121498103
Homotopy types of line arrangements
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.
92994378
A note on discriminantal arrangements
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 .
14391043
Arrangements and cohomology
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.
26463172
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.
123363269
A simple approach to the generation of uniformly distributed random variables with prescribed correlations
In a simulation study we encountered the problem of generating uniformly on (0,1) distributed random variables with a prescribed correlation matrix. This article reports our solution to this problem.We show that the following simple device leads to uniformly on (0,1)distributed random variables, which have a correlation matrix that isfor practical purposes sufficiently close to the prescribed one: Let(X1.., Xm) be a normal vector with theprescribed correlation matrix S = (pij:). Such a vector caneasily be generated. Suppose for simplicity that each Xi is standard normal and denote by Φ the standard normal distribution function. It turns out that the vector(V1:…,Vm:):= (φ(X1),…,φ(Xm)) with uniformly on (0,1)distributed components Vi; then has a correlation matrix S′, whose entries are for practical purposes sufficiently close to that of the target matrix S. This is a consequence of the grade correlation (or Spearman's rho) of normal vectors. Suppose, in addition, that the matrix is positive semidefinite. T...
120350643
Local asymptotic normality of intermediate and central order statistics
Consider n independent random variables Zi,…, Zn on R with common distribution function F, whose upper tail belongs to a parametric family F(t) = Fθ(t),t ≥ x0, where θ ∈ ⊝ ⊂ R d. A necessary and sufficient condition for the family Fθ, θ ∈ ⊝, is established such that the k-th largest order statistic Zn−k+1:n alone constitutes the central sequence yielding local asymptotic normality ( LAN ) of the loglikelihood ratio of the vector (Zn−i+1:n)1 i=kof the k largest order statistics. This is achieved for k = k(n)→n→∞∞ with k/n→n→∞ 0. In the case of vectors of central order statistics ( Zr:n, Zr+1:n,…, Zs:n ), with r/n and s/n both converging to q ∈ ( 0,1 ), it turns out that under fairly general conditions any order statistic Zm:n with r ≤ m ≤s builds the central sequence in a pertaining LAN expansion.These results lead to asymptotically optimal tests and estimators of the underlying parameter, which depend on single order statistics only
73549405
LAN of extreme order statistics
Consider an iid sampleZ1,...,Zn with common distribution functionF on the real line, whose upper tail belongs to a parametric family {Fβ: β∈⊝}. We establish local asymptotic normality (LAN) of the loglikelihood process pertaining to the vector(Zn−i+1∶n)i=1k of the upperk=k(n)→n→∞∞ order statistics in the sample, if the family {Fβ:β∈⊝} is in a neighborhood of the family of generalized Pareto distributions. It turns out that, except in one particular location case, thekth-largest order statisticZn−k+1∶n is the central sequence generating LAN. This implies thatZn−k+1∶n is asymptotically sufficient and that asymptotically optimal tests for the underlying parameter β can be based on the single order statisticZn−k+1∶n. The rate at whichZn−k+1∶n becomes asymptotically sufficient is however quite poor.
119786717
Extreme quantile estimation in δ-neighborhoods of generalized Pareto distributions
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.
120645444
Asymptotic Normality of the Kernel Quantile Estimator