A NOTE ON DOMAINS OF ATTRACTION OF P-MAX STABLE LAWS
A distribution function (df) F is said to belong to the p-max domain of attraction of a nondegenerate df G, iff there exist [alpha]n > 0, [beta]n > 0 such that the df Fn([alpha]n x [beta]n sign(x)) converges weakly to G. The class of possible limiting dfs G and their domains of attraction are characterized by results from classical extreme value theory. This main result unifies earlier ones obtained by Pantcheva et al.

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 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: 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: 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: 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: 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.

title: The cohomology and fundamental group of a hyperplane complement
abstract: An electrical game having a board including a representation of an athletic field adjacent to a grid which represents the results of a plurality of athletic plays and which includes a first coordinate representing said plurality of athletic plays and a second coordinate having a plurality of indicator number. A light bar having a plurality of bulbs mounted thereon may be movably oriented or fixed with respect to said grid and may be selectively coupled in circuit relationship with offensive and defensive play result selection switches, the activation of which determines which of the bulbs is illuminated. In the fixed bar embodiment, by first selecting one of said plurality of athletic plays in said first coordinate and by activating one offensive and one defensive play result selection switch, one of the indicator numbers is selected so that the result of the athletic play can be determined by reading the grid at the coodinates as determined by the play selection and the selection of one of said indicator numbers. In the movable bar embodiment the light bar is mounted beneath the grid and may have an indicator which is viewable from the side of the game and can be aligned with the selected play. The bar is mounted on a pair of paralled guide rods. One rod is rotatable and has knobs fixed at its opposite ends. A cord is connected by its opposite ends on opposite sides of the bar and is looped around the knobs so that rotation of either knob will cause the bar to move along the rods.