An Application of a Density Transform and the Local Limit Theorem

Cacoullos, T.; Papadatos, Nickos; Papathanasiou, V.

doi:10.4213/tvp3828

Cited by 3 publications

(4 citation statements)

References 10 publications

(14 reference statements)

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“…where τ i is the Stein kernel of X i . From here one readily recovers the key inequalities from [15,11]. This is also the starting point of the Nourdin-Peccati approach to Stein's method [66].…”

Section: Comparing Probability Densities By Comparing Stein Operatorsmentioning

confidence: 99%

“…Indeed [65,66,68] (among others) refer to τ as the "Stein factor" despite the fact that this term also refers to the bounds on the solutions of the Stein equations, see [81,23,7]. Other authors, including [14,13,11], rather refer to this function as the "ω-function" or the "covariance kernel" of X. We prefer to unify the terminology by calling τ a Stein kernel.…”

Section: Stein Operators Via the Stein Kernelmentioning

confidence: 99%

“…with τ j , j = 1, 2, the Stein kernel of X j (defined in (49)) and κ H,2 an explicit constant that can be computed in several cases. In [11], and references therein cited, consequences of (65) are explored in quite some detail. In particular in the Gaussian and central Gamma cases, (65) has been exploited fruitfully in conjunction with Malliavin calculus, leading to an important new stream of research known as "Nourdin-Peccati analysis", see [66,65].…”

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confidence: 99%

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Stein’s method for comparison of univariate distributions

Ley¹,

Reinert²,

Swan³

2017

Probab. Surveys

114

125

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We propose a new general version of Stein's method for univariate distributions. In particular we propose a canonical definition of the Stein operator of a probability distribution which is based on a linear difference or differential-type operator. The resulting Stein identity highlights the unifying theme behind the literature on Stein's method (both for continuous and discrete distributions). Viewing the Stein operator as an operator acting on pairs of functions, we provide an extensive toolkit for distributional comparisons. Several abstract approximation theorems are provided. Our approach is illustrated for comparison of several pairs of distributions : normal vs normal, sums of independent Rademacher vs normal, normal vs Student, and maximum of random variables vs exponential, Fréchet and Gumbel.

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Section: Comparing Probability Densities By Comparing Stein Operatorsmentioning

confidence: 99%

Section: Stein Operators Via the Stein Kernelmentioning

confidence: 99%

mentioning

confidence: 99%

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Stein’s method for comparison of univariate distributions

Ley¹,

Reinert²,

Swan³

2017

Probab. Surveys

114

125

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show abstract

“…Thus, we obtain δ ≤ min{4/(k − 2), 3/(2π(k − 2)) 1/3 } and the minimum is always equal to 4/(k − 2) for k ≥ 6. Theorem 2.1 of Cacoullos et al (1997) combined with Theorem 2.1 of Cacoullos et al (2001) yields the following upper bound for the total variation distance:…”

mentioning

confidence: 99%

Bounds for the Distance Between the Distributions of Sums of Absolutely Continuous i.i.d. Convex-Ordered Random Variables with Applications

Christofides

Vaggelatou²

2009

Journal of Applied Probability

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Let X1, X2,… and Y1, Y2,… be two sequences of absolutely continuous, independent and identically distributed (i.i.d.) random variables with equal means E(Xi)=E(Yi), i=1,2,… In this work we provide upper bounds for the total variation and Kolmogorov distances between the distributions of the partial sums ∑i=1nXi and ∑i=1nYi. In the case where the distributions of the Xis and the Yis are compared with respect to the convex order, the proposed upper bounds are further refined. Finally, in order to illustrate the applicability of the results presented, we consider specific examples concerning gamma and normal approximations.

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An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds

Afendras¹,

Papadatos²,

Papathanasiou³

2011

Bernoulli

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For an absolutely continuous (integer-valued) r.v. X of the Pearson (Ord) family, we show that, under natural moment conditions, a Stein-type covariance identity of order k holds (cf. [Goldstein and Reinert, J. Theoret. Probab. 18 (2005) 237-260]). This identity is closely related to the corresponding sequence of orthogonal polynomials, obtained by a Rodrigues-type formula, and provides convenient expressions for the Fourier coefficients of an arbitrary function. Application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v. X, expressions that seem to be known only in particular cases (for the Normal, see [Houdré and Kagan, J. Theoret. Probab. 8 (1995) 23-30]; see also [Houdré and Pérez-Abreu, Ann. Probab. 23 (1995) 400-419] for corresponding results related to the Wiener and Poisson processes). Some applications are also given.

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An Application of a Density Transform and the Local Limit Theorem

Cited by 3 publications

References 10 publications

Stein’s method for comparison of univariate distributions

Stein’s method for comparison of univariate distributions

Bounds for the Distance Between the Distributions of Sums of Absolutely Continuous i.i.d. Convex-Ordered Random Variables with Applications

An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds

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