5,483 research outputs found
Monotonicity, asymptotic normality and vertex degrees in random graphs
We exploit a result by Nerman which shows that conditional limit theorems
hold when a certain monotonicity condition is satisfied. Our main result is an
application to vertex degrees in random graphs, where we obtain asymptotic
normality for the number of vertices with a given degree in the random graph
with a fixed number of edges from the corresponding result for the
random graph with independent edges. We also give some simple
applications to random allocations and to spacings. Finally, inspired by these
results, but logically independent of them, we investigate whether a one-sided
version of the Cram\'{e}r--Wold theorem holds. We show that such a version
holds under a weak supplementary condition, but not without it.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6103 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Distributed Constrained Recursive Nonlinear Least-Squares Estimation: Algorithms and Asymptotics
This paper focuses on the problem of recursive nonlinear least squares
parameter estimation in multi-agent networks, in which the individual agents
observe sequentially over time an independent and identically distributed
(i.i.d.) time-series consisting of a nonlinear function of the true but unknown
parameter corrupted by noise. A distributed recursive estimator of the
\emph{consensus} + \emph{innovations} type, namely , is
proposed, in which the agents update their parameter estimates at each
observation sampling epoch in a collaborative way by simultaneously processing
the latest locally sensed information~(\emph{innovations}) and the parameter
estimates from other agents~(\emph{consensus}) in the local neighborhood
conforming to a pre-specified inter-agent communication topology. Under rather
weak conditions on the connectivity of the inter-agent communication and a
\emph{global observability} criterion, it is shown that at every network agent,
the proposed algorithm leads to consistent parameter estimates. Furthermore,
under standard smoothness assumptions on the local observation functions, the
distributed estimator is shown to yield order-optimal convergence rates, i.e.,
as far as the order of pathwise convergence is concerned, the local parameter
estimates at each agent are as good as the optimal centralized nonlinear least
squares estimator which would require access to all the observations across all
the agents at all times. In order to benchmark the performance of the proposed
distributed estimator with that of the centralized nonlinear
least squares estimator, the asymptotic normality of the estimate sequence is
established and the asymptotic covariance of the distributed estimator is
evaluated. Finally, simulation results are presented which illustrate and
verify the analytical findings.Comment: 28 pages. Initial Submission: Feb. 2016, Revised: July 2016,
Accepted: September 2016, To appear in IEEE Transactions on Signal and
Information Processing over Networks: Special Issue on Inference and Learning
over Network
Asymptotics in directed exponential random graph models with an increasing bi-degree sequence
Although asymptotic analyses of undirected network models based on degree
sequences have started to appear in recent literature, it remains an open
problem to study statistical properties of directed network models. In this
paper, we provide for the first time a rigorous analysis of directed
exponential random graph models using the in-degrees and out-degrees as
sufficient statistics with binary as well as continuous weighted edges. We
establish the uniform consistency and the asymptotic normality for the maximum
likelihood estimate, when the number of parameters grows and only one realized
observation of the graph is available. One key technique in the proofs is to
approximate the inverse of the Fisher information matrix using a simple matrix
with high accuracy. Numerical studies confirm our theoretical findings.Comment: Published at http://dx.doi.org/10.1214/15-AOS1343 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Confidence intervals for test information and relative efficiency
In latent theory the measurement properties of a mental test can be expressed in the test information function. The relative merits of two tests for the same latent trait can be described by the relative efficiency function, i.e. the ratio of the test information functions. It is argued that these functions have to be estimated if the values of the item difficulties are unknown. Using conditional maximum likelihood estimation as indicated by Andersen (1973), pointwise asymptotic distributions of the test information and relative efficiency function are derived for the case of dichotomously scored Rasch homogeneous items. Formulas for confidence intervals are derived from the asymptotic distributions. An application to a mathematics test is given and extensions to other latent trait models are discussed
Mod-phi convergence I: Normality zones and precise deviations
In this paper, we use the framework of mod- convergence to prove
precise large or moderate deviations for quite general sequences of real valued
random variables , which can be lattice or
non-lattice distributed. We establish precise estimates of the fluctuations
, instead of the usual estimates for the rate of
exponential decay . Our approach provides us with a
systematic way to characterise the normality zone, that is the zone in which
the Gaussian approximation for the tails is still valid. Besides, the residue
function measures the extent to which this approximation fails to hold at the
edge of the normality zone.
The first sections of the article are devoted to a proof of these abstract
results and comparisons with existing results. We then propose new examples
covered by this theory and coming from various areas of mathematics: classical
probability theory, number theory (statistics of additive arithmetic
functions), combinatorics (statistics of random permutations), random matrix
theory (characteristic polynomials of random matrices in compact Lie groups),
graph theory (number of subgraphs in a random Erd\H{o}s-R\'enyi graph), and
non-commutative probability theory (asymptotics of random character values of
symmetric groups). In particular, we complete our theory of precise deviations
by a concrete method of cumulants and dependency graphs, which applies to many
examples of sums of "weakly dependent" random variables. The large number as
well as the variety of examples hint at a universality class for second order
fluctuations.Comment: 103 pages. New (final) version: multiple small improvements ; a new
section on mod-Gaussian convergence coming from the factorization of the
generating function ; the multi-dimensional results have been moved to a
forthcoming paper ; and the introduction has been reworke
Degree-based goodness-of-fit tests for heterogeneous random graph models : independent and exchangeable cases
The degrees are a classical and relevant way to study the topology of a
network. They can be used to assess the goodness-of-fit for a given random
graph model. In this paper we introduce goodness-of-fit tests for two classes
of models. First, we consider the case of independent graph models such as the
heterogeneous Erd\"os-R\'enyi model in which the edges have different
connection probabilities. Second, we consider a generic model for exchangeable
random graphs called the W-graph. The stochastic block model and the expected
degree distribution model fall within this framework. We prove the asymptotic
normality of the degree mean square under these independent and exchangeable
models and derive formal tests. We study the power of the proposed tests and we
prove the asymptotic normality under specific sparsity regimes. The tests are
illustrated on real networks from social sciences and ecology, and their
performances are assessed via a simulation study
- …