Unified Variance Bounds and a Stein-Type Identity

Papadatos, Nickos; Papathanasiou, V.

doi:10.1201/9781420036084.ch5

Cited by 10 publications

(18 citation statements)

References 15 publications

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“…Whereas our interests here lie in normal approximation and the associated couplings, the emphasis in [3] and in [12] is to derive variance lower bounds.…”

Section: Introductionmentioning

confidence: 99%

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Zero biasing in one and higher dimensions, and applications

Goldstein¹,

Reinert²

2005

Stein's Method and Applications

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Given any mean zero, finite variance σ 2 random variable W , there exists a unique distribution on a variable W * such that EW f (W ) = σ 2 Ef (W * ) for all absolutely continuous functions f for which these expectations exist. This distributional 'zero bias' transformation of W to W * , of which the normal is the unique fixed point, was introduced in [9] to obtain bounds in normal approximations. After providing some background on the zero bias transformation in one dimension, we extend its definition to higher dimension and illustrate with an application to the normal approximation of sums of vectors obtained by simple random sampling.

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“…Whereas our interests here lie in normal approximation and the associated couplings, the emphasis in [3] and in [12] is to derive variance lower bounds.…”

Section: Introductionmentioning

confidence: 99%

“…In [12] the relationship between the w-function approach and the zero-bias coupling is exploited. Whereas our interests here lie in normal approximation and the associated couplings, the emphasis in [3] and in [12] is to derive variance lower bounds.…”

Section: Introductionmentioning

confidence: 99%

Zero biasing in one and higher dimensions, and applications

Goldstein¹,

Reinert²

2005

Stein's Method and Applications

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“…Therefore, it is useful to express the L 2 -norm of each P k in terms of the parameters δ , β , γ and µ and, in view of (5.5) and (5.6), it remains to obtain an expression for Eq k (X ). To this end, we first recall a definition from [20]; cf. [10].…”

Section: Orthogonality Of the Rodrigues-type Polynomials And Of Theirmentioning

confidence: 99%

Integrated Pearson family and orthogonality of the Rodrigues polynomials: A review including new results and an alternative classification of the Pearson system

Afendras

Papadatos

2015

Appl. Math. (Warsaw)

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An alternative classification of the Pearson family of probability densities is related to the orthogonality of the corresponding Rodrigues polynomials. This leads to a subset of the ordinary Pearson system, the Integrated Pearson Family. Basic properties of this family are discussed and reviewed, and some new results are presented. A detailed comparison between the integrated Pearson family and the ordinary Pearson system is presented, including an algorithm that enables to decide whether a given Pearson density belongs to the integrated system, or not. Recurrences between the derivatives of the corresponding orthonormal polynomial systems are also given.MSC: Primary 62E15, 60E05; Secondary 62-00. /∼npapadat/ This fact will be denoted by X ∼ IP(µ; q) or f ∼ IP(µ; q) or, more explicitly, X or f ∼ IP(µ; δ , β , γ).Despite the fact that the integrated Pearson family is quite restricted, compared to the usual Pearson system -see Proposition 2.1(iii), below -we believe that the reader will find here some interesting observations that are worth to be highlighted. The integrated Pearson system satisfies many interesting properties, like recurrences on moments and on Rodrigues polynomials, covariance identities, closeness of each type under particularly useful transformations etc.; such properties are by far more complicated (if they are, at all, true) for distributions outside the Integrated Pearson system. These features should be combined with the fact that the Rodrigues polynomials form an orthogonal system for the corresponding Pearson density if and only if the density belongs to the Integrated Pearson family. In other words, the Rodrigues polynomials and, consequently, the ordinary Pearson densities, are useful only if they are considered in the framework of the Integrated Pearson system. To our knowledge, these facts have not been written explicitly elsewhere.The paper is organized as follows: In Section 2 we provide a detailed classification of the integrated Pearson family. It turns out that, up to an affine transformation, there are six different types of densities, included in Table 2.1. We also provide conditions guaranteeing the existence of moments, and we give recurrences as long as these moments exist. In Section 3, a detailed comparison between the integrated Pearson family and the ordinary Pearson system is presented. Interestingly enough, there exist a simple algorithm that enables one to decide whether a given ordinary Pearson density belongs to the integrated

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“…Some applications of (1.2), concerning variance bounds and characterizations of dis tributions, are discussed in [15]. Moreover, Goldstein and Reinert ( [9]) extended Stein's method and they presented very interesting applications of (1.2) in the rate of convergence in the CLT.…”

Section: Identity (12) Generalizes the Well-known Stein Identity Formentioning

confidence: 99%