Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition

By Gesine Reinert, Adrian Röllin. 2007

191 citations recorded by the source corpus.

In this paper we establish a multivariate exchangeable pairs approach within the framework of Stein's method to assess distributional distances to potentially singular multivariate normal distributions. By extending the statistics into a higher-dimensional space, we also propose an embedding method which allows for a normal approximation even when the corresponding statistics of interest do not lend themselves easily to Stein's exchangeable pairs approach. To illustrate the method, we provide the examples of runs on the line as well as double-indexed permutation statistics.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e012cda984b868186941392379a01985bd284ba1

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Citations in the corpus

Nearly optimal central limit theorem and bootstrap approximations in high dimensions (2020, methodology): “hen Xi’s have independent components with vanishing odd moments. Our results are built using the exchangeable pair approach coupled with the SlepianStein method and employ ideas of many authors, e.g. [7, 28, 3, 8, 11, 36, 12, 15, 23, 32, 31], but the key technical tool behind our results is a set of”
GAUSSIAN FLUCTUATIONS OF YOUNG DIAGRAMS AND STRUCTURE CONSTANTS OF JACK CHARACTERS (2014, methodology): “nce in this theorem (Theorem 1.3). 7.1. Multivariate Stein’s method. As explained in introduction, our main tool will be a multivariate analog of the so-calledStein’s method due to Reinert and Röllin [RR09]. For any discrete random variables W,W∗ with values in Rd, we say that the pair (W,W∗) is exc”
Variance-Gamma approximation via Stein's method (2013, methodology): “For large m and n we have S ≈ N(0, 1) and T ≈ N(0, 1), so we can apply the O(m−1/2+n−1/2) bivariate central limit convergence rate (see, for example, Reinert and Röllin [38]) to bound the quantities |Egk(S, T )−Egk(Z1, Z2)|, k = 1, 2, 3.”
On Stein's method for multivariate normal approximation (2009, methodology): “It should be emphasized that showing that the number of d-runs on the line is asymptotically Gaussian seems infeasible with Stein’s original method of exchangeable pairs because of the failure of condition (1) from the introduction, but in [ 16 ], the random variable of interest is embedded in a ran”
A CLT and tight lower bounds for estimating entropy (2010, background): “Together with the canonical estimator constructed in the companion paper [33], this settles the longstanding open question of the sample complexities of these estimation problems, up to constant factors.”
Measuring Sample Quality with Stein's Method (2015, methodology): “We restrict attention to the unconstrained domain X = R d in Sections 4.1-4.3 and present extensions for constrained domains in Section 4.4.”
MULTIVARIATE NORMAL APPROXIMATION USING EXCHANGEABLE PAIRS (2007, unknown): “Subsequently, an extension of one of our main results (Theorem 4) to the case of multivariate normal approximation with non-identity covariance was formulated by Reinert and R¨ollin [ 37 ].”
Approximation of stable law in Wasserstein-1 distance by Stein’s method (2017, methodology): “Nowadays, Stein’s method has been extended and refined by many authors and become a very important tool for getting bounds of measure approximations, see [6, 10, 14, 22, 24, 29, 39, 46, 26, 27, 28, 33, 42, 43, 44, 47].”
Gaussian approximation of suprema of empirical processes (2012, methodology): “1), where Stein’s method for normal approximation (building here on [7] and originally due to [54, 55]) plays an important role (see also [9, 43, 50]); (ii) a deviation inequality for suprema of empirical processes that only requires finite moments of envelope functions (Theorem 5.”
Distribution-Free Consistent Independence Tests via Center-Outward Ranks and Signs (2019, background): “2.4), Shapiro and Hubert (1979), Barbour and Eagleson (1986), Pham, Möcks, and Sroka (1989), and the Berry–Esseen bound was established in Zhao et al. (1997), Barbour and Chen (2005), and Reinert and Röllin (2009).”
Measuring Sample Quality with Diffusions (2016, background): “11 Example 1 (Bayesian logistic regression with Gaussian prior).”
Multivariate normal approximation using Stein's method and Malliavin calculus (2008, methodology): “[4] or [20]), is the d-dimensional counterpart of Stein’s Lemma 1.”
Multi-Dimensional Gaussian Fluctuations on the Poisson Space (2010, methodology): “The following result is a d-dimensional version of Stein’s Lemma; analogous statements can be found in [4, 10, 22] – see also Barbour [1] and Götze [5], in connection with the so-called “generator approach” to Stein’s method.”
Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function (2010, methodology): “1 of Reinert and Röllin [19], we calculate two further quantities:”
Nonparametric inference in generalized functional linear models (2014, background): “So ( , ′) is an exchangeable pair (see [32]).”
Stein's method for comparison of univariate distributions (2014, background): “Also there exist many ways by which one can evaluate E [Af(W )] or E[AW f(W ) − AZf(W )] (even under unfavorable assumptions on W ) including exchangeable pairs (as for example in [91, 52, 85, 21, 19, 28]), biasing mechanisms (as in [4, 44, 41, 79, 37]), and other couplings (as in [22, 9]); see [84,”