8 search hits
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Higher Order Asymptotics for the MSE of Robust M-Estimators of Location on Shrinking Total Variation Neighborhoods
(2008)
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Matthias Brandl
- In the setup of shrinking neighbourhoods in sample size n about an ideal (“smooth”) central model, Rieder (1994) determines the optimal asymptotic linear estimator w.r.t. the asymptotic MSE evaluated uniformly on these neighbourhoods. We answer the question to which degree the asymptotic optimality carries over to finite sample size in the context of total variation neighbourhoods. In contrast to usual higher order asymptotics, instead of giving approximations to distribution functions (or densities), we expand the risk directly by application of Edgeworth and Taylor expansions. In the context of determining the exact finite sample risk for sample size n>2 M. Kohl showed in Kohl (2005) that the speed of convergence towards the asymptotic risk is faster by an order in case of total variation compared to convex contamination neighbourhoods. M. Kohl conjectured that this is caused by the higher symmetry of total variation neighbourhoods. Looking at the MSE in the asymptotic optimal setup we get the same results as M. Kohl for finite sample risk. Furthermore we can show by direct expansion of the MSE that for a higher speed of convergence symmetry of the ideal distribution F is essential. We confirm our result by a cross-check in the ideal model and illustrate our theoretical investigations for F = N(0;1). We also deal with the question of an actual realization of a least favourable deviation from the ideal model in a finite context. We settle on the symmetric case and monotone odd influence curves of Hampel-type form, and show that for a certain kind of manipulation mechanism we gain our theoretical results up to the desired order. It shows up that we only get access to the results in the finite context if we require the finite sample to attain the minimum and maximum of the given influence curve with a certain probability already. Depending on this probability we derive a lower bound on the sample length n. We substantiate the sufficiently exact algorithm by determining the amount K of observations to be manipulated as well as the bound c on the observations for having maximum influence on the MSE according to the value of the influence curve. We give a restrictive condition on the distribution of K that lets the probability of the (K:n)-quantile exceeding a now concrete bound c be exponentially negligible. The bound c is explicitly calculated for F = N(0;1) and suitable four-point distributions for K are given that satisfy all the previous claimed conditions. Thus we gain an algorithm generating observations from a least favourable distribution out of a finite total variation neighbourhood w.r.t. an asymptotically optimal risk.
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On the benefits of using NP-hard problems in Branch & Bound
(2008)
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Jörg Rambau
Cornelius Schwarz
- We present a Brand-and-Bound (B&B) method using combinatorial bounds for solving makespan minimization problems with sequence dependent setup costs. As an application we present a laser source sharing problem arising in car manufacturing.
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Double and bordered alpha-circulant self-dual codes over finite commutative chain rings
(2008)
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Michael Kiermaier
Alfred Wassermann
- In this paper we investigate codes over finite commutative rings R, whose generator matrices are built from alpha-circulant matrices. For a non-trivial ideal I < R we give a method to lift such codes over R/I to codes over R, such that some isomorphic copies are avoided. For the case where I is the minimal ideal of a finite chain ring we refine this lifting method: We impose the additional restriction that lifting preserves self-duality. It will be shown that this can be achieved by solving a linear system of equations over a finite field. Finally we apply this technique to Z_4-linear double nega-circulant and bordered circulant self-dual codes. We determine the best minimum Lee distance of these codes up to length 64.
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Existence and stability of stellardynamic models
(2008)
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Achim Schulze
- We examine existence and stability of stationary solutions to the Vlasov-Poisson system. This system is used in stellardynmaics to describe the evolution of galaxies where collissions are neglected and the evolution is determined by the self-consistent gravitational field which is created by the particles, e.g. the stars . In the first part we examine steady states which decsribe static shells under the influence of a fixed point mass. These solutions can be used as a model for a galaxy with a massive black hole in its center. For the Vlasov--Poisson system under the influence of such a point mass, we prove a global existence result. In the second part, we construct axially symmetric solutions depending on Jacobis integral. The presented results are in accordance with the numerical examinations of the P.O. Vandervoort.
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Stability of flat galaxies
(2008)
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Roman Firt
- In this thesis we investigate the existence and properties of stationary solutions of the flat Vlasov-Poisson system. This system of partial differential equations can be used as a model of extremely flat astronomical objects and is a combination between the two-dimensional motion of particles and the three-dimensional interaction through their gravitational potential.
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Bounds for the minimum oriented diameter
(2008)
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Sascha Kurz
Martin Lätsch
- We consider the problem of finding an orientation with minimum diameter of a connected bridgeless graph. Fomin et. al. discovered a relation between the minimum oriented diameter an the size of a minimal dominating set. We improve their upper bound.
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The Top-Dog Index: A New Measurement for the Demand Consistency of the Size Distribution in Pre-Pack Orders for a Fashion Discounter with Many Small Branches
(2008)
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Sascha Kurz
Jörg Rambau
Jörg Schlüchtermann
Rainer Wolf
- We propose the new Top-Dog-Index, a measure for the branch-dependent historic deviation of the supply data of apparel sizes from the sales data of a fashion discounter. A common approach is to estimate demand for sizes directly from the sales data. This approach may yield information for the demand for sizes if aggregated over all branches and products. However, as we will show in a real-world business case, this direct approach is in general not capable to provide information about each branchs individual demand for sizes: the supply per branch is so small that either the number of sales is statistically too small for a good estimate (early measurement) or there will be too much unsatisfied demand neglected in the sales data (late measurement). Moreover, in our real-world data we could not verify any of the demand distribution assumptions suggested in the literature. Our approach cannot estimate the demand for sizes directly. It can, however, individually measure for each branch the scarcest and the amplest sizes, aggregated over all products. This measurement can iteratively be used to adapt the size distributions in the pre-pack orders for the future. A real-world blind study shows the potential of this distribution free heuristic optimization approach: The gross yield measured in percent of gross value was almost one percentage point higher in the test-group branches than in the control-group branches.
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Maximal integral point sets over Z^2
(2008)
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Sascha Kurz
Andrey Radoslavov Antonov
- Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set P={p(1),...,p(n)} in Z^2 a maximal integral point set over Z^2 if all pairwise distances are integral and every additional point p(n+1) destroys this property. Here we consider such sets for a given cardinality and with minimum possible diameter. We determine some exact values via exhaustive search and give several constructions for arbitrary cardinalities. Since we cannot guarantee the maximality in these cases we describe an algorithm to prove or disprove the maximality of a given integral point set. We additionally consider restrictions as no three points on a line and no four points on a circle.
