Moderate Deviations for Ewens-Pitman Sampling Models

Favaro, Stefano; Feng, Shui; Gao, Fuqing

doi:10.1007/s13171-018-0124-z

Cited by 6 publications

(6 citation statements)

References 22 publications

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“…l,m have already been obtained in [9] and [1], the fluctuation scale is n α for K n , M l,n , and m α for K The corresponding large deviation principle (LDP for short) of ( 1) have been discussed thoroughly in [4], [1] and [2]. Moreover, the moderate deviation principle (MDP for short) are also discussed in [3].…”

Section: Introductionmentioning

confidence: 98%

“…In this article, we will show that this restriction is actually unnecessary. To establish MDP, we will adopt the scheme in [3]. First, we find the asymptotic log-Laplace transform; then we apply Gärtner-Ellis theorem to complete the proof.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, we manage to find a closed integral representation for their moment generating functions.Then we can apply complex asymptotic analysis to these integral representations to get limiting log-Laplace transform. In [3], the upper bound and lower bound estimation scheme seems hard to establish limiting log-Laplace transform when β n does not satisfy lim n→∞ βn log 1−α n = ∞. This article will focus on the establishment of limiting log-Laplace transform.…”

Section: Introductionmentioning

confidence: 99%

“…Their asymptotic behavior, such as large deviation principle, has already been discussed in [4],[1] and [2]. Moderate deviation principle is also discussed in [3] with some speed restriction. In this article, we will apply complex asymptotic analysis to show that this speed restriction is unnecessary.…”

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confidence: 99%

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Some Moderate Deviations for Ewens-Pitman Sampling Model

Zhou

2018

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Ewens-Pitman model has been successfully applied to various fields including Bayesian statistics. There are four important estimators Kn, M l,n ,K1,m are related to discovery probability. Their asymptotic behavior, such as large deviation principle, has already been discussed in [4],[1] and [2]. Moderate deviation principle is also discussed in [3] with some speed restriction. In this article, we will apply complex asymptotic analysis to show that this speed restriction is unnecessary. βn log 1−α n = ∞, and there are also similar

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Some Moderate Deviations for Ewens-Pitman Sampling Model

Zhou

2018

Preprint

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“…For α ∈ (0, 1) and θ > −α, the random variable S α,θ is referred to as Pitman's α-diversity. Large and moderate deviations results for K n are established in Feng and Hoppe [11] and Favaro et al [10], whereas a concentration inequality for K n is obtained in Pereira et al [19] by relying on certain concentration inequalities for martingales.…”

Section: Introductionmentioning

confidence: 99%

A Berry-Esseen theorem for Pitman's $α$-diversity

Dolera,

Favaro

2018

Preprint

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This paper is concerned with the study of the random variable Kn denoting the number of distinct elements in a random sample (X1, . . . , Xn) of exchangeable random variables driven by the two parameter Poisson-Dirichlet distribution, PD(α, θ). For α ∈ (0, 1), Theorem 3.8 in [23] shows that Kn n α a.s.−→ S α,θ as n → +∞. Here, S α,θ is a random variable distributed according to the so-called scaled Mittag-Leffler distribution. Our main result states thatholds with an explicit constant C(α, θ). The key ingredients of the proof are a novel probabilistic representation of Kn as compound distribution and new, refined versions of certain quantitative bounds for the Poisson approximation and the compound Poisson distribution. Finally, we present the following application in the context of Bayesian nonparametric inference for species sampling problems: given an initial (observable) random sample (X1, . . . , Xn) from the population, estimate of the number K (n) m of hitherto unseen species that would be observed in m additional (unobservable) samples. In the approach proposed in [8], (X1, . . . , Xn) is a random sample from PD(α, θ) featuring Kn = j ≤ n species (blocks), for which there holds K (n) m m α | (X1, . . . , Xn) a.s. −→ S α,θ (n, j) as m → +∞, where S α,θ (n, j) is related to S α,θ+n . Thus, we combine the previous main result with a new Berry-Esseen bound for de Finetti's theorem recently obtained in [6], to obtain another Berry-Esseen theorem for the convergence of the distribution of K (n) m m α | (X1, . . . , Xn).

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