625 research outputs found
On Multivariate Records from Random Vectors with Independent Components
Let be independent copies of a
random vector with values in and with a
continuous distribution function. The random vector is a
complete record, if each of its components is a record. As we require
to have independent components, crucial results for univariate
records clearly carry over. But there are substantial differences as well:
While there are infinitely many records in case , there occur only
finitely many in the series if . Consequently, there is a terminal
complete record with probability one. We compute the distribution of the random
total number of complete records and investigate the distribution of the
terminal record. For complete records, the sequence of waiting times forms a
Markov chain, but differently from the univariate case, now the state infinity
is an absorbing element of the state space
Some Results on Joint Record Events
Let be independent and identically distributed random
variables on the real line with a joint continuous distribution function .
The stochastic behavior of the sequence of subsequent records is well known.
Alternatively to that, we investigate the stochastic behavior of arbitrary
, under the condition that they are records, without knowing their
orders in the sequence of records. The results are completely different. In
particular it turns out that the distribution of , being a record, is not
affected by the additional knowledge that is a record as well. On the
contrary, the distribution of , being a record, is affected by the
additional knowledge that is a record as well. If has a density, then
the gain of this additional information, measured by the corresponding
Kullback-Leibler distance, is , independent of . We derive the limiting
joint distribution of two records, which is not a bivariate extreme value
distribution. We extend this result to the case of three records. In a special
case we also derive the limiting joint distribution of increments among
records
Statistical Modeling of Spatial Extremes
The areal modeling of the extremes of a natural process such as rainfall or
temperature is important in environmental statistics; for example,
understanding extreme areal rainfall is crucial in flood protection. This
article reviews recent progress in the statistical modeling of spatial
extremes, starting with sketches of the necessary elements of extreme value
statistics and geostatistics. The main types of statistical models thus far
proposed, based on latent variables, on copulas and on spatial max-stable
processes, are described and then are compared by application to a data set on
rainfall in Switzerland. Whereas latent variable modeling allows a better fit
to marginal distributions, it fits the joint distributions of extremes poorly,
so appropriately-chosen copula or max-stable models seem essential for
successful spatial modeling of extremes.Comment: Published in at http://dx.doi.org/10.1214/11-STS376 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Rejoinder to "Statistical Modeling of Spatial Extremes"
Rejoinder to "Statistical Modeling of Spatial Extremes" by A. C. Davison, S.
A. Padoan and M. Ribatet [arXiv:1208.3378].Comment: Published in at http://dx.doi.org/10.1214/12-STS376REJ the
Statistical Science (http://www.imstat.org/sts/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Multivariate Nonparametric Estimation of the Pickands Dependence Function using Bernstein Polynomials
Many applications in risk analysis, especially in environmental sciences,
require the estimation of the dependence among multivariate maxima. A way to do
this is by inferring the Pickands dependence function of the underlying
extreme-value copula. A nonparametric estimator is constructed as the sample
equivalent of a multivariate extension of the madogram. Shape constraints on
the family of Pickands dependence functions are taken into account by means of
a representation in terms of a specific type of Bernstein polynomials. The
large-sample theory of the estimator is developed and its finite-sample
performance is evaluated with a simulation study. The approach is illustrated
by analyzing clusters consisting of seven weather stations that have recorded
weekly maxima of hourly rainfall in France from 1993 to 2011
Shock fragmentation model for gravitational collapse
A cloud of gas collapsing under gravity will fragment. We present a new
theory for this process, in which layers shocked gas fragment due to their
gravitational instability. Our model explains why angular momentum does not
inhibit the collapse process. The theory predicts that the fragmentation
process produces objects which are significantly smaller than most stars,
implying that accretion onto the fragments plays an essential role in
determining the initial masses of stars. This prediction is also consistent
with the hypothesis that planets can be produced by gravitational collapse.Comment: 22 pages, 3 figure
Simulating Supersonic Turbulence in Magnetized Molecular Clouds
We present results of large-scale three-dimensional simulations of weakly
magnetized supersonic turbulence at grid resolutions up to 1024^3 cells. Our
numerical experiments are carried out with the Piecewise Parabolic Method on a
Local Stencil and assume an isothermal equation of state. The turbulence is
driven by a large-scale isotropic solenoidal force in a periodic computational
domain and fully develops in a few flow crossing times. We then evolve the flow
for a number of flow crossing times and analyze various statistical properties
of the saturated turbulent state. We show that the energy transfer rate in the
inertial range of scales is surprisingly close to a constant, indicating that
Kolmogorov's phenomenology for incompressible turbulence can be extended to
magnetized supersonic flows. We also discuss numerical dissipation effects and
convergence of different turbulence diagnostics as grid resolution refines from
256^3 to 1024^3 cells.Comment: 10 pages, 3 figures, to appear in the proceedings of the DOE/SciDAC
2009 conferenc
Structure Function Scaling in Compressible Super-Alfvenic MHD Turbulence
Supersonic turbulent flows of magnetized gas are believed to play an
important role in the dynamics of star-forming clouds in galaxies.
