Upper Bounds for Stein-Type Operators

Daly, Fraser

doi:10.1214/ejp.v13-479

Cited by 25 publications

(32 citation statements)

References 21 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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“…The constant 4 is worse than the constant 2 in the bound (3.27) obtained by [8], but our derivation is considerably simpler. This example also illustrates how the iterative technique can be used to deduce bounds of the form f (n+1) ≤ C h (n) from bounds of the form f (n) ≤ K h (n) .…”

Section: )mentioning

confidence: 69%

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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“…We next need some properties of the solution (4.3). Some preliminary results can be found in Weinberg (2005), Bon (2006), Chatterjee, Fulman and Röllin (2006) and Daly (2008). We give self-contained proofs of the following bounds.…”

Section: Applicationsmentioning

confidence: 86%

New rates for exponential approximation and the theorems of Rényi and Yaglom

Peköz¹,

Röllin²

2011

Ann. Probab.

136

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We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain new rates of convergence with respect to the Wasserstein and Kolmogorov metrics for the theorem of Rényi on random sums and generalizations of it, hitting times for Markov chains, and to obtain a new rate for the classical theorem of Yaglom on the exponential asymptotic behavior of a critical Galton-Watson process conditioned on nonextinction. The primary tools are an adaptation of Stein's method, Stein couplings, as well as the equilibrium distributional transformation from renewal theory.

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Upper Bounds for Stein-Type Operators

Cited by 25 publications

References 21 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

New rates for exponential approximation and the theorems of Rényi and Yaglom

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