67,163 research outputs found
Parametric modelling of cost data: some simulation evidence
Recently, commentators have suggested that the distributional form of cost data should be explicitly modelled to gain efficiency in estimating the population mean. We perform a series of simulation experiments to evaluate the usual sample mean and the mean estimator of a lognormal distribution, in the context of both theoretical distributions and three large empirical datasets. The sample mean is always unbiased, but is somewhat less efficient when the population distribution is truly lognormal. However the lognormal estimator can perform appallingly when the true distribution is not lognormal. In practical situations, where the true distribution is unknown, the sample mean generally remains the estimator of choice, especially when limited sample size prohibits detailed modelling of the cost data distribution
Extracting Galaxy Cluster Gas Inhomogeneity from X-ray Surface Brightness: A Statistical Approach and Application to Abell 3667
Our previous analysis indicates that small-scale fluctuations in the
intracluster medium (ICM) from cosmological hydrodynamic simulations follow the
lognormal distribution. In order to test the lognormal nature of the ICM
directly against X-ray observations of galaxy clusters, we develop a method of
extracting statistical information about the three-dimensional properties of
the fluctuations from the two-dimensional X-ray surface brightness.
We first create a set of synthetic clusters with lognormal fluctuations.
Performing mock observations of these synthetic clusters, we find that the
resulting X-ray surface brightness fluctuations also follow the lognormal
distribution fairly well. Systematic analysis of the synthetic clusters
provides an empirical relation between the density fluctuations and the X-ray
surface brightness. We analyze \chandra observations of the galaxy cluster
Abell 3667, and find that its X-ray surface brightness fluctuations follow the
lognormal distribution. While the lognormal model was originally motivated by
cosmological hydrodynamic simulations, this is the first observational
confirmation of the lognormal signature in a real cluster. Finally we check the
synthetic cluster results against clusters from cosmological hydrodynamic
simulations. As a result of the complex structure exhibited by simulated
clusters, the empirical relation shows large scatter. Nevertheless we are able
to reproduce the true value of the fluctuation amplitude of simulated clusters
within a factor of two from their X-ray surface brightness alone.
Our current methodology combined with existing observational data is useful
in describing and inferring the statistical properties of the three dimensional
inhomogeneity in galaxy clusters.Comment: 34 pages, 17 figures, accepted for publication in Ap
On the inadequacy of N-point correlation functions to describe nonlinear cosmological fields: explicit examples and connection to simulations
Motivated by recent results on lognormal statistics showing that the moment
hierarchy of a lognormal variable completely fails at capturing its information
content in the large variance regime, we discuss in this work the inadequacy of
the hierarchy of correlation functions to describe a correlated lognormal
field, which provides a roughly accurate description of the non-linear
cosmological matter density field. We present families of fields having the
same hierarchy of correlation functions than the lognormal field at all orders.
This explicitly demonstrates the little studied though known fact that the
correlation function hierarchy never provides a complete description of a
lognormal field, and that it fails to capture information in the non-linear
regime, where other simple observables are left totally unconstrained. We
discuss why perturbative, Edgeworth-like approaches to statistics in the
non-linear regime, common in cosmology, can never reproduce or predict that
effect, and why it is however generic for tailed fields, hinting at a breakdown
of the perturbation theory based on the field fluctuations. We make a rough but
successful quantitative connection to N-body simulations results, that showed
that the spectrum of the log-density field carries more information than the
spectrum of the field entering the non-linear regime.Comment: 10 pages, 3 figures, matches version accepted for publication by ApJ.
Some editing in the conclusion and minor changes w.r.t. v
Bayesian weak lensing tomography: Reconstructing the 3D large-scale distribution of matter with a lognormal prior
We present a Bayesian reconstruction algorithm that infers the
three-dimensional large-scale matter distribution from the weak gravitational
lensing effects measured in the image shapes of galaxies. The algorithm is
designed to also work with non-Gaussian posterior distributions which arise,
for example, from a non-Gaussian prior distribution. In this work, we use a
lognormal prior and compare the reconstruction results to a Gaussian prior in a
suite of increasingly realistic tests on mock data. We find that in cases of
high noise levels (i.e. for low source galaxy densities and/or high shape
measurement uncertainties), both normal and lognormal priors lead to
reconstructions of comparable quality, but with the lognormal reconstruction
being prone to mass-sheet degeneracy. In the low-noise regime and on small
scales, the lognormal model produces better reconstructions than the normal
model: The lognormal model 1) enforces non-negative densities, while negative
densities are present when a normal prior is employed, 2) better traces the
extremal values and the skewness of the true underlying distribution, and 3)
yields a higher pixel-wise correlation between the reconstruction and the true
density.Comment: 23 pages, 12 figures; updated to match version accepted for
publication in PR
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