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 $n^{-1}$ for smooth test functions. These general bounds are applied to Friedman's statistic for comparing $r$ treatments across $n$ trials and the family of power divergence statistics for goodness-of-fit across $n$ trials and $r$ classifications, with index parameter $\lambda\in\mathbb{R}$ (Pearson's statistic corresponds to $\lambda=1$). We obtain a $O(n^{-1})$ bound for the rate of convergence of Friedman's statistic for any number of treatments $r\geq2$. We also obtain a $O(n^{-1})$ bound on the rate of convergence of the power divergence statistics for any $r\geq2$ when $\lambda$ is a positive integer or any real number greater than 5. We conjecture that the $O(n^{-1})$ rate holds for any $\lambda\in\mathbb{R}$.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 $p_1$ and $p_2$ having nested supports. These explicit bounds are expressed in terms of the derivative of the likelihood ratio $p_1/p_2$ as well as the Stein kernel $\tau_1$ of $p_1$. 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-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

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

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