3,232 research outputs found
Gravitational lensing with gravity in accordance with astrophysical observations
In this article we perform a second order perturbation analysis of the
gravitational metric theory of gravity developed by
Bernal et al. (2011). We show that the theory accounts in detail for two
observational facts: (1) the phenomenology of flattened rotation curves
associated to the Tully-Fisher relation observed in spiral galaxies, and (2)
the details of observations of gravitational lensing in galaxies and groups of
galaxies, without the need of any dark matter. We show how all dynamical
observations on flat rotation curves and gravitational lensing can be
synthesised in terms of the empirically required metric coefficients of any
metric theory of gravity. We construct the corresponding metric components for
the theory presented at second order in perturbation, which are shown to be
perfectly compatible with the empirically derived ones. It is also shown that
under the theory being presented, in order to obtain a complete full agreement
with the observational results, a specific signature of Riemann's tensor has to
be chosen. This signature corresponds to the one most widely used nowadays in
relativity theory. Also, a computational program, the MEXICAS (Metric
EXtended-gravity Incorporated through a Computer Algebraic System) code,
developed for its usage in the Computer Algebraic System (CAS) Maxima for
working out perturbations on a metric theory of gravity, is presented and made
publicly available.Comment: 13 pages, 1 table. Accepted for publication in Monthly Notices of the
Royal Astronomical Society (MNRAS
On the Efficient Calculation of a Linear Combination of Chi-Square Random Variables with an Application in Counting String Vacua
Linear combinations of chi square random variables occur in a wide range of
fields. Unfortunately, a closed, analytic expression for the pdf is not yet
known. As a first result of this work, an explicit analytic expression for the
density of the sum of two gamma random variables is derived. Then a
computationally efficient algorithm to numerically calculate the linear
combination of chi square random variables is developed. An explicit expression
for the error bound is obtained. The proposed technique is shown to be
computationally efficient, i.e. only polynomial in growth in the number of
terms compared to the exponential growth of most other methods. It provides a
vast improvement in accuracy and shows only logarithmic growth in the required
precision. In addition, it is applicable to a much greater number of terms and
currently the only way of computing the distribution for hundreds of terms. As
an application, the exponential dependence of the eigenvalue fluctuation
probability of a random matrix model for 4d supergravity with N scalar fields
is found to be of the asymptotic form exp(-0.35N).Comment: 21 pages, 19 figures. 3rd versio
Adjusted chi-square test for degree-corrected block models
We propose a goodness-of-fit test for degree-corrected stochastic block
models (DCSBM). The test is based on an adjusted chi-square statistic for
measuring equality of means among groups of multinomial distributions with
observations. In the context of network models, the number of
multinomials, , grows much faster than the number of observations, ,
hence the setting deviates from classical asymptotics. We show that a simple
adjustment allows the statistic to converge in distribution, under null, as
long as the harmonic mean of grows to infinity. This result applies
to large sparse networks where the role of is played by the degree of
node . Our distributional results are nonasymptotic, with explicit
constants, providing finite-sample bounds on the Kolmogorov-Smirnov distance to
the target distribution. When applied sequentially, the test can also be used
to determine the number of communities. The test operates on a (row) compressed
version of the adjacency matrix, conditional on the degrees, and as a result is
highly scalable to large sparse networks. We incorporate a novel idea of
compressing the columns based on a -community assignment when testing
for communities. This approach increases the power in sequential
applications without sacrificing computational efficiency, and we prove its
consistency in recovering the number of communities. Since the test statistic
does not rely on a specific alternative, its utility goes beyond sequential
testing and can be used to simultaneously test against a wide range of
alternatives outside the DCSBM family. We show the effectiveness of the
approach by extensive numerical experiments with simulated and real data. In
particular, applying the test to the Facebook-100 dataset, we find that a DCSBM
with a small number of communities is far from a good fit in almost all cases
On multivariate chi-square distributions and their applications in testing multiple hypotheses
We are considered with three different types of multivariate chi-square distributions. Their members play important roles as limiting distributions of vectors of test statistics in several applications of multiple hypotheses testing. We explain these applications and provide formulas for computing multiplicity-adjusted -values under the respective global hypothesis
Heteroskedasticity and Spatiotemporal Dependence Robust Inference for Linear Panel Models with Fixed Effects
This paper studies robust inference for linear panel models with fixed effects in the presence of heteroskedasticity and spatiotemporal dependence of unknown forms. We propose a bivariate kernel covariance estimator that is
flexible to nest existing estimators as special cases with certain choices of bandwidths. For distributional approximations, we embed the level of smoothing and the sample size in two different limiting sequences. In the first case where the level of smoothing increases with the sample size, the proposed covariance estimator is consistent and the associated Wald statistic converges to a chi square distribution. We show that our covariance estimator improves upon existing estimators in terms of robustness and efficiency. In the second case where the level of smoothing is fixed, the covariance estimator has a random limit and we show by asymptotic expansion that the limiting distribution of the Wald statistic depends on the bandwidth parameters, the kernel function, and the number of restrictions being tested. As this distribution is nonstandard, we establish the validity of a convenient F-approximation to this distribution. For bandwidth selection, we employ and optimize a modified asymptotic mean square error criterion. The fl
exibility of our estimator and the proposed bandwidth selection procedure make our estimator adaptive to the dependence structure. This adaptiveness effectively automates the selection of covariance estimators. Simulation results show that our proposed testing procedure works reasonably well in finite samples.Adaptiveness, HAC estimator, F-approximation, Fixed-smoothing asymptotics, Increasing-smoothing asymptotics, Panel data, Optimal bandwidth, Robust inference, Spatiotemporal dependence
Stein's method on the second Wiener chaos : 2-Wasserstein distance
In the first part of the paper we use a new Fourier technique to obtain a
Stein characterizations for random variables in the second Wiener chaos. We
provide the connection between this result and similar conclusions that can be
derived using Malliavin calculus. We also introduce a new form of discrepancy
which we use, in the second part of the paper, to provide bounds on the
2-Wasserstein distance between linear combinations of independent centered
random variables. Our method of proof is entirely original. In particular it
does not rely on estimation of bounds on solutions of the so-called Stein
equations at the heart of Stein's method. We provide several applications, and
discuss comparison with recent similar results on the same topic
An overview of the goodness-of-fit test problem for copulas
We review the main "omnibus procedures" for goodness-of-fit testing for
copulas: tests based on the empirical copula process, on probability integral
transformations, on Kendall's dependence function, etc, and some corresponding
reductions of dimension techniques. The problems of finding asymptotic
distribution-free test statistics and the calculation of reliable p-values are
discussed. Some particular cases, like convenient tests for time-dependent
copulas, for Archimedean or extreme-value copulas, etc, are dealt with.
Finally, the practical performances of the proposed approaches are briefly
summarized
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