Bootstrap optimal bandwidth selection for kernel density estimates
The accuracy of bootstrap optimal bandwidth selection in kernel density estimates with respect to L 2 -error is investigated: The asymptotic distribution of the bootstrap squared error estimate with L 2 -optimal resampling bandwidth is computed under minimum conditions; the bootstrap approach thus yields the target optimal bandwidth for increasing sample sizes iff the bandwidth of the resampling density estimate is chosen superoptimal. This theoretical result therefore supplements a simulation study by Taylor (1989) on this subject.

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: 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: 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: 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 minimal model of the complement of an arrangement of hyperplanes
abstract: 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.  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 . In , 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  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.  (1985 Revitsion).

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: 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.

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.