Stein's method and the zero bias transformation with application to simple random sampling

Goldstein, Larry; Reinert, Gesine

doi:10.1214/aoap/1043862419

Cited by 168 publications

(273 citation statements)

References 14 publications

Supporting

Mentioning

263

Contrasting

Unclassified

Order By: Relevance

“…The first is the notion of the 'zerobias transform' of W , as defined by Goldstein and Reinert [25]. A random variable W * is called a zerobias transform of W if for all ϕ, we have…”

Section: 2mentioning

confidence: 99%

Fluctuations of eigenvalues and second order Poincaré inequalities

Chatterjee

2007

Probab. Theory Relat. Fields

181

263

View full text Add to dashboard Cite

Abstract. Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of 'second order Poincaré inequalities': just as ordinary Poincaré inequalities give variance bounds, second order Poincaré inequalities give central limit theorems. The proof of the main result employs Stein's method of normal approximation. A number of examples are worked out, some of which are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.

show abstract

“…The first is the notion of the 'zerobias transform' of W , as defined by Goldstein and Reinert [25]. A random variable W * is called a zerobias transform of W if for all ϕ, we have…”

Section: 2mentioning

confidence: 99%

Fluctuations of eigenvalues and second order Poincaré inequalities

Chatterjee

2007

Probab. Theory Relat. Fields

181

263

View full text Add to dashboard Cite

show abstract

“…This faster rate arises from our use of smooth test functions; a related result is obtained in [15] using a coupling approach.…”

Section: First Examples: Rademacher Averagesmentioning

confidence: 82%

“…Its domain, noted domD, is given by the class of random variables F ∈ L 2 (σ{X}) such that the kernels f n ∈ ℓ 2 0 (N) •n in the chaotic expansion F = E(F ) + n 1 J n (f n ) (see (2.9)) verify the relation 15) where the symbol f n (·, k) indicates that the integration is performed with respect to n − 1 variables. According e.g.…”

Section: Discrete Malliavin Calculus and A New Chain Rulementioning

confidence: 99%

Stein's Method and Stochastic Analysis of Rademacher Functionals

Reinert

Nourdin

Peccati

2010

Electron. J. Probab.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…This distribution has been called the zero bias distribution by Goldstein and Reinert [12], but has appeared many times before in the literature in disguise; see Goldstein and Reinert [12] for references.…”

Section: Proof Of Sufficiencymentioning

confidence: 99%

Use of exchangeable pairs in the analysis of simulations

Stein¹,

Diaconis²,

Holmes³

et al. 2004

Institute of Mathematical Statistics Lecture Notes - Monograph Series

Self Cite

View full text Add to dashboard Cite

The method of exchangeable pairs has emerged as an important tool in proving limit theorems for Poisson, normal and other classical approximations. Here the method is used in a simulation context. We estimate transition probabilitites from the simulations and use these to reduce variances. Exchangeable pairs are used as control variates.Finally, a general approximation theorem is developed that can be complemented by simualtions to provide actual estimates of approximation errors.

show abstract

Stein's method and the zero bias transformation with application to simple random sampling

Abstract: Let W be a random variable with mean zero and variance σ 2 .

Cited by 168 publications

References 14 publications

Fluctuations of eigenvalues and second order Poincaré inequalities

Fluctuations of eigenvalues and second order Poincaré inequalities

Stein's Method and Stochastic Analysis of Rademacher Functionals

Use of exchangeable pairs in the analysis of simulations

Contact Info

Product

Resources

About