216 research outputs found
Asymptotic inference for high-dimensional data
In this paper, we study inference for high-dimensional data characterized by
small sample sizes relative to the dimension of the data. In particular, we
provide an infinite-dimensional framework to study statistical models that
involve situations in which (i) the number of parameters increase with the
sample size (that is, allowed to be random) and (ii) there is a possibility of
missing data. Under a variety of tail conditions on the components of the data,
we provide precise conditions for the joint consistency of the estimators of
the mean. In the process, we clarify and improve some of the recent consistency
results that appeared in the literature. An important aspect of the work
presented is the development of asymptotic normality results for these models.
As a consequence, we construct different test statistics for one-sample and
two-sample problems concerning the mean vector and obtain their asymptotic
distributions as a corollary of the infinite-dimensional results. Finally, we
use these theoretical results to develop an asymptotically justifiable
methodology for data analyses. Simulation results presented here describe
situations where the methodology can be successfully applied. They also
evaluate its robustness under a variety of conditions, some of which are
substantially different from the technical conditions. Comparisons to other
methods used in the literature are provided. Analyses of real-life data is also
included.Comment: Published in at http://dx.doi.org/10.1214/09-AOS718 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Corrections and acknowledgment for ``Local limit theory and large deviations for supercritical branching processes''
Corrections and acknowledgment for ``Local limit theory and large deviations
for supercritical branching processes'' [math.PR/0407059]Comment: Published at http://dx.doi.org/10.1214/105051606000000574 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Local limit theory and large deviations for supercritical Branching processes
In this paper we study several aspects of the growth of a supercritical
Galton-Watson process {Z_n:n\ge1}, and bring out some criticality phenomena
determined by the Schroder constant. We develop the local limit theory of Z_n,
that is, the behavior of P(Z_n=v_n) as v_n\nearrow \infty, and use this to
study conditional large deviations of {Y_{Z_n}:n\ge1}, where Y_n satisfies an
LDP, particularly of {Z_n^{-1}Z_{n+1}:n\ge1} conditioned on Z_n\ge v_n
Estimation with Norm Regularization
Analysis of non-asymptotic estimation error and structured statistical
recovery based on norm regularized regression, such as Lasso, needs to consider
four aspects: the norm, the loss function, the design matrix, and the noise
model. This paper presents generalizations of such estimation error analysis on
all four aspects compared to the existing literature. We characterize the
restricted error set where the estimation error vector lies, establish
relations between error sets for the constrained and regularized problems, and
present an estimation error bound applicable to any norm. Precise
characterizations of the bound is presented for isotropic as well as
anisotropic subGaussian design matrices, subGaussian noise models, and convex
loss functions, including least squares and generalized linear models. Generic
chaining and associated results play an important role in the analysis. A key
result from the analysis is that the sample complexity of all such estimators
depends on the Gaussian width of a spherical cap corresponding to the
restricted error set. Further, once the number of samples crosses the
required sample complexity, the estimation error decreases as
, where depends on the Gaussian width of the unit norm
ball.Comment: Fixed technical issues. Generalized some result
Large deviation results for branching processes in fixed and random environments
This thesis considers three different aspects of large deviations for branching processes. First, we study the deviation between the empirical mean and the true mean. Second, we investigate the large deviation behavior exhibited by the tail of the random variable W occurring in multi-type branching processes. Finally, we discuss the large deviations as they apply to a branching random walk in stationary ergodic environments
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