27,105 research outputs found
Effect of Dimensionality on the Continuum Percolation of Overlapping Hyperspheres and Hypercubes: II. Simulation Results and Analyses
In the first paper of this series [S. Torquato, J. Chem. Phys. {\bf 136},
054106 (2012)], analytical results concerning the continuum percolation of
overlapping hyperparticles in -dimensional Euclidean space
were obtained, including lower bounds on the percolation threshold. In the
present investigation, we provide additional analytical results for certain
cluster statistics, such as the concentration of -mers and related
quantities, and obtain an upper bound on the percolation threshold . We
utilize the tightest lower bound obtained in the first paper to formulate an
efficient simulation method, called the {\it rescaled-particle} algorithm, to
estimate continuum percolation properties across many space dimensions with
heretofore unattained accuracy. This simulation procedure is applied to compute
the threshold and associated mean number of overlaps per particle
for both overlapping hyperspheres and oriented hypercubes for . These simulations results are compared to corresponding upper
and lower bounds on these percolation properties. We find that the bounds
converge to one another as the space dimension increases, but the lower bound
provides an excellent estimate of and , even for
relatively low dimensions. We confirm a prediction of the first paper in this
series that low-dimensional percolation properties encode high-dimensional
information. We also show that the concentration of monomers dominate over
concentration values for higher-order clusters (dimers, trimers, etc.) as the
space dimension becomes large. Finally, we provide accurate analytical
estimates of the pair connectedness function and blocking function at their
contact values for any as a function of density.Comment: 24 pages, 10 figure
Finite sample performance of linear least squares estimators under sub-Gaussian martingale difference noise
Linear Least Squares is a very well known technique for parameter estimation,
which is used even when sub-optimal, because of its very low computational
requirements and the fact that exact knowledge of the noise statistics is not
required. Surprisingly, bounding the probability of large errors with finitely
many samples has been left open, especially when dealing with correlated noise
with unknown covariance. In this paper we analyze the finite sample performance
of the linear least squares estimator under sub-Gaussian martingale difference
noise. In order to analyze this important question we used concentration of
measure bounds. When applying these bounds we obtained tight bounds on the tail
of the estimator's distribution. We show the fast exponential convergence of
the number of samples required to ensure a given accuracy with high
probability. We provide probability tail bounds on the estimation error's norm.
Our analysis method is simple and uses simple type bounds on the
estimation error. The tightness of the bounds is tested through simulation. The
proposed bounds make it possible to predict the number of samples required for
least squares estimation even when least squares is sub-optimal and used for
computational simplicity. The finite sample analysis of least squares models
with this general noise model is novel
Learning the dependence structure of rare events: a non-asymptotic study
Assessing the probability of occurrence of extreme events is a crucial issue
in various fields like finance, insurance, telecommunication or environmental
sciences. In a multivariate framework, the tail dependence is characterized by
the so-called stable tail dependence function (STDF). Learning this structure
is the keystone of multivariate extremes. Although extensive studies have
proved consistency and asymptotic normality for the empirical version of the
STDF, non-asymptotic bounds are still missing. The main purpose of this paper
is to fill this gap. Taking advantage of adapted VC-type concentration
inequalities, upper bounds are derived with expected rate of convergence in
O(k^-1/2). The concentration tools involved in this analysis rely on a more
general study of maximal deviations in low probability regions, and thus
directly apply to the classification of extreme data
On Modeling and Estimation for the Relative Risk and Risk Difference
A common problem in formulating models for the relative risk and risk
difference is the variation dependence between these parameters and the
baseline risk, which is a nuisance model. We address this problem by proposing
the conditional log odds-product as a preferred nuisance model. This novel
nuisance model facilitates maximum-likelihood estimation, but also permits
doubly-robust estimation for the parameters of interest. Our approach is
illustrated via simulations and a data analysis.Comment: To appear in Journal of the American Statistical Association: Theory
and Method
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