Understanding statistical properties of such flows is crucial for developing a
theory of star formation. In this letter we propose a unified approach for
obtaining the velocity scaling in compressible and super--Alfv\'{e}nic
turbulence, valid for arbitrary sonic Mach number, \ms. We demonstrate with
numerical simulations that the scaling can be described with the
She--L\'{e}v\^{e}que formalism, where only one parameter, interpreted as the
Hausdorff dimension of the most intense dissipative structures, needs to be
varied as a function of \ms. Our results thus provide a method for obtaining
the velocity scaling in interstellar clouds once their Mach numbers have been
inferred from observations.Comment: published in Physical Review Letter
Structure Function Scaling of a 2MASS Extinction Map of Taurus
We compute the structure function scaling of a 2MASS extinction map of the
Taurus molecular cloud complex. The scaling exponents of the structure
functions of the extinction map follow the Boldyrev's velocity structure
function scaling of supersonic turbulence. This confirms our previous result
based on a spectral map of 13CO J=1-0 covering the same region and suggests
that supersonic turbulence is important in the fragmentation of this
star--forming cloud.Comment: submitted to Ap
The Statistics of Supersonic Isothermal Turbulence
We present results of large-scale three-dimensional simulations of supersonic
Euler turbulence with the piecewise parabolic method and multiple grid
resolutions up to 2048^3 points. Our numerical experiments describe
non-magnetized driven turbulent flows with an isothermal equation of state and
an rms Mach number of 6. We discuss numerical resolution issues and demonstrate
convergence, in a statistical sense, of the inertial range dynamics in
simulations on grids larger than 512^3 points. The simulations allowed us to
measure the absolute velocity scaling exponents for the first time. The
inertial range velocity scaling in this strongly compressible regime deviates
substantially from the incompressible Kolmogorov laws. The slope of the
velocity power spectrum, for instance, is -1.95 compared to -5/3 in the
incompressible case. The exponent of the third-order velocity structure
function is 1.28, while in incompressible turbulence it is known to be unity.
We propose a natural extension of Kolmogorov's phenomenology that takes into
account compressibility by mixing the velocity and density statistics and
preserves the Kolmogorov scaling of the power spectrum and structure functions
of the density-weighted velocity v=\rho^{1/3}u. The low-order statistics of v
appear to be invariant with respect to changes in the Mach number. For
instance, at Mach 6 the slope of the power spectrum of v is -1.69, and the
exponent of the third-order structure function of v is unity. We also directly
measure the mass dimension of the "fractal" density distribution in the
inertial subrange, D_m = 2.4, which is similar to the observed fractal
dimension of molecular clouds and agrees well with the cascade phenomenology.Comment: 15 pages, 19 figures, ApJ v665, n2, 200
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