Zero biasing in one and higher dimensions, and applications

Goldstein, Larry; Reinert, Gesine

doi:10.1142/9789812567673_0001

Cited by 19 publications

(23 citation statements)

References 13 publications

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“…Instead, however, we elect to follow [18] and use the multivariate normal Stein equation in conjunction with the chi-square Stein equation. Indeed, there is powerful array of tools for proving approximation theorems using the multivariate normal Stein equation (see [9,21,22,32,38] for coupling techniques for multivariate normal approximation). In particular, we use the multivariate normal exchangeable pair coupling of [38].…”

Section: Elements Of Stein's Methodsmentioning

confidence: 99%

Bounds for the chi-square approximation of Friedman's statistic by Stein's method

Gaunt¹,

Reinert²

2021

Preprint

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Friedman's chi-square test is a non-parametric statistical test for r ≥ 2 treatments across n ≥ 1 trials to assess the null hypothesis that there is no treatment effect. We use Stein's method with an exchangeable pair coupling to derive an explicit bound on the distance between the distribution of Friedman's statistic and its limiting chi-square distribution, measured using smooth test functions. Our bound is of the optimal order n −1 , and also has an optimal dependence on the parameter r, in that the bound tends to zero if and only if r/n → 0. From this bound, we deduce a Kolmogorov distance bound that decays to zero under the weaker condition r 1/2 /n → 0.

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Section: Elements Of Stein's Methodsmentioning

confidence: 99%

Bounds for the chi-square approximation of Friedman's statistic by Stein's method

Gaunt¹,

Reinert²

2021

Preprint

Self Cite

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“…We see the expectation on the right hand exists since F and all its derivatives are bounded, V k+1 is independent of Y for all k, EU −k i < ∞ for i ≥ k + 1, and use of (14).…”

Section: Transformations In Generalmentioning

confidence: 94%

“…The zero bias transformation was introduced and used in [13] to obtain bounds of order n −1 in normal approximations for smooth test functions under third order moment conditions, in the presence of dependence induced by simple random sampling. In [11] it is used to provide bounds to the normal distribution for hierarchical sequences generated by the iteration of a so called averaging function, in [12] for normal approximation in combinatorial central limit theorems with random permutations having distribution constant over cycle type, and in [14] the extension of the zero bias transformation to higher dimension is considered.…”

Section: Introductionmentioning

confidence: 99%

Distributional Transformations, Orthogonal Polynomials, and Stein Characterizations

Goldstein

Reinert

2005

J Theor Probab

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A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test function's higher order derivatives. The class includes the size and zero bias transformations, and when specializing to weighting by polynomial functions, relates distributional families closed under independent addition, and in particular the infinitely divisible distributions, to the family of transformations induced by their associated orthogonal polynomial systems. For these families, generalizing a well known property of size biasing, sums of independent variables are transformed by replacing summands chosen according to a multivariate distribution on its index set by independent variables whose distributions are transformed by members of that same family. A variety of the transformations associated with the classical orthogonal polynomial systems have as fixed points the original distribution, or a member of the same family with different parameter.

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“…using the various coupling techniques developed for multivariate normal approximation (see [13], [14], [26] and [22]). However, in general the derivatives of the test function h(g(w)) will be unbounded (for the χ 2 (1) distribution, g(w) = w 2 and g ′ (w) = 2w) and therefore the derivatives of the solution (1.10) will also in general be unbounded.…”

Section: Introductionmentioning

confidence: 99%

Stein’s method for functions of multivariate normal random variables

Gaunt¹

2020

Ann. Inst. H. Poincaré Probab. Statist.

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It is a well-known fact that if the random vector W converges in distribution to a multivariate normal random variable Σ 1/2 Z, then g(W) converges in distribution to g(Σ 1/2 Z) if g is continuous. In this paper, we develop a general method for deriving bounds on the distributional distance between g(W) and g(Σ 1/2 Z). To illustrate this method, we obtain several bounds for the case that the j-component of W is given byXij , where the Xij are independent. In particular, provided g satisfies certain differentiability and growth rate conditions, we obtain an order n −(p−1)/2 bound, for smooth test functions, if the first p moments of the Xij agree with those of the normal distribution. If p is an even integer and g is an even function, this convergence rate can be improved further to order n −p/2 . We apply these general bounds to some examples concerning asymptotically chi-square, variance-gamma and chi distributed statistics. Primary 60F05.

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Zero biasing in one and higher dimensions, and applications

Cited by 19 publications

References 13 publications

Bounds for the chi-square approximation of Friedman's statistic by Stein's method

Bounds for the chi-square approximation of Friedman's statistic by Stein's method

Distributional Transformations, Orthogonal Polynomials, and Stein Characterizations

Stein’s method for functions of multivariate normal random variables

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