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The paper applies the theory developed in Part I to the discrete normal approximation in total variation of random vectors in ${\mathbb Z}^d$. We illustrate the use of the method for sums of independent integer valued random vectors, and…

Probability · Mathematics 2016-12-23 A. D. Barbour , Malwina J. Luczak , Aihua Xia

For integer valued random variables, the translated Poisson distributions form a flexible family for approximation in total variation, in much the same way that the normal family is used for approximation in Kolmogorov distance. Using the…

Probability · Mathematics 2016-12-26 A. D. Barbour , Malwina J. Luczak , Aihua Xia

Given a random sample from a parametric model, we show how indirect inference estimators based on appropriate nonparametric density estimators (i.e., simulation-based minimum distance estimators) can be constructed that, under mild…

Statistics Theory · Mathematics 2011-01-10 Richard Nickl , Benedikt M. Pötscher

We consider the problem of quantum multi-parameter estimation with experimental constraints and formulate the solution in terms of a convex optimization. Specifically, we outline an efficient method to identify the optimal strategy for…

Quantum Physics · Physics 2013-05-29 Kevin C. Young , Mohan Sarovar , Robert Kosut , K. Birgitta Whaley

The field of quantum metrology seeks to apply quantum techniques and/or resources to classical sensing approaches with the goal of enhancing the precision in the estimation of a parameter beyond what can be achieved with classical…

A usual assumption in quantum estimation is that the unknown parameter labels the possible states of the system, while it influences neither the sample space of outcomes nor the measurement aimed at extracting information on the parameter…

Quantum Physics · Physics 2017-01-18 Luigi Seveso , Matteo A. C. Rossi , Matteo G. A. Paris

Unbiased estimators are introduced for averaged Bregman divergences which generalize Stein's Unbiased (Predictive) Risk Estimator, and the minimization of these estimators is proposed as a regularization parameter selection method for…

Numerical Analysis · Mathematics 2021-11-22 Elias S. Helou , Sandra A. Santos , Lucas E. A. Simões

We develop a variant of Stein's method of comparison of generators to bound the Kolmogorov, total variation, and Wasserstein-1 distances between distributions on the real line. Our discrepancy is expressed in terms of the ratio of reverse…

Probability · Mathematics 2025-10-28 Paul Mansanarez , Guillaume Poly , Yvik Swan

Applying Stein's method, an inductive technique and size bias coupling yields a Berry-Esseen theorem for normal approximation without the usual restriction that the coupling be bounded. The theorem is applied to counting the number of…

Probability · Mathematics 2013-02-26 Larry Goldstein

We provide a new general theorem for multivariate normal approximation on convex sets. The theorem is formulated in terms of a multivariate extension of Stein couplings. We apply the results to a homogeneity test in dense random graphs and…

Probability · Mathematics 2016-08-14 Xiao Fang , Adrian Röllin

By using the conjugate distribution technique of Cram\'er, we obtain some expansions of large deviation probabilities for martingales with differences satisfying the conditional Bernstein's condition. The expansions are of the same order as…

Probability · Mathematics 2014-09-16 Xiequan Fan , Ion Grama , Quansheng Liu

We derive normal approximation bounds in the Wasserstein distance for sums of weighted U-statistics, based on a general distance bound for functionals of independent random variables of arbitrary distributions. Those bounds are applied to…

Probability · Mathematics 2020-07-28 Nicolas Privault , Grzegorz Serafin

We develop Stein's method for the half-normal distribution and apply it to derive rates of convergence in distributional limit theorems for three statistics of the simple symmetric random walk: the maximum value, the number of returns to…

Probability · Mathematics 2015-11-24 Christian Döbler

We derive Stein approximation bounds for functionals of uniform random variables, using chaos expansions and the Clark-Ocone representation formula combined with derivation and finite difference operators. This approach covers sums and…

Probability · Mathematics 2018-02-28 Nicolas Privault , Grzegorz Serafin

We establish a quantitative normal approximation result for sums of random variables with multilevel local dependencies. As a corollary, we obtain a quantitative normal approximation result for linear functionals of random fields which may…

Probability · Mathematics 2019-05-27 Julian Fischer

Stein's method provides a way of bounding the distance of a probability distribution to a target distribution $\mu$. Here we develop Stein's method for the class of discrete Gibbs measures with a density $e^V$, where $V$ is the energy…

Probability · Mathematics 2008-08-22 Peter Eichelsbacher , Gesine Reinert

We study the mutual information estimation for mixed-pair random variables. One random variable is discrete and the other one is continuous. We develop a kernel method to estimate the mutual information between the two random variables. The…

Statistics Theory · Mathematics 2018-12-31 Aleksandr Beknazaryan , Xin Dang , Hailin Sang

The purpose of the present paper is to establish moderate deviation principles for a rather general class of random variables fulfilling certain bounds of the cumulants. We apply a celebrated lemma of the theory of large deviations…

Probability · Mathematics 2012-09-28 Hanna Doering , Peter Eichelsbacher

Let $(X_{i}, i\in J)$ be a family of locally dependent nonnegative integer-valued random variables, and consider the sum $W=\sum\nolimits_{i\in J}X_i$. We first establish a general error upper bound for $d_{TV}(W, M)$ using Stein's method,…

Probability · Mathematics 2023-12-12 Zhonggen Su , Vladimir V. Ulyanov , Xiaolin Wang

In this work, we generalize the Cram\'er-von Mises statistic via projection-averaging to obtain a robust test for the multivariate two-sample problem. The proposed test is consistent against all fixed alternatives, robust to heavy-tailed…

Statistics Theory · Mathematics 2019-05-22 Ilmun Kim , Sivaraman Balakrishnan , Larry Wasserman