hpdq " Opdq{G with some finite group G Ă Opdq, the space Hpdq with this topology is again a compact and separable Hausdorff space.
Theorem 2 Suppose that (13) holds. (a) If the quarter median for P is unique and given by θ P R d , then θn converges almost surely to θ as n Ñ 8.
(b) If the solution of the quarter median problem for P is unique and given by pθ, Hq P S d´1ˆH pdq, then pθn, rUnsq converges almost surely to pθ, Hq as n Ñ 8.
In connection with distributional asymptotics we require (13) and also that P is absolutely continuous. Let Ma " M paq be the (unique) median of P a , a P S d´1 . As in the paper of Grübel (1996) we further assume that P a has a density fa such that pa, tq Ñ faptq is continuous at each pa, tq P D :" and, in addition, that there is a point θ P R d such that Ma " a t θ for each a P S d´1 .
For example, (16) is satisfied if P is symmetric about θ in the sense that pX 1´θ q and´pX 1´θ q have the same distribution. Of course, if (13) and (16) hold then the quarter median for P is unique and given by θ.
Of course, fa and Ma also depend on P . Finally, we write Xn Ñd Q for convergence in distribution of a sequence pXnq nPN of random variables to a random variable with distribution Q, and N d pµ, Σq for the d-dimensional normal distribution with mean vector µ and covariance matrix Σ.
Theorem 3 Suppose that the conditions (13)-(16) hold and that the solution of the quarter median problem for the absolutely continuous probability measure P is unique and given by pθ, Hq. Then, in the above setup, and with U P H, where ΣpP, U q :" U t ∆pP, U qU and ∆pP, U q is given by (17).
As part of the proof we will show that ΣpP, U q does not depend on the choice of the element U of H.

Elliptical distributions
We now consider a special family of distributions in more detail. Let h : R`Ñ R`be a function with the property Then, for each µ P R d and each positive definite matrix Σ P R dˆd , is the density of a probability measure P on pR d , B d q, the elliptical distribution with location vector µ and dispersion matrix Σ. We abbreviate this to P " Ell d pµ, Σ; hq. If µ " 0 and Σ " I d we write Sym d phq for Ell d p0, I d ; hq and speak of a spherically symmetric distribution with the density h`}x} 2˘{ c d phq, x P R d . A transformation to spherical coordinates leads to The function h is said to be the density generator of the elliptical distributions Ell d pµ, Σ; hq. For hptq " expp´t{2q the associated constant is c d phq " p2πq d{2 , and we obtain the multivariate normal distributions N d pµ, Σq. For later purposes we list some properties of elliptical distributions. Clearly, if P " Ell d pµ, Σ; hq and if T : Further, for d ě 2 the univariate marginal densities of P " Sym d phq coincide and are given by The function g is nonincreasing and continuous on p0, 8q. Hence the univariate marginal distributions are unimodal with mode 0 and their density is continuous on Rzt0u; see e.g. (Fang et al 1990, p. 37).
In what follows we deal with generators h that satisfy the condition sup 0ďtď hptq ă 8 for some finite interval r0, s.
Then g and g 1 are bounded and continuous. Additionally, with for k " 1, . . . , d´1, we have 0 ă c k phq ă 8; especially, gp0q " c d´1 phq. Finally, the transformation law mentioned above can be generalized to kˆd-matrices with rank k ă d at the cost of changing the function h. In particular, the first marginal distribution associated with Ell d p0, Σ; hq is Ell 1 p0, σ 11 ; gq if Σ " pσ ij q d i,j"1 , and has a density that is bounded, continuous and strictly positive at 0. More generally, if P " Ell d pµ, Σ; hq then the density fa of P a for a P S d´1 is given by Ma " a t µ is the unique median of P a , fapMaq is positive for all a P S d´1 , and the function pa, tq Ñ faptq is continuous. Hence the assumptions (13) -(16) of Theorem 3 are satisfied. As Σ is positive definite and symmetric the principal axes theorem applies and gives the representation with some U P Opdq and λ 1 ě λ 2 ě¨¨¨ě λ d ą 0, where the λ i 's are the eigenvalues of Σ. The distribution P " Ell d pµ, Σ; hq is said to be strictly elliptical if no two of these eigenvalues coincide. The next theorem deals with the uniqueness of (the solution of) the quarter median (problem) for elliptical distributions. Note that condition (21) is not imposed there.
Theorem 4 If P " Ell d pµ, Σ; hq, then the quarter median of P is unique and given by µ.
Moreover, if P is strictly elliptical, then the solution of the quarter median problem is unique and given by pµ, rU sq with U as in (23).
For samples from strictly elliptical distributions with density generators satisfying (21) we obtain asymptotic normality for a sequence of associated quarter medians θn " θnpX 1 , . . . , Xnq. Part (a) of the following result is essentially a corollary to Theorem 3. In the symmetric case we only have weak uniqueness, hence we need to consider this situation separately. Intermediate cases, where only some of the eigenvalues coincide, can be treated similarly, but require a certain amount of bookkeeping.
Theorem 5 Let Xn, n P N, be independent random vectors with distribution P " Ell d pµ, Σ; hq, with h satisfying (21). Let c d´1 phq, c d phq be as in (19) and let (a) Suppose that pθn, Unq nPN is a sequence of random variables with values in R dˆO pdq such that, for all n P N, pθn, Unq solves the quarter median problem for Pn. Then, if P is (b) Suppose that Σ " λI d for some λ ą 0. Let pUnq nPN be an arbitrary sequence of elements of Opdq and suppose that, for all n P N, θn is a marginal median of P Un n . Then ?
Remarkably, the covariance matrix of the limiting normal distribution always is a fixed multiple of the dispersion matrix Σ. This is in contrast to other familiar estimators of µ such as the empirical spatial median SMedpX 1 , . . . , X 1 q, for example. In fact, with reference to Bai et al (1990) where the asymptotics of the spatial median are considered in an even more general context, Somorčík (2006) states that in the d-dimensional case under some weak conditions where the asymptotic covariance matrix is given by In the special (spherical) symmetric case Σ " I d the identity (27) It was shown by Grübel (1996) that in this symmetric case the empirical orthomedian has the same asymptotic covariance matrix as the spatial median. In what follows, we give a comparison of the covariances of the limit distributions for some estimators of µ, in particular for QMedpX 1 , . . . , Xnq as the empirical quarter median, the sample mean 1 n ř n j"1 X j , and the maximum likelihood estimator MLpX 1 , . . . , Xnq, for some special distribution families, where our main focus is on strictly elliptical distribution families. In all cases, the limit distribution of the standardized estimators is a centered multivariate normal, and by asymptotic covariance we mean the respective covariance matrix.
