152 research outputs found
Approximating stationary distributions of fast mixing Glauber dynamics, with applications to exponential random graphs
We provide a general bound on the Wasserstein distance between two arbitrary
distributions of sequences of Bernoulli random variables. The bound is in terms
of a mixing quantity for the Glauber dynamics of one of the sequences, and a
simple expectation of the other. The result is applied to estimate, with
explicit error, expectations of functions of random vectors for some Ising
models and exponential random graphs in "high temperature" regimes.Comment: Ver3: 24 pages, major revision with new results; Ver2: updated
reference; Ver1: 19 pages, 1 figur
Bounds for the normal approximation of the maximum likelihood estimator
While the asymptotic normality of the maximum likelihood estimator under
regularity conditions is long established, this paper derives explicit bounds
for the bounded Wasserstein distance between the distribution of the maximum
likelihood estimator (MLE) and the normal distribution. For this task, we
employ Stein's method. We focus on independent and identically distributed
random variables, covering both discrete and continuous distributions as well
as exponential and non-exponential families. In particular, a closed form
expression of the MLE is not required. We also use a perturbation method to
treat cases where the MLE has positive probability of being on the boundary of
the parameter space.Comment: Published at http://dx.doi.org/10.3150/15-BEJ741 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
The rate of convergence of some asymptotically chi-square distributed statistics by Stein's method
We build on recent works on Stein's method for functions of multivariate
normal random variables to derive bounds for the rate of convergence of some
asymptotically chi-square distributed statistics. We obtain some general bounds
and establish some simple sufficient conditions for convergence rates of order
for smooth test functions. These general bounds are applied to
Friedman's statistic for comparing treatments across trials and the
family of power divergence statistics for goodness-of-fit across trials and
classifications, with index parameter (Pearson's
statistic corresponds to ). We obtain a bound for the
rate of convergence of Friedman's statistic for any number of treatments
. We also obtain a bound on the rate of convergence of the
power divergence statistics for any when is a positive
integer or any real number greater than 5. We conjecture that the
rate holds for any .Comment: 32 page
Distances between nested densities and a measure of the impact of the prior in Bayesian statistics
In this paper we propose tight upper and lower bounds for the Wasserstein
distance between any two {{univariate continuous distributions}} with
probability densities and having nested supports. These explicit
bounds are expressed in terms of the derivative of the likelihood ratio
as well as the Stein kernel of . The method of proof
relies on a new variant of Stein's method which manipulates Stein operators.
We give several applications of these bounds. Our main application is in
Bayesian statistics : we derive explicit data-driven bounds on the Wasserstein
distance between the posterior distribution based on a given prior and the
no-prior posterior based uniquely on the sampling distribution. This is the
first finite sample result confirming the well-known fact that with
well-identified parameters and large sample sizes, reasonable choices of prior
distributions will have only minor effects on posterior inferences if the data
are benign
Stein's method and stochastic analysis of Rademacher functionals
We compute explicit bounds in the Gaussian approximation of functionals of
infinite Rademacher sequences. Our tools involve Stein's method, as well as the
use of appropriate discrete Malliavin operators. Although our approach does not
require the classical use of exchangeable pairs, we employ a chaos expansion in
order to construct an explicit exchangeable pair vector for any random variable
which depends on a finite set of Rademacher variables. Among several examples,
which include random variables which depend on infinitely many Rademacher
variables, we provide three main applications: (i) to CLTs for multilinear
forms belonging to a fixed chaos, (ii) to the Gaussian approximation of
weighted infinite 2-runs, and (iii) to the computation of explicit bounds in
CLTs for multiple integrals over sparse sets. This last application provides an
alternate proof (and several refinements) of a recent result by Blei and
Janson.Comment: 35 pages + Appendix. New version: some inaccuracies in Sect. 6
correcte
Stein-type covariance identities: Klaassen, Papathanasiou and Olkin-Shepp type bounds for arbitrary target distributions
In this paper, we present a minimal formalism for Stein operators which leads
to different probabilistic representations of solutions to Stein equations.
These in turn provide a wide family of Stein-Covariance identities which we put
to use for revisiting the very classical topic of bounding the variance of
functionals of random variables. Applying the Cauchy-Schwarz inequality yields
first order upper and lower Klaassen-type variance bounds. A probabilistic
representation of Lagrange's identity (i.e. Cauchy-Schwarz with remainder)
leads to Papathanasiou-type variance expansions of arbitrary order. A matrix
Cauchy-Schwarz inequality leads to Olkin-Shepp type covariance bounds. All
results hold for univariate target distribution under very weak assumptions (in
particular they hold for continuous and discrete distributions alike). Many
concrete illustrations are provided
Risk in a large claims insurance market with bipartite graph structure
We model the influence of sharing large exogeneous losses to the reinsurance
market by a bipartite graph. Using Pareto-tailed claims and multivariate
regular variation we obtain asymptotic results for the Value-at-Risk and the
Conditional Tail Expectation. We show that the dependence on the network
structure plays a fundamental role in their asymptotic behaviour. As is
well-known in a non-network setting, if the Pareto exponent is larger than 1,
then for the individual agent (reinsurance company) diversification is
beneficial, whereas when it is less than 1, concentration on a few objects is
the better strategy. An additional aspect of this paper is the amount of
uninsured losses which have to be convered by society. In the situation of
networks of agents, in our setting diversification is never detrimental
concerning the amount of uninsured losses. If the Pareto-tailed claims have
finite mean, diversification turns out to be never detrimental, both for
society and for individual agents. In contrast, if the Pareto-tailed claims
have infinite mean, a conflicting situation may arise between the incentives of
individual agents and the interest of some regulator to keep risk for society
small. We explain the influence of the network structure on diversification
effects in different network scenarios
- …