121496414
On testing the extreme value index via the pot-method
n alternatives, based on the point processes N of the exceedances among Y n i . over a sequence of thresholds t . It turns out that the random number of n . exceedances t n over t is the central sequence for the log-likelihood n . . . ratio d L N rd L N , yielding its local asymptotic normality LAN . b n b n n 0 . This result implies in particular that t n carries asymptotically all the information about the underlying parameter b, which is contained in N. n . We establish sharp bounds for the rate at which t n becomes asymptoti- cally sufficient, which show, however, that this is quite a poor rate. These results remain true if we add an unknown scale parameter. 1 n . A model assumption on the upper tail of the underlying d.f. such as M is, for example, indispensable if one is interested in extreme quantities of the underlying d.f., that is, such quantities of F which are usually outside the range of the given data Y , . . . , Y. A typical example is inference about 1 n y1 . . 4 extreme quantiles F p ( inf t g R: F t G p with p being close to 1. This problem is typically tackled by hydrologists for the prediction of large w
122237156
A Note on the Comedian for Elliptical Distributions
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 ?1,1] 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.
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
121269184
Local asymptotic normality of truncated empirical processes
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.
120059149
A note on the inverse bootstrap process for large quantiles
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.
122299017
Asymptotic independence of median and MAD
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.
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 .
53346307
The minimal model of the complement of an arrangement of hyperplanes
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. Introduction. 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 [14]. In [6], 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 [5] 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. 1980 Mathematicts Subject Clatstsification (1985 Revitsion). Primary 55P62, 57M05; Secondary 06C10. (r)1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page
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.
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.
8300586
bases for cohomology of local systems on hyperplane complements
We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes A. The cohomology plays an important role for the theory of generalized hy- pergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only non-vanishing cohomol- ogy group, under some nonresonance conditions on the local system, for any arrangement A. The bases are determined by a linear ordering of the hyperplanes, and are indexed by certain "no-broken-circuits" bases of A. The basic forms depend on the local system, but any two bases con- structed in this way are related by a matrix of integer constants which depend only on the linear orders and not on the local system. In certain special cases we show the existence of bases of monomial logarithmic forms.