A note on generalized Pareto distributions and thek upper extremes
SummaryDenote byD(n, k) the maximum distance between the distribution function of thekth-largest order statistic in aniid sample of sizen, equally standardized ink, and of its corresponding limiting extreme value distribution.Suppose that the underlying distribution function is ultimately continuous and strictly increasing. It is shown in the present paper thatD(n,k) converges to zero for any sequencek=k (n) with
$$\mathop {\lim }\limits_{n \to \infty } k/n = 0$$
 if and only if the underlying distribution is ultimately a generalized Pareto distribution. ThusgPds do not only yield the best rate of joint convergence of extremes, but they are also the only distributions where there actually is convergence.

title: Local asymptotic normality of intermediate and central order statistics
abstract: 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

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: 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: 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: 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: 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: 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.

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 geometric duality for order complexes and hyperplane complements
abstract: 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.