Rates of Convergence in Normal Approximation Under Moment Conditions Via New Bounds on Solutions of the Stein Equation

Gaunt, Robert E.

doi:10.1007/s10959-014-0562-z

Cited by 25 publications

(34 citation statements)

References 19 publications

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“…The O(p −1 ) bound (1.6) is the first faster than O(p −1/2 ) bound for the random sum S n in the literature. The faster convergence rate is a result of the vanishing third moment assumption, and as such complements a number of other 'matching moments' limit theorems that are found in the Stein's method literature, see, for example, [4,12,15,18,21,28]. Theorem 1.3 gives the first bounds in the literature on the rate of convergence of the deterministic sum T n to its asymptotic Laplace distribution.…”

Section: Introductionsupporting

confidence: 56%

Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

Gaunt

2018

J Theor Probab

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We use Stein's method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren [37] for Stein's method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and it first two derivatives, of the Rayleigh Stein equation.where the test function h is real-valued. The second step is to solve (1.2) for f h and obtain suitable bounds for the solution. Finally, to approximate the distribution of a random variable of interest W by the target distribution q, one may evaluate both sides of (1.2) at W , take expectations and finally take the supremum of both sides over a class of functions H to obtain d H (W, Y ) := sup h∈H |E[h(W )] − E[h(Y )]| = sup h∈H E[Af h (W )].

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Section: Introductionsupporting

confidence: 56%

Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

Gaunt

2018

J Theor Probab

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“…Analogous bounds to (3.65) and (3.67) were obtained for the n-th derivatives of the solution f , viewed as n-linear forms, by [41] and [23] respectively.…”

Section: The Stein Equation Is (Seementioning

confidence: 61%

An iterative technique for bounding derivatives of solutions of Stein equations

Döbler¹,

Gaunt²,

Vollmer³

2017

Electron. J. Probab.

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We introduce a simple iterative technique for bounding derivatives of solutions of Stein equations Lf = h − Eh(Z), where L is a linear differential operator and Z is the limit random variable. Given bounds on just the solutions or certain lower order derivatives of the solution, the technique allows one to deduce bounds for derivatives of any order, in terms of supremum norms of derivatives of the test function h. This approach can be readily applied to many Stein equations from the literature. We consider a number of applications; in particular, we derive new bounds for derivatives of any order of the solution of the general variancegamma Stein equation. Finally, we present a connection between Stein equations and Poisson equations, from which we first recognised the importance of the iterative technique to Stein's method.1 for all functions f belonging to some measure determining class. For continuous random variables, L is a differential operator; for discrete random variables, L is a difference operator. Such characterisations have often been derived via Stein's density approach [53], [54] or the generator approach of Barbour and Götze [3], [32]. The scope of the density approach has recently been extended by [38], and other techniques for obtaining Stein characterisations are discussed in that work.The characterisation (1.1) leads to the so-called Stein equation:where the test function h is real-valued. The second step of Stein's method, which will be the focus of this paper, concerns the problem of obtaining a solution f to the Stein equation (1.2) and then establishing estimates for f and some of its lower order derivatives (for continuous distributions). Evaluating (1.2) at a random variable W and taking expectations gives3) and thus the problem of bounding the quantity Eh(W ) − Eh(Z) reduces to solving (1.2) and bounding the right-hand side of (1.3). The third step of Stein's method concerns the problem of bounding the expectation E[Lf (W )]. For continuous limit distributions, such bounds are usually obtained via Taylor expansions and coupling techniques. For many classical distributions, the problem of obtaining the first two necessary ingredients is relatively tractable. As a result, over the years, Stein's method has been adapted to many standard distributions, including the beta [11], [30], gamma [28], [39], exponential [5], [19], [45], Laplace [48] and, more generally, the class of variance-gamma distributions [21]. The method has also been adapted to distributions arising from specific problems, such as preferential attachment graphs [46], the Curie-Weiss model [6] and statistical mechanics [13], [14]. For a comprehensive overview of the literature, see [38]. Estimates for solutions of Stein equationsThe Stein equations of many classical distributions, such as the normal, beta and gamma, are linear first order ODEs with simple coefficients. As a result, the problem of solving the Stein equation and bounding the derivatives of the solution is reasonably tractable.In fact, for Stein characterisation...

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“…Bounds for non-smooth test functions are often more informative (see, for example, [15]), although an advantage of working with smooth test functions is that it is sometimes possible to obtain improved error bounds that may not hold for non-smooth test functions. Indeed, a standardised sum of independent random variables with first p moments equal to those of the standard normal distribution converges to this distribution at a rate of order n −(p−1)/2 for smooth test functions (see [8], [12] and [17]). In fact, it has also been shown by [7] and [11] that certain asymptotically chi-square and variance-gamma distributed statistics converge to their limiting distribution at a rate of order n −1 for smooth test functions even when the third moments are non-zero.…”

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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Rates of Convergence in Normal Approximation Under Moment Conditions Via New Bounds on Solutions of the Stein Equation

Cited by 25 publications

References 19 publications

Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

An iterative technique for bounding derivatives of solutions of Stein equations

Stein’s method for functions of multivariate normal random variables

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