178 research outputs found
On Hoeffding's inequalities
In a celebrated work by Hoeffding [J. Amer. Statist. Assoc. 58 (1963) 13-30],
several inequalities for tail probabilities of sums M_n=X_1+... +X_n of bounded
independent random variables X_j were proved. These inequalities had a
considerable impact on the development of probability and statistics, and
remained unimproved until 1995 when Talagrand [Inst. Hautes Etudes Sci. Publ.
Math. 81 (1995a) 73-205] inserted certain missing factors in the bounds of two
theorems. By similar factors, a third theorem was refined by Pinelis [Progress
in Probability 43 (1998) 257-314] and refined (and extended) by me. In this
article, I introduce a new type of inequality.
Namely, I show that P{M_n\geq x}\leq cP{S_n\geq x}, where c is an absolute
constant and S_n=\epsilon_1+... +\epsilon_n is a sum of independent identically
distributed Bernoulli random variables (a random variable is called Bernoulli
if it assumes at most two values). The inequality holds for those x\in R where
the survival function x\mapsto P{S_n\geq x} has a jump down. For the remaining
x the inequality still holds provided that the function between the adjacent
jump points is interpolated linearly or \log-linearly. If it is necessary, to
estimate P{S_n\geq x} special bounds can be used for binomial probabilities.
The results extend to martingales with bounded differences. It is apparent that
Theorem 1.1 of this article is the most important.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000036
Lattice point problems and distribution of values of quadratic forms
For d-dimensional irrational ellipsoids E with d >= 9 we show that the number
of lattice points in rE is approximated by the volume of rE, as r tends to
infinity, up to an error of order o(r^{d-2}). The estimate refines an earlier
authors' bound of order O(r^{d-2}) which holds for arbitrary ellipsoids, and is
optimal for rational ellipsoids. As an application we prove a conjecture of
Davenport and Lewis that the gaps between successive values, say s<n(s), s,n(s)
in Q[Z^d], of a positive definite irrational quadratic form Q[x], x in R^d, are
shrinking, i.e., that n(s) - s -> 0 as s -> \infty, for d >= 9. For comparison
note that sup_s (n(s)-s) 0, for rational Q[x] and
d>= 5. As a corollary we derive Oppenheim's conjecture for indefinite
irrational quadratic forms, i.e., the set Q[Z^d] is dense in R, for d >= 9,
which was proved for d >= 3 by G. Margulis in 1986 using other methods.
Finally, we provide explicit bounds for errors in terms of certain
characteristics of trigonometric sums.Comment: 51 pages, published versio
Quasi-likelihood-based higher-order spectral estimation of random fields with possible long-range dependence
An Edgeworth expansion for finite population L-statistics
In this paper, we consider the one-term Edgeworth expansion for finite
population L-statistics. We provide an explicit formula for the Edgeworth
correction term and give sufficient conditions for the validity of the
expansion which are expressed in terms of the weight function that defines the
statistics and moment conditions.Comment: 14 pages. Minor revisions. Some explanatory comments and a numerical
example were added. Lith. Math. J. (to appear
Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in random environment
The objective of the present paper is to establish exponential large
deviation inequalities, and to use them to show exponential concentration
inequalities for the free energy of a polymer in general random environment,
its rate of convergence, and an expression of its limit value in terms of those
of some multiplicative cascades.Comment: 25 page
Optimal Uncertainty Quantification
We propose a rigorous framework for Uncertainty Quantification (UQ) in which
the UQ objectives and the assumptions/information set are brought to the
forefront. This framework, which we call \emph{Optimal Uncertainty
Quantification} (OUQ), is based on the observation that, given a set of
assumptions and information about the problem, there exist optimal bounds on
uncertainties: these are obtained as values of well-defined optimization
problems corresponding to extremizing probabilities of failure, or of
deviations, subject to the constraints imposed by the scenarios compatible with
the assumptions and information. In particular, this framework does not
implicitly impose inappropriate assumptions, nor does it repudiate relevant
information. Although OUQ optimization problems are extremely large, we show
that under general conditions they have finite-dimensional reductions. As an
application, we develop \emph{Optimal Concentration Inequalities} (OCI) of
Hoeffding and McDiarmid type. Surprisingly, these results show that
uncertainties in input parameters, which propagate to output uncertainties in
the classical sensitivity analysis paradigm, may fail to do so if the transfer
functions (or probability distributions) are imperfectly known. We show how,
for hierarchical structures, this phenomenon may lead to the non-propagation of
uncertainties or information across scales. In addition, a general algorithmic
framework is developed for OUQ and is tested on the Caltech surrogate model for
hypervelocity impact and on the seismic safety assessment of truss structures,
suggesting the feasibility of the framework for important complex systems. The
introduction of this paper provides both an overview of the paper and a
self-contained mini-tutorial about basic concepts and issues of UQ.Comment: 90 pages. Accepted for publication in SIAM Review (Expository
Research Papers). See SIAM Review for higher quality figure
On the error term in Weyl's law for the Heisenberg manifolds (II)
In this paper we study the mean square of the error term in the Weyl's law of
an irrational -dimensional Heisenberg manifold . An asymptotic formula
is established
Towards Machine Wald
The past century has seen a steady increase in the need of estimating and
predicting complex systems and making (possibly critical) decisions with
limited information. Although computers have made possible the numerical
evaluation of sophisticated statistical models, these models are still designed
\emph{by humans} because there is currently no known recipe or algorithm for
dividing the design of a statistical model into a sequence of arithmetic
operations. Indeed enabling computers to \emph{think} as \emph{humans} have the
ability to do when faced with uncertainty is challenging in several major ways:
(1) Finding optimal statistical models remains to be formulated as a well posed
problem when information on the system of interest is incomplete and comes in
the form of a complex combination of sample data, partial knowledge of
constitutive relations and a limited description of the distribution of input
random variables. (2) The space of admissible scenarios along with the space of
relevant information, assumptions, and/or beliefs, tend to be infinite
dimensional, whereas calculus on a computer is necessarily discrete and finite.
With this purpose, this paper explores the foundations of a rigorous framework
for the scientific computation of optimal statistical estimators/models and
reviews their connections with Decision Theory, Machine Learning, Bayesian
Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty
Quantification and Information Based Complexity.Comment: 37 page
An improvement of the Berry--Esseen inequality with applications to Poisson and mixed Poisson random sums
By a modification of the method that was applied in (Korolev and Shevtsova,
2009), here the inequalities
and
are proved for the
uniform distance between the standard normal distribution
function and the distribution function of the normalized sum of an
arbitrary number of independent identically distributed random
variables with zero mean, unit variance and finite third absolute moment
. The first of these inequalities sharpens the best known version of
the classical Berry--Esseen inequality since
by virtue of
the condition , and 0.4785 is the best known upper estimate of the
absolute constant in the classical Berry--Esseen inequality. The second
inequality is applied to lowering the upper estimate of the absolute constant
in the analog of the Berry--Esseen inequality for Poisson random sums to 0.3051
which is strictly less than the least possible value of the absolute constant
in the classical Berry--Esseen inequality. As a corollary, the estimates of the
rate of convergence in limit theorems for compound mixed Poisson distributions
are refined.Comment: 33 page
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