Example 1 We work out the details for some specific distribution families in the case of dimension d " 2.
(a) For normal distributions P " N 2 pµ, Σq Theorem 5 (a) leads to The maximum likelihood estimator of µ is the sample mean; its (asymptotic) covariance is simply Σ itself, hence the asymptotic efficiency of the quarter median is about 64%. Of course, the quarter median is more robust with respect to gross outliers in the data. In the symmetric case Σ " I 2 both orthomedian and spatial median have asymptotic covariance 4 π I 2 . For non-symmetric two-dimensional normal distributions with Σ " diagp1, λq, 0 ă λ ă 1, Brown (1983) and Grübel (1996) obtained expressions for the asymptotic covariance matrices of the spatial median and the orthomedian. Both are of diagonal form, with the two diagonal entries given by expressions involving infinite series and double integrals respectively. These can be used numerically, see (Brown 1983, Table  1) and (Grübel 1996, Table 1). For the second value, referring to the shorter axis, we have the simple explicit formula λπ{2 for the quarter median, which is smaller than the value for the spatial median if λ ă 0.038, and for the orthomedian if λ ă 0.1.
(c) With hptq " p1`tq´3 {2 we obtain a class of bivariate Cauchy distributions MC 2 pµ, Σq with densities The associated marginal distributions are univariate Cauchy distributions. For this distribution family the sample mean is not even consistent, hence we do not take it into consideration as an estimator for µ. Again we use (Mitchell 1989, formulas (3.2), (3.3)) to see that for given fixed positive definite Σ the Fisher information matrix is 3 5 Σ´1. Thus, by the general asymptotic theory of maximum likelihood estimators, we arrive at the value π 2 4 Σ for the asymptotic covariance matrix of the sample quarter median. In the symmetric case Σ " I d , using (26) and (28), it follows that (d) With hptq " p1´tq 2 for 0 ď t ď 1 and 0 elsewhere we have a special class of symmetric bivariate Pearson type II distributions SMPII 2 pµ, Σq with densities The asymptotic covariance matrix for the sample mean of independent twodimensional random vectors X 1 , . . . , Xn with the density f p¨; µ, Σq is covpX 1 q " 1 see (Fang et al 1990, p. 89). The centered bivariate limit normal distribution of the maximum likelihood estimator has the covariance matrix 1 12 Σ. Using it follows that 25 1024 π 2 Σ is the asymptotic covariance matrix for the sample quarter median. In the symmetric case Σ " I d we obtain (e) Taking hptq " expp´tq p1`expp´tqq 2 , t ě 0, we obtain the symmetric bivariate logistic distributions SML 2 pµ, Σq, with densities given by For X " SML 2 p0, I 2 q the distribution of R 2 " }X} 2 is the univariate half-logistic distribution with density 2 expp´tq p1`expp´tqq 2 , t ě 0. Then, and covpXq " m SML2 I 2 . From this we deduce that the asymptotic covariance matrix for the sample mean of independent two-dimensional random vectors X 1 , . . . , Xn with the above density f p¨; µ, Σq is covpX 1 q " m SML2 Σ. With the Fisher information matrix is k SML2 Σ´1, and the centered bivariate limit normal distribution of the maximum likelihood estimator has the covariance matrix k´1 SML2 Σ. Further, with c 1 phq " we obtain that c2phq 2 p2c1phqq 2 Σ « 1.35901156 Σ is the covariance matrix of the centered limiting normal distribution of the quarter median. In the symmetric case Σ " I d we get In contrast to the other multivariate versions of the median that appear in the above example the quarter median goes beyond providing a location estimate as it comes with an equipartition basis. In the elliptical case P " Ell d pµ, Σ; hq the basis elements are the unit vectors proportional to the eigenvectors of Σ and thus contain information about the dispersion parameter. In Theorem 3 the asymptotic behavior of pUnq nPN is important. In the strictly elliptical case we have consistency and we may assume that convergence takes place in SOpdq; see also the beginning of the proof of Theorem 3. We aim at a second order statement about the distributional asymptotics of this sequence.
In the following theorem we only consider d " 2, but see the ensuing remarks. We can then use the parametrization of SOp2q given in Remark 1 (b) by an angle from an half-open interval I of length π{2. We assume that U " Uα with I chosen such that α is in its interior. Then consistency implies that Un " Uα n with I-valued random variables αn that converge to α with probability 1. It then makes sense to consider the distributional asymptotics of ? npαn´αq as n Ñ 8.
In particular, the asymptotic variance in (34) is large if the eigenvalues are close to each other. This is to be expected as Ell 2 pµ, Σ; hq is then close to a symmetric distribution, where U is not unique.
In dimensions higher than two a possible approach could be based on the representation of the Lie group SOpdq by its Lie algebra sopdq, which consists of the skew-symmetric matrices A P R dˆd , where U P SOpdq is written as the matrix exponential U " exppAq of some A P sopdq; see (Muirhead 1982, Theorem A9.7) for a proof of the latter assertion, and Hall (2004) for an elementary introduction to Lie groups, Lie algebras, and representations. This leads to an asymptotic distribution Q for pU t Unq ? n as n Ñ 8, where Q is a probability measure on (the Borel subsets of) SOpdq, and it avoids any ambiguities caused by parametrization. In fact, Q is the distribution of the random orthogonal matrix exppAq, where A is a random skew symmetric matrix with jointly normal subdiagonal entries. If d " 2, as in the above theorem, then Uα " exp`Apαq˘with A :"ˆ0´α α 0˙, and A is then specified by its one subdiagonal entry, which has a central normal distribution with variance given by (34).

