Rates of uniform convergence of extreme order statistics
SummaryBounds for the convergence uniformly over all Borel sets of the largest order statistic as well as of the joint distribution of extremes are established which reveal in which way these rates are determined by the distance of the underlying density from the density of the corresponding generalized Pareto distribution.The results are highlighted by several examples among which there is a bound for the rate at which the joint distribution of thek largest order statistics from a normal distribution converges uniformly to its limit.

title: A note on the inverse bootstrap process for large quantiles
abstract: It is shown that for qn = 1 - j(n)/n with j(n) --> n-->[infinity] [infinity], j(n)/n --> n-->[infinity] 0, the resulting bootstrap estimate of the quantile function of the sample qn-quantile has exact accuracy Op(1/j(n)1/4). This is a consequence of a functional central limit theorem for the corresponding inverse bootstrap processes.

title: A Note on the Comedian for Elliptical Distributions
abstract: The comedianCOM(X,Y) of random variablesX,Yis a median based robust alternative to the covariance ofXofY. For the bivariate normal case it is known thatCOM(X,Y), standardized by the median absolute deviations ofXandY, is a symmetric, strictly increasing and continuous function of the correlation coefficient?with range ?] and can therefore serve as a robust alternative to?. We show that this result, which is not true in general, extends to elliptical distributions even in the case where moments ofX,Ydo not exist.

title: A note on uniform asymptotic normality of intermediate order statistics
abstract: It is proved that under fairly general von Mises-type conditions on the underlying distribution, the intermediate order statistics, properly standardized, converge uniformly over all Borel sets to the standard normal distribution. This closes the gap between central order statistics and extremes, where uniform convergence under mild conditions is well-known.

title: Asymptotic independence of median and MAD
abstract: The asymptotic joint normality of the sample median and the median absolute deviation from the median (MAD) as robust counterparts of sample mean and standard deviation is established. This characterizes their asymptotic independence, paralleling the asymptotic independence of mean and standard deviation. Analogous results are established for the median and the interquartile range.

title: Local asymptotic normality of truncated empirical processes
abstract: Given n iid copies X 1 ,..., X n of a random element X in some arbitrary measurable space S, we are only interested in those observations that fall into some subset D having but a small probability of occurrence. It is assumed that the distribution P x of X belongs on D to a parametric family P x (.  D) = P V , V ∈ Θ ⊂ R d . Nonlinear regression analysis and the peaks-over-threshold (POT) approach in extreme value analysis are prominent examples. For the POT approach on S = R and P V being a generalized Pareto distribution, it is known that the complete information about the underlying parameter V 0 is asymptotically contained in the number τ(n) of observations in D among X 1 ,..., X n , but not in their actual values. This result is formulated in terms of local asymptotic normality of the log-likelihood ratio of the point process of exceedances with τ(n) being the central sequence. In this paper we establish a necessary and sufficient condition such that τ(n) has this property for a general truncated empirical process in an arbitrary sample space and for an arbitrary parametric family. The known results are then consequences of this result. We can, moreover, characterize the influence of the actual observations in D on the central sequence, if this condition is violated. Immediate applications are asymptotically optimal tests for testing V 0 and, if Θ ⊂ R, asymptotic efficiency of the ML-estimator? n satisfying P Vn (D) = τ(n)/n, where these statistics are based on τ(n) only.

title: Arrangements and cohomology
abstract: To a matroidM is associated a graded commutative algebraA=A(M), the Orlik-Solomon algebra ofM. Motivated by its role in the construction of generalized hypergeometric functions, we study the cohomologyH* (A, dω) ofA(M) with coboundary mapdω given by multiplication by a fixed element ω ofA1. Using a description of decomposable relations inA, we construct new examples of “resonant” values of ω, and give a precise calculation ofH1 (A, dω) as a function of ω. We describe the setR1(A={ω|H1(A(M,dω)≠}, and use it as a tool in the classification of Orlik-Solomon algebras, with applications to the topology of complex hyperplane complements. We show thatR1(A) is a complete invariant of the quadratic closure ofA, and show under various hypotheses that one can reconstruct the matroidM, or at least its Tutte polynomial, from the varietyR1(A). We demonstrate with several examples thatR1 is easily calculable, may contain nonlocal components, and that combinatorial properties ofR1(A) are often sufficient to distinguish nonisomorphic rank three Orlik-Solomon algebras.

