166 research outputs found
A Note on Perfect Slice Sampling
Perfect slice sampling is a method to turn Markov Chain Monte Carlo (MCMC) samplers into exact generators for independent random variates. We show that the simplest version of the perfect slice sampler suggested in the literature does not always sample from the target distribution. (author's abstract)Series: Research Report Series / Department of Statistics and Mathematic
A Note on the Folding Coupler
Perfect Gibbs sampling is a method to turn Markov Chain Monte Carlo (MCMC) samplers into exact generators for independent random vectors. We show that a perfect Gibbs sampling algorithm suggested in the literature is not always generating from the correct distribution. (author's abstract)Series: Research Report Series / Department of Statistics and Mathematic
Algebraic Connectivity and Degree Sequences of Trees
We investigate the structure of trees that have minimal algebraic
connectivity among all trees with a given degree sequence. We show that such
trees are caterpillars and that the vertex degrees are non-decreasing on every
path on non-pendant vertices starting at the characteristic set of the Fiedler
vector.Comment: 8 page
Graphs with Given Degree Sequence and Maximal Spectral Radius
We describe the structure of those graphs that have largest spectral radius
in the class of all connected graphs with a given degree sequence. We show that
in such a graph the degree sequence is non-increasing with respect to an
ordering of the vertices induced by breadth-first search. For trees the
resulting structure is uniquely determined up to isomorphism. We also show that
the largest spectral radius in such classes of trees is strictly monotone with
respect to majorization.Comment: 12 pages, 4 figures; revised version. Important change: Theorem 3
(formely Theorem 7) now states (and correctly proofs) the majorization result
only for "degree sequences of trees" (instead for general connected graphs).
Bo Zhou from the South China Normal University in Guangzhou, P.R. China, has
found a counter-example to the stronger resul
Faber-Krahn Type Inequalities for Trees
The Faber-Krahn theorem states that among all bounded domains with the same
volume in (with the standard Euclidean metric), a ball that has
lowest first Dirichlet eigenvalue. Recently it has been shown that a similar
result holds for (semi-)regular trees. In this article we show that such a
theorem also hold for other classes of (not necessarily non-regular) trees.
However, for these new results no couterparts in the world of the
Laplace-Beltrami-operator on manifolds are known.Comment: 19 pages, 5 figure
Sampling from Linear Multivariate Densities
It is well known that the generation of random vectors with non-independent components is difficult. Nevertheless, we propose a new and very simple generation algorithm for multivariate linear densities over point-symmetric domains.
Among other applications it can be used to design a simple decomposition-rejection algorithm for multivariate concave distributions.Series: Research Report Series / Department of Statistics and Mathematic
Automatic Markov Chain Monte Carlo Procedures for Sampling from Multivariate Distributions
Generating samples from multivariate distributions efficiently is an important task in Monte Carlo integration and many other stochastic simulation problems. Markov chain Monte Carlo has been shown to be very efficient compared to "conventional methods", especially when many dimensions are involved. In this article we propose a Hit-and-Run sampler in combination with the Ratio-of-Uniforms method. We show that it is well suited for an algorithm to generate points from quite arbitrary distributions, which include all log-concave distributions. The algorithm works automatically in the sense that only the mode (or an approximation of it) and an oracle is required, i.e., a subroutine that returns the value of the density function at any point x. We show that the number of evaluations of the density increases slowly with dimension. (author's abstract)Series: Preprint Series / Department of Applied Statistics and Data Processin
rstream: Streams of Random Numbers for Stochastic Simulation
The package rstream provides a unified interface to streams of random numbers for the R statistical computing language. Features are: * independent streams of random numbers * substreams * easy handling of streams (initialize, reset) * antithetic random variates The paper describes this packages and demonstrates an simple example the usefulness of this approach.Series: Preprint Series / Department of Applied Statistics and Data Processin
Largest Laplacian Eigenvalue and Degree Sequences of Trees
We investigate the structure of trees that have greatest maximum eigenvalue
among all trees with a given degree sequence. We show that in such an extremal
tree the degree sequence is non-increasing with respect to an ordering of the
vertices that is obtained by breadth-first search. This structure is uniquely
determined up to isomorphism. We also show that the maximum eigenvalue in such
classes of trees is strictly monotone with respect to majorization.Comment: 9 pages, 2 figure
A Faber-Krahn-type Inequality for Regular Trees
In the last years some results for the Laplacian on manifolds have been shown to hold also for the graph Laplacian, e.g. Courant's nodal domain theorem or Cheeger's inequality. Friedman (Some geometric aspects of graphs and their eigenfunctions, Duke Math. J. 69 (3), pp. 487-525, 1993) described the idea of a ``graph with boundary". With this concept it is possible to formulate Dirichlet and Neumann eigenvalue problems. Friedman also conjectured another ``classical" result for manifolds, the Faber-Krahn theorem, for regular bounded trees with boundary. The Faber-Krahn theorem states that among all bounded domains with fixed volume, a ball has lowest first Dirichlet eigenvalue. In this paper we show such a result for regular trees by using a rearrangement technique. We give restrictive conditions for trees with boundary where the first Dirichlet eigenvalue is minimized for a given "volume". Amazingly Friedman's conjecture is false, i.e. in general these trees are not ``balls". But we will show that these are similar to ``balls". (author's abstract)Series: Preprint Series / Department of Applied Statistics and Data Processin
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