Measurability, selection and algorithms
Let S be a topological space with Borel σ-field BpSq. We write C b pSq for the set of bounded continuous functions f : S Ñ R. Let Z, Z 1 , Z 2 , . . . be S-valued random variables, i.e. BpSq-measurable functions on some probability space (which may depend on the respective variable). In its classical form, convergence in distribution Zn Ñd Z of Zn to Z as n Ñ 8 means that lim nÑ8 Ef pZnq " Ef pZq for all f P C b pSq.
This is the notion that we used in Section 2.4. An extension of this concept, due to Hoffmann-Jørgensen, can be applied if the Zn's are not measurable with respect to the full Borel σ-field on S; roughly, the expectations Ef pZnq in (35) are then replaced by outer expectations E ‹ f pZnq. Similarly, almost sure convergence now refers to outer probabilities. The research monographs of Dudley (1999) and van der Vaart and Wellner (1996) give an in-depth treatment of this circle of ideas, together with a variety of applications. This extension appears in our proof of Theorem 3 in connection with empirical processes; it can also be used if the estimators pθn, Unq are not measurable. We refer the reader to the paper of Kuelbs and Zinn (2013), where this is worked out in detail for the related situation of quantile processes. While this avoids the problem of choosing an estimator from the respective solution set in a measurable way, it is of independent interest whether such a selection is possible. For the classical one-dimensional median and the marginal median this can obviously be done by choosing the midpoint of the respective (component) interval, but no such simple rule seems to exist for the quarter median.
We think of estimators as functions of the random variables X i , i P N, and it is then enough to establish measurability for these functions. To be precise, for a given n P N and an n-tuple px 1 , . . . , xnq P pR d q n we denote by`θnpx 1 , . . . , xnq, Unpx 1 , . . . , xnq˘P R dˆS Opdq a solution of the quarter median problem for the empirical distribution Pn;x 1 ,...,xn " n´1 ř n k"1 δx k .
Permutation invariance means that the estimates depend on the data only through the respective empirical distribution.
Selecting a solution in a measurable way from the respective set of all solutions may seem as a corollary to establishing an algorithm that returns an estimate for every data input x 1 , . . . , xn. Indeed, without such an algorithm a statistical procedure would seem to be of limited use. In dimension two, and using the parametrization in Remark 1 (b), we obtain a real function on the interval r0, π{2q by counting the number of data points in the upper right quadrant specified by Uα and MMedmpP Uα q; see Figure 3 for a plot of this function for the city data in Figure 1, where the counts refer to the open quarter space. By construction, all half-spaces contain at least half of the data points, so each angle that leads to a count of n{4 (if n is divisible by 4) would give a quarter median. For absolutely continuous distributions this approach, together In what follows, still considering the case d " 2, we present a different algorithm. Let x 1 , . . . , xn be n ě 2 pairwise distinct data points in R 2 . For 1 ď i ă j ď n let b ij " xj´xi }xj´xi} P S 1 be the direction of the line through x i and x j , and let b 1 ij P S 1 be orthogonal to b ij . With the empirical distribution Pn " n´1 ř n k"1 δx k we associate the finite set L pPnq of pairs pθ ij , U ij q, with U ij P Op2q as the matrix with row vectors b t ij and b 1 ij t , and θ ij " U t ij η with some η P M`P Uij n˘.
Theorem 7 There is an element pθ ij , U ij q P L pPnq that solves the quarter median problem for Pn.
Hence, checking successively the conditions (5) and (6) for all`θ ij , U ij˘P L pPnq, 1 ď i ă j ď n, provides a solution of the quarter median problem for Pn. This procedure, however, may lead to an estimator that is not permutation invariant in the sense of (36), meaning that it would not be a function of the empirical distribution Pn. For our introductory data example, see Figure 1, this happens for both of the above algorithms: Figure 3 shows that different solutions appear depending on whether the angles are scanned clockwise or counterclockwise. Similarly, the algorithm based on Theorem 7 could lead to the pi, jq-pair corresponding to the cities Berlin and Frankfurt am Main, or München and Dortmund, for example.

Numerical comparison
We present a small simulation study on the performance of the quarter median (QMed) as a location estimator, comparing it with that of three other proceduresthe spatial median (SMed); Oja's simplicial median (OMed), see Oja (1983); and Tukey's half-space median (TMed), see Tukey (1975). For a detailed description of these classical estimators we refer to the survey papers of Small (1990) and Oja (2013).
A Monte Carlo study on the performance of the three (and some other bivariate location) estimators is given by Massé and Plante (2003).
Samples are drawn from the strictly elliptical distributions N 2 pµ, Σq (bivariate normal), MDE 2 pµ, Σq (bivariate doubly exponential), SMPII 2 pµ, Σq (symmetric bivariate Pearson type II), and SML 2 pµ, Σq (symmetric bivariate logistic) introduced in Example 1. The bivariate Cauchy distributions MC 2 pµ, Σq also discussed there are not taken into consideration for the simulation study, as the Oja median does not exist for this distribution family. In fact, for a given distribution P on R 2 , independent X j " P , j " 1, 2, and θ P R 2 , let ∆pX 1 , X 2 , θq be the area of the triangle in R 2 with vertices X 1 , X 2 , θ. If the expectation γ P pθq :" E p∆pX 1 , X 2 , θqq is finite for all θ P R 2 , the Oja median is a point θ Oja pP q P R 2 that minimizes the function θ Þ Ñ γ P pθq. Obviously, γ P " 8 for P " MC 2 p0, Σq, while for the other elliptical distributions P under consideration θ Oja pP q " µ.