title: The minimal model of the complement of an arrangement of hyperplanes
abstract: In this paper the methods of rational homotopy theory are applied to a family of examples from singularity theory. Let A be a finite collection of hyperplanes in Cl, and let M = Cl-UHEA H. We say A is a rational K(w, 1) arrangement if the rational completion of M is aspherical. For these arrangements an identity (the LCS formula) is established relating the lower central series of 7r1 (M) to the cohomology of M. This identity was established by group-theoretic means for the class of fiber-type arrangements in previous work. We reproduce this result by showing that the class of rational K(w, 1) arrangements contains all fiber-type arrangements. This class includes the reflection arrangements of types Al and Bl. There is much interest in arrangements for which M is a K(w, 1) space. The methods developed here do not apply directly because M is rarely a nilpotent space. We give examples of K(w, 1) arrangements which are not rational K(w, 1) for which the LCS formula fails, and K(w, 1) arrangements which are not rational K(w, 1) where the LCS formula holds. It remains an open question whether rational K(w, 1) arrangements are necessarily K(w, 1). l.  An arrangement of hyperplanes is a finite collection of Clinear subspaces of dimension (1-l) in C1. To such an arrangement A is associated an open 21-manifold, the complement M = Cl-U{HIH E A}. The connections between the topology of M and the combinatorial geometry of A are the source of much current research in this area. The most successful investigations concern the cohomology of M [13, 15], whereas the most difficult unsolved problems involve the homotopy groups of M [5]. In this paper we study the link between cohomology and homotopy provided by Sullivan's theory of minimal models . In , a numerical relationship was established between 1(M) and H*(M) for the class of fiber-type arrangements. This LCS formula reads as follows: rI ( 1-tn ) Y7n (M) = PM (-t) n>l where the (>n(M) are the ranks of successive quotients in the lower central series of 1(M)) and PM(t) is the Poincare polynomial of H*(M). Because the sequence (>n(M) is related to the l-minimal model y of M, we conjecture in  that the LdS formula holds precisely when R determines H* (M). With the methods developed in this paper we can resolve this conjecture. Specifically, we show (Corollary 3.8) that the LCS formula holds when H*(5°) is isomorphic to H*(M). Arrangements satisfying the latter condition are called rational K(7r, l). Corollary 3.8 may be Received by the editors June 1, 1987.  (1985 Revitsion).

title: On the algebra associated with a geometric lattice
abstract: Let L be a geometric lattice. Following P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. math.56 (1980), 167–189, we associate with L a graded commutative algebra A(L). In this paper we introduce a new invariant ψ of the algebra A(L) which suffices to distinguish algebras for which all other known invariants coincide. This result is applied to the study of arrangements of complex hyperplanes, with L being the intersection lattice. In this case A(L) is isomorphic to the cohomology algebra of the associated hyperplane complement. The goal is to find examples of arrangements with non-isomorphic lattices but homotopy equivalent complements. The invariant introduced here effectively narrows the list of candidates. Nevertheless, we exhibit combinatorially inequivalent arrangements for which all known invariants, including ψ, coincide.

title: A note on discriminantal arrangements
abstract: Let v O be a fixed affine arrangement of n hyperplanes in general position in Kk . Let U(n, k) denote the set of general position arrangements whose elements are parallel translates of the hyperplanes of v o . Then U(n, k) is the complement of a central arrangement R (n, k). These are the well-known discriminantal arrangements introduced by Y. I. Manin and V. V. Schechtman. In this note we give an explicit description of R (n, k) in terms of the original arrangement v o . In terms of the underlying matroids, _q (n, k) realizes an adjoint of the dual of the matroid associated with v o . Using this description we show that, contrary to the conventional wisdom, neither the intersection lattice of S7 (n, k) nor the topology of U(n, k) is independent of the original arrangement v o .

title: Homotopy types of line arrangements
abstract: SummaryWe prove that the complement of a real affine line arrangement inC2 is homotopy equivalent to the canonical 2-complex associated with Randell's presentation of the fundamental group. This provides a much smaller model for the homotopy type of the complement of a real affine 2- or central 3-arrangement than the Salvetti complex and its cousins. As an application we prove that these exist (infinitely many) pairs of central arrangements inC3 with different underlying matroids whose complements are homotopy equivalent. We also show that two real 3-arrangements whose oriented matroids are connected by a sequence of flips are homotopy equivalent.