A geometric duality for order complexes and hyperplane complements
The top cohomology of the complement of an arrangement of complex hyperplanes is canonically isomorphic to the order homology of the associated intersection lattice. We put this result into a geometric framework by constructing a realization of the order complex of the intersection lattice inside the link of the arrangement. The canonical isomorphism is shown to coincide with the restriction of the classical Alexander duality mapping. Thus the isomorphism is a consequence of the linking of ( l − 2)-spheres and l -tori in the (2 l − 1)-sphere.

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: 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: Rates of uniform convergence of extreme order statistics
abstract: 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 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: Some Best Parameter Estimates for Distributions with Finite Endpoint
abstract: Consider n observations on, ,generated independently by a distribution function F which is only partially specified: We require that F has in its upper tail a density f such that as x tends to the right endpoint θ of F. The parameters of interest are a > 0 and b, . Based on the largest order statistics in the sample, we can define in case α > 1 /2 estimates of α, b and θ that behave asymptotically like the BLUE of θ as if α was known and like the UMVU estimates of and θin some ideal model as if θ was known. We show in addition that the BLUE estimate of θ can be outperformed by a bias corrected largest order statistic.

title: On the estimation of the quantile density function
abstract: Under fairly general conditions, asymptotic normality is achieved for the histogram type estimator of the underlying quantile density function as well as for a kernel type competitor. This entails the comparison of the asymptotic relative efficiencies of these estimators.

title: Extreme quantile estimation in δ-neighborhoods of generalized Pareto distributions
abstract: Suppose that we are given an i.i.d. sample of size n from a distribution function F, which lies in a certain neigborhood of a generalized Pareto distribution. A suitable data transformation then reduces the estimation of extreme quantiles of F i.e., quantiles outside the range of our data, to the problem of estimating the location and scale parameter in the family of exponential distributions. We establish bounds for this statistical model reduction, define the pertaining UMVU estimators and discuss their asymptotic behavior.