Without loss of generality we take µ " 0. The dispersion matrix chosen is Σ "ˆ1 0 0 λ˙, with λ P t0.01, 0.1, 0.5, 0.9u. Repeatedly, with r " 10000 replications, samples of size n " 100 are drawn from the underlying distribution. Let SMed n,i , OMed n,i , TMed n,i , QMed n,i be the observed values of the corresponding estimators obtained in the ith repetition. Table 2 shows the numerical values of the componentsm 1 ,m 2 of the empirical means and the numerical values of the eigenvaluesl 1 ,l 2 of the empirical covariance matrices of ? nSMed n,i , ? nOMed n,i , ? nTMed n,i , ?
nQMed n,i , i " 1, . . . , r. The simulations are conducted by using the statistical software environment R; the calculation of the estimators SMed, OMed, and TMed is easily done by applying the corresponding functions med(...,method="Spatial"), med(...,method="Oja",...) and TukeyMedian(...) provided with the additional R software packages depth and TukeyRegion. Note that, in the later package, the Tukey median is defined to be the barycenter of the Tukey region.   Oja and Niinima (1985), and Σ QMed "ˆπ {2 0 0 πλ{2˙. The limit distribution of TMed is the distribution of arg sup tPR 2 inf uPS1´Z puq´1 2πλ 1{2 u t t¯, where pZpuq, u P S 1 q is a centered Gaussian process with the covariance function cov pZpuq, Zpvqq " P pV``p0; u, vq´1{4, u, v P S 1 ; see Nolan (1999). To the best of our knowledge, nothing is known about the type of this distribution. In this respect, see for example also the interesting results on the asymptotic behaviour of the empirical Tukey depth process given by Massé (2004). Table 3 shows the values ofl 1 ,l 2 and, for comparison, that of the diagonal elements l 1 , l 2 of Σ SMed , Σ OMed , and Σ QMed .

Proof of Theorem 1
Using Makeev's result, we only need to remove the smoothness assumption.
For an arbitrary probability measure P on pR d , B d q let P " P˚N d p0, I d q be the convolution of P with a centered d-dimensional normal distribution with independent components that all have variance ą 0. Then P is absolutely continuous with an everywhere positive density, so that a solution`θ , U q of the equipartition problem exists. Let~¨~be a matrix norm on R dˆd that is compatible with the Euclidean norm }¨} on R d . From θ " U t U θ it then follows that }θ } ď~U t ~}U θ } ď s}U θ }, where s " sup U PSOpdq~U t~ă 8. Let t " lim sup Ñ0 }U θ }. Hence there exists a sequence p nqnPN with n Ñ 0 such that }U n θ n } Ñ t and U n Ñ U P Opdq as n Ñ 8. As a consequence, P U n n Ñ P U weakly. Due to the fact that U n θ n is a marginal median of P U n n we deduce from this that the sequence p}U n θ n }q 8 n"1 is bounded; hence, by (37), the sequence pθ n q 8 n"1 is also bounded. Therefore, θ n 1 Ñ θ P R d and U n 1 Ñ U P Opdq along some subsequence p n 1 q of p nq. To see that pθ, U q solves the quarter median problem for P we use that P U n 1 n 1 Ñ P U weakly and argue as follows: With b t n 1 ,1 , . . . , b t n 1 ,d and b t 1 , . . . , b t d as the row vectors of U n 1 and U, respectively, we have that, for each 1 ď i ă j ď d and each δ ą 0, 1 4 ď lim sup n 1 Ñ8 P n 1`V`´p θ n 1 ; b n 1 ,i , b n 1 ,j qď lim sup Let δ Ó 0 to obtain P`V`´pθ; b i , b j q˘ě 1 4 . The other inequalities stated in (5) and (6) can be verified in the same way.

Proof of Proposition 1
(a) Let θ be a quarter median for P with b 1 , . . . , b d an associated orthonormal basis of R d such that the constraints (5) and (6) are satisfied. Further, let Spxq " Ax`c with A P Opdq, c P R d , be a Euclidean motion. Then b 1 is again an orthonormal basis for R d , and it is easily checked that In view of the definition of push-forwards this implies that Spθq is a quarter median for P S , with b 1 1 , . . . , b 1 d an associated equipartition basis.
where we have used (9) and G Ă Opdq.
(c) Let b t 1 , . . . , b t d be the row vectors of U . With θ U " pθ U,1 , . . . , θ U,d q t :" U θ " pb t 1 θ, . . . , b t d θq t it holds that θ " U t θ U " ř d i"1 θ U,i b i and that, by definition, θ U is a marginal median of P U . We can now choose an elementθ U " pθ U,1 , . . . ,θ U,d q t P M´P U¯s uch that or 1 ď i ă j ď d. Thus, pθ, U q solves the quarter median problem for P .
(d) The first statement may be regarded as an equivariance property of solution pairs for the quarter median problem; its proof proceeds as in (a). For the second statement we note that A P Opdq and then calculate, using the shift equivariance of the marginal median, (e) Let U P Opdq and pUnq 8 n"1 be a sequence of elements in Opdq converging to U. Due to P Un Ñ P U weakly and the fact that P b has a unique median for all b P S d , it follows that MMedmpP Un q Ñ MMedmpP U q. Then, from and the compactness of Opdq we deduce that U t n MMedmpP Un q Ñ U t MMedmpP U q.

Proof of Theorem 2
We will make use of the fact that if mn, n P N, are medians of univariate distributions that converge weakly as n Ñ 8 to a univariate distribution with unique median m, then limnÑ8 mn " m. Further, we only prove the second part of the theorem as the arguments for (a) are similar. Let pΩ, A, Pq be a probability space on which the random vectors X 1 , X 2 , . . . are defined, so that P Xj " P for all j P N. Then there is a P-null set N P A such that for each ω P N c Pn;ω :" 1 n n ÿ j"1 δ Xj pωq Ñ P weakly.
Fix ω P N c ; until further notice we omit the argument ω below. By definition, the quarter median θn of Pn can be written as θn " U t n ηn with some Un P Opdq and some ηn P r MMed´pP Un n q, MMed`pP Un n qs (both may depend on ω P N c ). Let pU n 1 q be a subsequence of pUnq. As Opdq is compact a subsubsequence pU n 2 q converges to some V P Opdq, and we have weak convergence P U n 2 n 2 Ñ P V by the extended continuous mapping theorem; see (Billingsley 1968, Theorem 5.5). As the distribution P b has a unique median for all b P S d´1 , the marginal median η 2 n of P U n 2 n 2 converges to the marginal median η of P V . Now fix i, j with 1 ď i ă j ď d. Let a 2 n , b 2 n , a, b be the corresponding rows in U 2 n and V respectively. Using the same arguments as in the proof of Theorem 1 we then obtain for each ą 0, and with m instead of n 2 , where in the first line we have used that pθm, Umq solves the quarter median problem for Pm. Letting Ó 0 we get P pV´`pη; a, bqq ě 1{4. More generally, and using the same argument, we obtain P`V˘˘pη; a, bq˘ě 1{4 as well as P`H˘pη; aq˘ě 1{2 and P`H˘pηpωq; bq˘q ě 1{2. Recall that some of the quantities depend on the ω chosen above so that, for example, P`H`pη; aq˘" P`H`pηpωq; apωqq˘" P`tx P R d : xapωq, x´ηpωqy ě 0u˘.
