Quantile coupling inequalities and their applications

Mason, David M.; Zhou, Harrison H.

doi:10.1214/12-ps198

Cited by 25 publications

(22 citation statements)

References 44 publications

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“…where |θ| ≤ 1. By Theorem 1 of Mason and Zhou [19] with ε n = α and K n = C ln n, then whenever n ≥ 64C 2 (ln n) 2 and |W n | ≤ √ n 8 ln n , we have…”

Section: Proof Of Lower Bound In Theoremmentioning

confidence: 97%

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Cramér Moderate Deviation Expansion for Martingales with One-Sided Sakhanenko’s Condition and Its Applications

2019

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We give a Cramér moderate deviation expansion for martingales with differences having finite conditional moments of order 2 + ρ, ρ ∈ (0, 1], and finite one-sided conditional exponential moments. The upper bound of the range of validity and the remainder of our expansion are both optimal. Consequently, it leads to a "half-side" moderate deviation principle for martingales. Moreover, applications to quantile coupling inequality, β-mixing and ψ-mixing sequences are discussed.Keywords Martingales · Cramér moderate deviations · quantile coupling inequality · β-mixing sequences · ψ-mixing sequencesis the standard normal distribution function. Cramér type moderate deviations for sums of independent r.v.s have been obtained by many authors. See, for instance, Feller [15], Petrov [20], Sakhanenko [25] and [12]. We refer to the monographs of Petrov [21], Saulis and Statulevičius [26] and the references therein.

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“…where |θ| ≤ 1. By Theorem 1 of Mason and Zhou [19] with ε n = α and K n = C ln n, then whenever n ≥ 64C 2 (ln n) 2 and |W n | ≤ √ n 8 ln n , we have…”

Section: Proof Of Lower Bound In Theoremmentioning

confidence: 97%

“…When the martingale differences are bounded, Mason and Zhou [19] proved that (3.3) holds whenever |W n | ≤ α 4 √ n. Notice that the bounded martingale differences satisfy the conditional Sakhanenko condition. Moreover, the range |W n | ≤ α 4 √ n has been extended to a much larger one |W n | ≤ α √ n in our theorem.…”

Section: )mentioning

confidence: 99%

Cramér Moderate Deviation Expansion for Martingales with One-Sided Sakhanenko’s Condition and Its Applications

2019

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“…We refer to Csörgő and Hall (1984), Csörgő (2007) and Mason and Zhou (2012) for a survey of possible applications of the strong approximation and many references. There is a huge literature on the strong approximations and their applications.…”

Section: Introductionmentioning

confidence: 99%

Strong approximations for the p-fold integrated empirical process with applications to statistical tests

Alvarez-Andrade

Bouzebda

Lachal

2017

TEST

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The main purpose of this paper is to investigate the strong approximation of the p-fold integrated empirical process, p being a fixed positive integer. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on the Komlós et al. (1975)'s results. We obtain an exponential bound for the tail probability of the weighted approximation to the p-fold integrated empirical process. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of integrated empirical processes when the parameters are estimated. We study the behavior of the self-intersection local time of the partial sum process representation of integrated empirical processes. Finally, simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the integrated empirical processes.

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“…We refer to Csörgő and Hall (1984), Csörgő (2007) and Mason and Zhou (2012) for a survey of some applications of the strong approximation and many references. There is a huge literature on the strong approximations and their applications.…”

Section: Introductionmentioning

confidence: 99%

Some asymptotic results for the integrated empirical process with applications to statistical tests

Alvarez-Andrade

Bouzebda

Lachal

2016

Communications in Statistics - Theory and Methods

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The main purpose of this paper is to investigate the strong approximation of the integrated empirical process. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on the Komlós et al. (1975)'s results. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of the integrated empirical process when the parameters are estimated. Finally, we study the behavior of the self-intersection local time of the partial sum process representation of the integrated empirical process.

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Quantile coupling inequalities and their applications

Cited by 25 publications

References 44 publications

Cramér Moderate Deviation Expansion for Martingales with One-Sided Sakhanenko’s Condition and Its Applications

Cramér Moderate Deviation Expansion for Martingales with One-Sided Sakhanenko’s Condition and Its Applications

Strong approximations for the p-fold integrated empirical process with applications to statistical tests

Some asymptotic results for the integrated empirical process with applications to statistical tests

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