12,318 research outputs found
Asymptotics of the discrete log-concave maximum likelihood estimator and related applications
The assumption of log-concavity is a flexible and appealing nonparametric
shape constraint in distribution modelling. In this work, we study the
log-concave maximum likelihood estimator (MLE) of a probability mass function
(pmf). We show that the MLE is strongly consistent and derive its pointwise
asymptotic theory under both the well- and misspecified setting. Our asymptotic
results are used to calculate confidence intervals for the true log-concave
pmf. Both the MLE and the associated confidence intervals may be easily
computed using the R package logcondiscr. We illustrate our theoretical results
using recent data from the H1N1 pandemic in Ontario, Canada.Comment: 21 pages, 7 Figure
Central limit theorems for double Poisson integrals
Motivated by second order asymptotic results, we characterize the convergence
in law of double integrals, with respect to Poisson random measures, toward a
standard Gaussian distribution. Our conditions are expressed in terms of
contractions of the kernels. To prove our main results, we use the theory of
stable convergence of generalized stochastic integrals developed by Peccati and
Taqqu. One of the advantages of our approach is that the conditions are
expressed directly in terms of the kernel appearing in the multiple integral
and do not make any explicit use of asymptotic dependence properties such as
mixing. We illustrate our techniques by an application involving linear and
quadratic functionals of generalized Ornstein--Uhlenbeck processes, as well as
examples concerning random hazard rates.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ123 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Random Geometric Series
Integer sequences where each element is determined by a previous randomly
chosen element are investigated analytically. In particular, the random
geometric series x_n=2x_p with 0<=p<=n-1 is studied. At large n, the moments
grow algebraically, n^beta(s) with beta(s)=2^s-1, while the typical
behavior is x_n n^ln 2. The probability distribution is obtained explicitly in
terms of the Stirling numbers of the first kind and it approaches a log-normal
distribution asymptotically.Comment: 6 pages, 2 figure
A detailed investigation into near degenerate exponential random graphs
The exponential family of random graphs has been a topic of continued
research interest. Despite the relative simplicity, these models capture a
variety of interesting features displayed by large-scale networks and allow us
to better understand how phases transition between one another as tuning
parameters vary. As the parameters cross certain lines, the model
asymptotically transitions from a very sparse graph to a very dense graph,
completely skipping all intermediate structures. We delve deeper into this near
degenerate tendency and give an explicit characterization of the asymptotic
graph structure as a function of the parameters.Comment: 15 pages, 3 figures, 2 table
Inferring Pattern and Disorder in Close-Packed Structures from X-ray Diffraction Studies, Part II: Structure and Intrinsic Computation in Zinc Sulphide
In the previous paper of this series [D. P. Varn, G. S. Canright, and J. P.
Crutchfield, Physical Review B, submitted] we detailed a
procedure--epsilon-machine spectral reconstruction--to discover and analyze
patterns and disorder in close-packed structures as revealed in x-ray
diffraction spectra. We argued that this computational mechanics approach is
more general than the current alternative theory, the fault model, and that it
provides a unique characterization of the disorder present. We demonstrated the
efficacy of computational mechanics on four prototype spectra, finding that it
was able to recover a statistical description of the underlying modular-layer
stacking using epsilon-machine representations. Here we use this procedure to
analyze structure and disorder in four previously published zinc sulphide
diffraction spectra. We selected zinc sulphide not only for the theoretical
interest this material has attracted in an effort to develop an understanding
of polytypism, but also because it displays solid-state phase transitions and
experimental data is available.Comment: 15 pages, 14 figures, 4 tables, 57 citations;
http://www.santafe.edu/projects/CompMech/papers/ipdcpsii.htm
Performance analysis and optimal selection of large mean-variance portfolios under estimation risk
We study the consistency of sample mean-variance portfolios of arbitrarily
high dimension that are based on Bayesian or shrinkage estimation of the input
parameters as well as weighted sampling. In an asymptotic setting where the
number of assets remains comparable in magnitude to the sample size, we provide
a characterization of the estimation risk by providing deterministic
equivalents of the portfolio out-of-sample performance in terms of the
underlying investment scenario. The previous estimates represent a means of
quantifying the amount of risk underestimation and return overestimation of
improved portfolio constructions beyond standard ones. Well-known for the
latter, if not corrected, these deviations lead to inaccurate and overly
optimistic Sharpe-based investment decisions. Our results are based on recent
contributions in the field of random matrix theory. Along with the asymptotic
analysis, the analytical framework allows us to find bias corrections improving
on the achieved out-of-sample performance of typical portfolio constructions.
Some numerical simulations validate our theoretical findings
From Random Matrices to Quasiperiodic Jacobi Matrices via Orthogonal Polynomials
We present an informal review of results on asymptotics of orthogonal
polynomials, stressing their spectral aspects and similarity in two cases
considered. They are polynomials orthonormal on a finite union of disjoint
intervals with respect to the Szego weight and polynomials orthonormal on R
with respect to varying weights and having the same union of intervals as the
set of oscillations of asymptotics. In both cases we construct double infinite
Jacobi matrices with generically quasiperiodic coefficients and show that each
of them is an isospectral deformation of another. Related results on asymptotic
eigenvalue distribution of a class of random matrices of large size are also
shortly discussed
- …