In summary, we have shown that every limit point pηpωq, V pωqq of the sequence pθnpωq, Unpωqq nPN is a solution of the quarter median problem for P . Hence, if the solution is unique and given by pθ, Hq P R dˆH , then, on a set of probability 1, η " θ and rV s " H.

Proof of Theorem 3
Invariance with respect to shifts means that we may assume θ " 0. As already noticed in Section 4.3, the quarter median θn can be written as θn " U t n ηn with ηn P r MMed´pP Un n q, MMed`pP Un n qs. Theorem 2 implies that prUnsq nPN converges almost surely to H with respect to the quotient topology on Hpdq. The mapping p : Opdq Ñ Hpdq, U Þ Ñ rU s, associates to each H P Hpdq a finite fiber p´1ptHuq, and Opdq is a covering space for Hpdq. General results from algebraic topology, see e.g. (tom Dieck 1991, Section II.6), imply that p may locally be inverted to obtain homeomorphisms to open subsets of the individual sheets of the covering space. In particular, we may assume that the matrices Un converge in Opdq to some U P H almost surely as n Ñ 8.
For each a P S d´1 and n P N let Mn paq :" Med˘`P a n˘, and let M paq be the uniquely determined median of P a . We define two stochastic processes Yn "`Yn paq, a P S d´1w ith index set S d´1 by Yn paq " ?
n`Mn paq´M paq˘for all a P S d´1 , and will use results presented by Grübel (1996) and adopt arguments used by Kuelbs and Zinn (2013) to prove that Yn Ñd Y as n Ñ 8.
Here Y " pYa, a P S d´1 q is a centered Gaussian process with continuous paths and covariance function covpYa, Y b q " with We regard the Yn as processes with paths in 8pSd´1q, the space of real-valued bounded functions on S d´1 endowed with the supremum norm, and the symbol 'Ñd' refers to weak or distributional convergence in the sense of Hoffmann-Jørgensen; see e.g. van der Vaart and Wellner (1996) and Dudley (1999) for details. For the proof of (38) we start with a functional central limit theorem for the empirical processes and then use an almost sure representation in order to be able to work with individual paths in an -δ style.
Let F pa,¨q :" Fap¨q be the distribution function of a t X and let Fnpa,¨q " Fa,np¨q " 1 n n ÿ j"1 1pa t X j ď¨q be the (random) distribution function associated with P a n , a P S d´1 . Here 1p¨q denotes the indicator function of its (logical) argument. We introduce the empirical process Zn " p ?
n pFnpa, tq´F pa, tqq , pa, tq P S d´1ˆR q as a process with paths in 8pSd´1ˆRq. We obtain a semimetric d on S d´1ˆR via d`pa, sq, pb, tq˘2 " E´`1pa t X ď sq´P pa t X ď sq˘´`1pb t X ď tq´P pb t X ď tq˘¯2, pa, sq, pb, tq P S d´1ˆR . As noticed in (Grübel 1996, Proof of Theorem 1), we then have Zn Ñd Z 0 , where Z 0 " pZ 0 pa, tq, pa, tq P S d´1ˆR q is a centered Gaussian process with covariance function E`Z 0 pa, sqZ 0 pb, tq˘" E´`1pa t X ď sq´P pa t X ď sq˘`1pb t X ď tq´P pb t X ď tq˘¯, pa, sq, pb, tq P S d´1ˆR , and sample paths that are bounded and continuous with respect to d. The assumptions on P ensure that the covariance function is continuous with respect to the usual topology on S d´1ˆR . This implies that the process Z 0 has continuous sample paths.
For the remainder of the proof we abbreviate 'almost surely' to 'a.s.', and convergence refers to n Ñ 8 unless specified otherwise.
As Fn is a deterministic function of Zn it follows thatFn :" Fn˝gn has the same distribution underP as Fn under P, n P N. This extends to the associated onedimensional projections and their medians, and to the Yn-processes, whereMn paq :" Med˘`Fnpa,¨q˘andỸn :" Yn˝gn, n P N.
For the integral of the marginal medians needed by Grübel (1996) this was enough, but here we need the stronger stronger statement sup aPS d´1ˇỸn paq´Ỹ paqˇˇÑ 0P-a.s..
For the proof of (43) we adopt some arguments given in (Kuelbs and Zinn 2013, Proof of Proposition 1). First we note that F`a, M paq˘" 1{2 and then write F`a, M paq`h˘´1{2 " fa`M paq˘h`rpa, hqh, where rpa, hq " fa`M paq`ζ a,h h˘´fa`M paq˘with some ζ a,h P r0, 1s. Using compactness of S d´1 we see that the assumption (14) Absolute continuity of P implies that samples from P are in general position with probability 1, meaning that at most d points are on a hyperplane. Hence at most d of the variables a t X 1 , . . . , a t Xn coincide, and thus, given that the range ofFn " Fn˝gn is contained in the range of Fn,ˇF n`a,Mn paq˘´1{2ˇˇď d{n .
with probability 1. Using this and (45) we get sup aPS d´1ˇ?

From
?
n´F`a,Mn paq˘´1{2¯"Ỹn paq´fa`M paq˘`r`a,Mn paq´M paq˘¯, which holds by (44), we obtain that,P-a.s., paq´fa`M paq˘`r`a,Mn paq´M paq˘¯`Z 0`a , M paq˘ˇˇÑ 0.
Thus, sup aPS d´1ˇỸn paqfa`M paq˘`Z 0`a , M paq˘ˇˇďÃn`Bn Ñ 0P-a.s., which finishes the proof of (43) as a Þ Ñ fapM paqq is bounded away from 0 on S d´1 . Returning to the un-tilded variables we see that this establishes the functional limit theorems in (38).
This shows that the volume of the interval r MMed´pP Un n q, MMed`pP Un n qs shrinks fast enough so that the choice of a vector from this interval is irrelevant for the distributional asymptotics. Now let bnpiq t and bpiq t be the ith row of Un, n P N, and U respectively, for i " 1, . . . , d. In view of Un Ñ U in Opdq we have bnpiq Ñ bpiq P-a.s. for i " 1, . . . , d. Let Wn :" ?
nηn " Un ? npθn´θq and Wn :" Note that Wn is an element of the d-dimensional interval rWń , Wǹ s with probability 1. As the paths of the limit process are continuous the functional limit theorems (38) now imply Wn Ñd W , hence with W " N d p0, Ξq, where the entries of Ξ can be calculated from (39) and (40). Indeed, as M " 0 and as p0, U q solves the quarter median problem for P , we obtain Kpb i , b i q " 1{4 and Kpb i , b j q " 0 if i " j, so that Ξ " ∆pP, U q with ∆pP, U q the diagonal matrix given in (17).
We now apply U t n on the left hand side, U t on the right hand side of (47) and use (Billingsley 1968, Theorem 5.5) again to obtain the claimed asymptotic normality of ? npθn´θq, together with the mean and covariance matrix of the limit distribution. It remains to show that ΣpP, U q does not depend on the choice of U from the set H. This follows easily from the fact that, for diagonal matrices ∆, we always have P t ∆P " ∆ if P corresponds to the permutation of two rows or the multiplication bý 1 of some row. For the proof the assumption (16) is important. Without such an assumption we would need the decomposition of ? n`Mn pbnpiqqq´M pbpiqq˘into two terms ?
n`Mn pbnpiqqq´M pbnpiqq˘and ?
n`M pbnpiqqq´M pbpiqq˘. The first could still be analyzed with the above arguments, but for the second term the local behaviour of the function a Þ Ñ M paq and distributional asymptotics of the estimates Un would become important; see also Theorem 6.
Further, (46) and the ensuing comment show that the original estimate θn need not be permutation invariant. This may seem surprising as the proof relies on empirical process theory, which deals with the data through their empirical distribution. Here it turned out to be enough that the estimate can be squeezed between two other estimates that are such functions of the empirical distribution.

Proof of Theorem 4
The transformation law for d-dimensional elliptical distributions, together with the equivariance from Proposition 1 (a), means that we may assume that µ " 0 and that Σ " diagpλ 1 , . . . , λ d q. Recall that e 1 , . . . , e d is the canonical basis of R d . For a subset I of t1, . . . , du let The assumptions imply that P T " P for all reflections T : R d Ñ R d about hyperplanes Hpe i q, hence P pA I q does not depend on I; also, ř IĂt1,...,du P pA I q " 1. A quarter space such as tx P R d : x t e i ě 0, x t e j ě 0u with i " j can be written as the union of 2 d´2 such sets A I , where the intersections are all P -null sets, hence all of these have probability 2 d´2 2´d " 1{4. This shows that p0, Σq solves the quarter median problem for P .
In order to see that the quarter median is unique it is enough to show that MMedpP U q " 0 for all U P Opdq under the above assumptions on P . This follows from the fact that Ell d p0, Σ; hq is invariant under the reflection x Þ Ñ´x, x P R d .
It remains to show that the solution of the quarter median is unique in the strictly elliptical case. To prove this, we recall that µ " 0 and that Σ " diagpλ 1 , . . . , λ d q.
Assume that p0, rU sq with U P Opdq is a solution of the quarter median problem. Let b t 1 , . . . , b t d be the row vectors of U . Then P U " Ell d p0,Σ; hq, whereΣ "`σ ij˘1 ďi,jďd " U ΣU t . With U ij as the 2ˆd matrix with the rows b t nd density generator h 2 which is equal to h if d " 2 and given by h 2 puq " π d{2´1 Γ pd{2´1q ż 8 u pt´uq d{2´2 hptq dt for u ą 0, if d ě 3; see, e.g. (Fang et al 1990, Section 2.2.3). According to (Grübel 1996(Grübel , p. 1466 the probability assigned by P ij to the left lower quadrant is given by which is equal to 1 4 if and only ifσ ij " 0, i.e. if and only ifΣ ij is a diagonal matrix with positive diagonal elements. Therefore,Σ " U ΣU t is a diagonal matrix,Σ " diagpλ 1 , . . . ,λ d q say, or equivalently, Σ " U tΣ U " U t diagpλ 1 , . . . ,λ d qU . It follows from this that the set of eigenvalues λ 1 , . . . ,λ d ( ofΣ and the set of eigenvalues λ 1 , . . . , λ d ( of Σ coincide and, as a consequence, that there is a permutation matrix Π such that Π t ΣΠ "Σ, which in turn implies U t Π t ΣU Π " Σ. Thus, putting U t Π t ": Q " pq ij q we have QΣ " ΣQ t , i.e. q ij λ j " λ i q ij . As the λ j are pairwise distinct this gives q ij " 0 for i " j. Thus Q is a diagonal matrix with the entries˘1 in the diagonal so that U " I d . Taken together this shows that p0, rI d sq is the unique solution of the quarter median problem. 4.6 Proof of Theorem 5 (a) We first assume that µ " 0 and that Σ " diagpλ 1 , . . . , λ d q. Using Theorem 3 we only need to evaluate the diagonal elements of ∆pP, U q, and because of Theorem 4 we may take U to be the identity matrix. In particular, b i " e i for i " 1, . . . , d. Then fe i is the ith marginal density of P , and the calculations preceding Theorem 4 lead to Putting this together we see that, for such P , the asymptotic covariance matrix is indeed the specified multiple of Σ. We now use the equivariance properties of the quarter median to 'bootstrap' this to general strictly elliptical distributions. Hence suppose that P " Ell d pµ, Σ; hq and that λ 1 ą λ 2 ą¨¨¨ą λ d are the eigenvalues of Σ. Then, for some U P Opdq, Σ " U t diagpλ 1 , . . . , λ d qU . If X i , i P N, are independent with distribution P then the random variables Y i :" U pX i´µ q, i P N, are independent with distribution Ell d p0, diagpλ 1 , . . . , λ d q; hq. As U pθn´µ˘is a quarter median of Y 1 , . . . , Yn the first part of the proof leads to ? n U`θn´µ˘Ñd N d p0, Ξq, with Ξ " σ 2 QMed diagpλ 1 , . . . , λ d q. Using the well-known properties of weak convergence and the behavior of multivariate normal distributions under affine transformations we finally obtain (24).
(b) Suppose that the eigenvalues of Σ coincide and let pUnq nPN be as in the statement of the second part of the theorem. Then, for any subsequence pU n 1 q there exists a subsubsequence pU n 2 q such that U n 2 Ñ U 0 for some U 0 P Opdq as n 2 Ñ 8. With m for n 2 and bmpiq t , bpiq t the ith row of Um respectively U 0 , and using the same arguments as for (47), we obtain The symmetry of P implies that the limit distribution does not depend on U 0 .

Proof of Theorem 6
As the result is somewhat tangential to the topic of multivariate location estimation we only provide the main ideas together with some arguments specific to the quarter median application.
The general approach consists in writing the parameter θ " θpP q as the solution of an equation Ψ P pθq " z with a suitably chosen function Ψ P and known z. Let Ψn be the empirical version of Ψ P and suppose that Ψnpθnq sufficiently close to z for large n P N. Then, under certain conditions, properties of the estimator sequence pθnq nPN can be obtained by localizing Ψ P near θ and then using the delta method. An excellent general exposition of this approach can be found in the textbook of van der Vaart (1998), see also van der Vaart (2002). Specifically, (van der Vaart 1998, Theorem 5.9) together with (van der Vaart 2002, Theorem 6.17) may serve as a guide towards filling in the details. Ultimately, this would also lead to an alternative proof for Theorem 5 in the case d " 2 that does not make use of the results presented by Grübel (1996).
For a, b ą 0 let Epa, bq :" ! px, yq t P R 2 : x 2 a 2`y 2 b 2 " 1 ) " pa cosptq, b sinptqq t : 0 ď t ă 2π ( be the boundary of the two-dimensional centered ellipse with main half-axes parallel to the unit vectors e 1 and e 2 and of length a and b respectively. On this set we consider the push-forward Qpa, bq :" unifpS 1 q T a,b of the uniform distribution on S 1 " Ep1, 1q under the mapping T a,b : R 2 Ñ R 2 , px, yq t Þ Ñ pax, byq t . If a " b then the solution of the quarter median problem for Qpa, bq is unique and given by µ " p0, 0q t and U " diagp1, 1q " Uα with α " 0. Let P be a probability measure on the Borel subsets of R 2 . We introduce the function Ψ P : RˆRˆr´π{4, π{4q Ñ R 3 , where the components Ψ P,i are the respective probabilities of the upper right, upper left and lower left quarter spaces associated with the argument. Formally, and with bα, b 1 α the columns of U t α , Ψ P,1 px, y, αq " P`V``ppx, yq t ; bα, b 1 α q˘, and Ψ P,2 px, y, αq " P`V´`p¨¨¨q˘, Ψ P,3 px, y, αq " P`V´´p¨¨¨q˘. Suppose that P " Qpa, bq with a " b. Then elementary geometric considerations lead to the following matrix of partial derivatives of Ψ P at px, y, zq " p0, 0, 0q, In order to obtain the derivative for more general P we use some structural properties of symmetric and elliptical distributions. Note that Qpr, rq is the uniform distribution on the boundary of the sphere with radius r. If X " Sym 2 phq then }X} has density f }X} prq " 2π c 2 phq r hpr 2 q, r ą 0, and given }X} " r, X is uniformly distributed on tx P R 2 : }x} " ru. Taken together this implies the mixture representation Sym 2 phq " 2π Qpr, rq r hpr 2 q dr.
Inserting quarter spaces we obtain a relation between Ψ P and the Ψ -functions corresponding to the measures Qpa, bq, a, b ą 0. This in turn can be used to relate the derivative DpP q of Ψ P at p0, 0, 0q to the derivatives obtained earlier for Qpa, bq, and we arrive at If px, y, αq solves the quarter median problem for an absolutely continuous P , then Ψ P px, y, αq " p1{4, 1{4, 1{4q. For a sample of size n from P the random vector of observation counts in the quarter spaces associated with the solution of the quarter median problem for P has a multinomial distribution with parameters n and p1{4, 1{4, 1{4, 1{4q. The multivariate central limit theorem shows that the standardized counts in the three quarter spaces corresponding to the components of Ψ are asymptotically normal with mean vector 0 and covariance matrix We may now apply the delta method to obtain asymptotic normality for the triplet consisting of the quarter median coordinates and the rotation angle for a sequence of independent random variables with distribution P " Ell 2`p 0, 0q t , diagpa 2 , b 2 q; h˘. The limiting normal distribution is centered and has covariance matrix Σ 1 " pDpP q´1q t Σ 0 DpP q´1 " c 2 phq 2 4c 1 phq 2¨a It remains to extend this to P " Ell 2 pµ, Σ; hq. We use essentially the same argument as at the end of Section 4.6 (a). By assumption and Theorem 4, we have that Σ " U t α diagpλ 1 , λ 2 qUα, where Uα is the matrix representing the rotation by the angle α in the interior of the half-open interval I, see (8), and that pµ, Uαq is the unique solution of the quarter median problem for P . Theorem 2 implies that αn´α P p´π{4,`π{4q with probability 1 for n sufficiently large. Let Y i :" U t α pX i´µ q, i P N. Then, with probability 1 for n large enough,´U t α pQMedpX 1 , . . . , Xnq´µq, U αnpX1,...,Xnq´α¯i s a solution of the quarter median problem for Y 1 , . . . , Yn. Noting that λ 1 " a 2 and λ 2 " b 2 we may now apply the above asymptotic normality result to the Y -sequence to obtain U t α 0 0 1˙ˆ? n`QMedpX 1 , . . . , Xnq´µ? n`αnpX 1 , . . . , Xnq´α˘˙Ñ d N 3 p0, Σ 1 q.
which is the statement of the theorem.

Proof of Proposition 2
We denote by C the set of non-empty closed subsets of R dˆS Opdq. For x 1 , . . . , xn P R d let Γ px 1 , . . . , xnq be the (by Theorem 1 non-empty) set of solutions pθ, U q P R dŜ Opdq of the quarter median problem for Pn;x 1 ,...,xn . As permutations of x 1 , . . . , xn do not change Pn;x 1 ,...,xn we may regard Γ as a function on E :" pR d q n { ", where px 1 , . . . , xnq, py 1 , . . . , ynq P pR d q n are equivalent if y i " x πpiq for i " 1, . . . , n for some permutation π of the set t1, . . . , nu. We aim to prove that the function Γ has values in C and that it is measurable in the sense that the set is a Borel set for each C P C. Then, by the measurable selection theorem of Kuratowski and Ryll-Nardzewski (see, e.g. (Rockafellar 1976, 1.C Corollary)) there exists a measurable function τ : E Ñ R dˆS Opdq such that τ pzq P Γ pzq for all z P E. In fact, we will prove that the set ApCq is closed whenever C P C.
Arguing as in the first part of the proof of Theorem 1 in Section 4.1 we see that if pθ , U q PN is a sequence of elements in Γ pzq converging to pθ, U q P R dˆS Opdq as Ñ 8, then pθ, U q P Γ pzq. This shows that the function Γ has values in C. Now, more generally, fix C P C, let ApCq be as in (50), and let pz q PN be a sequence of elements of ApCq that converges to some element z P E as Ñ 8. As pointed out in connection with the transition from Opdq to Hpdq in Section 4.4 we can find px p q 1 , . . . , x p q n q P z , l P N, and px 1 , . . . , xnq P z such that px p q 1 , . . . , x p q n q converges to px 1 , . . . , xnq P z in the space pR d q n . Then for each P N there exists an associated element pθ , U q of Γ px p q 1 , . . . , x p q n q. Arguing as in Section 4.1 again, we see that along some subsequence p 1 q it holds that pθ 1 , U 1 q Ñ pθ, U q P R dˆS Opdq and P U 1 n;x p 1 q 1 ,...,x p 1 q n Ñ P U n;x1,...,xn weakly.
With b t 1 , . . . , b t d as the row vectors of U this gives Pn;x 1 ,...,xn`V˘˘pθ; b i , b j q˘ě 1 4 . Consequently, pθ, U q P Γ pzq; also, z P ApCq as C is closed.

Proof of Theorem 7
Let θ be a quarter median of Pn, with associated orthogonal directions b, b 1 P S 2 . Let L θ,b " tθ`tb : t P Ru, L θ,b 1 " tθ`tb 1 : t P Ru be the axes of the rectangular coordinate system with origin θ and directions b, b 1 . We consider two cases.
(i) θ P tx 1 , . . . , xnu. We may assume without loss of generality that θ " x 1 , and define ψ " min`min arccosp|b t 1j b|q : 2 ď j ď n ( , min arccosp|b t 1j b 1 |q : 2 ď j ď n (˘. There are orthogonal directions b˚, b 1 P S 1 obtained from b, b 1 by the same clockwise or counterclockwise rotation with the angle ψ, where b˚or b 1 is an element of t´b 12 , b 12 , . . . ,´b 1n , b 1n u. By construction, the number of data points x j lying in V˘˘pθ; b˚, b 1 q is greater than or equal to the number of data points x j lying in V˘˘pθ; b, b 1 q; also the number of data points x j lying in H˘pθ; b˚q and the number of data points x j lying in H˘pθ; b 1 q is greater than or equal to the number of data points x j lying in H˘pθ; bq and the number of data points x j lying in H˘pθ; b 1 q, respectively. It follows that pθ, U˚q with U˚as the matrix with the row vectors b˚, b 1 solves the quarter median problem for Pn. Thus by Proposition 3 (c) there is an η P M`P Uns uch that pθ˚, U˚q with θ˚" U t η also solves the quarter median problem for Pn. Note that η " αb˚`βb 1 , where α is median of P bn and β is a is median of P bn . Because b˚or b 1 is an element of t´b 12 , b 12 , . . . ,´b 1n , b 1n u, and Med˘´P´bn¯"´Med¯´P it follows that pθ˚, U˚q P L pPnq.
(ii) θ R tx 1 , . . . , xnu. Withθ as the intersection point of the line parallel to L θ,b through the data pointx P tx 1 , . . . , xnu with the shortest distance to L θ,b , i.e. inft}xý } : y P L θ,b u " min inft}x j´y } : y P L θ,b u : 1 ď j ď n ( , and the line parallel to L θ,b 1 through the data pointx P tx 1 , . . . , xnu with the shortest distance to L θ,b 1 , we have that pθ, U q with U as matrix with the row vectors b, b 1 is a solution of the quarter median problem for Pn. Ifx "x thenθ "x "x and we arrive at case (i). So, we can (and do) assume that there are two points, x 1 , x 2 say, such that x 1 P L θ,b , x 2 P L θ,b 1 . Now we put B :" tb 1j : 2 ď j ď nu Y tb 2j : 3 ď j ď nu and define ψ " min`min arccosp|d t b|q : d P B There are orthogonal directions b˚, b 1 P S 1 obtained from b, b 1 by the same clockwise or counterclockwise rotation with the angle ψ, where b˚or b 1 is an element of B Y p´Bq. With L 1 " tx 1`t b˚: t P Ru and L 2 " tx 2`t b 1 : t P Ru we obtain the axes of a new rectangular coordinate system. The origin θ of the coordinate system with the axes L θ,b and L θ,b 1 and the origin θ˚of the new system are both located on the Thales circle about the line segment connecting x 1 and x 2 . By construction, the number of data points x j lying in V˘˘pθ˚; b˚, b 1 q is greater than or equal to the number of data points x j lying in V˘˘pθ; b, b 1 q. Further, the number of data points x j lying in H˘pθ˚; b˚q and the number of data points x j lying in H˘pθ˚; b 1 q are greater than or equal to the number of data points x j lying in H˘pθ; bq and the number of data points x j lying in H˘pθ; b 1 q, respectively. Consequently, pθ˚, U˚q with U˚as matrix with the row vectors b˚, b 1 solves the quarter median for Pn. Again, by Proposition 3 (c), there is an η P M`P Un˘s uch that pθ˚˚, U˚q, with θ˚˚" U t η also solves the quarter median problem for Pn. Here, b˚or b 1 is an element of B Y p´Bq, so that, again by (51), we have`θ˚˚, U˚˘P L pPnq.