Self-Normalized Processes

Peña, Víctor H. de la; Lai, Tze Leung; Shao, Qi-Man

doi:10.1007/978-3-540-85636-8

Cited by 99 publications

(41 citation statements)

References 83 publications

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“…Remark 5.1. We would like to remark that if instead of the uniform distribution the cone measure on B n p is considered (recall the definition in Remark 4.3), probabilities of the type P(n 1/p−1/q Z q ≥ x) for sufficiently large x were already considered in [27] (see also [8,Chapter 3]) in the context of self-normalized large deviations. The result in [27] applies to far more general situations and its proof is in fact highly technical, while we are relying on elementary principles from large deviation theory.…”

Section: A Large Deviation Principle For the Q-normmentioning

confidence: 99%

High-dimensional limit theorems for random vectors in ℓpn-balls

Kabluchko

Prochno

Thäle

2019

Commun. Contemp. Math.

115

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In this paper, we prove a multivariate central limit theorem for ℓq-norms of highdimensional random vectors that are chosen uniformly at random in an ℓ n p -ball. As a consequence, we provide several applications on the intersections of ℓ n p -balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an ℓ n p -ball onto a line spanned by a random direction θ ∈ S n−1 . The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreover, we complement our central limit theorems by providing a complete description of the large deviation behavior, which covers fluctuations far beyond the Gaussian scale. In the regime 1 ≤ p < q this displays in speed and rate function deviations of the q-norm on an ℓ n p -ball obtained by Schechtman and Zinn, but we obtain explicit constants.2010 Mathematics Subject Classification. 52A22, 60D05, 60F05, 60F10.

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Section: A Large Deviation Principle For the Q-normmentioning

confidence: 99%

High-dimensional limit theorems for random vectors in ℓpn-balls

Kabluchko

Prochno

Thäle

2019

Commun. Contemp. Math.

115

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“…Putting together the results in Theorem 1, (15), (16), (34), (35), and the moment calculations of Gaussian distributions, we conclude that We clarify that LR γ (X) = (LR(X)) γ is the γ th moment of the likelihood ratio. With this result, if we choose m 3 = O(n), it is sufficient to guarantee that, with probability tending to 1, the ratio between the empirical moments and the theoretical moments are within an ε distance from 1.…”

Section: Lemma 5 Under Conditions (C1) and (C5) Let Y Be A Random Vmentioning

confidence: 87%

“…Case 2: n λ < x ≤ c √ n. Applying Theorem 2.18 of [16], for each δ > 0, there exist κ 1 (δ) and κ 2 (δ) such that…”

mentioning

confidence: 98%

The Convergence Rate and Asymptotic Distribution of the Bootstrap Quantile Variance Estimator for Importance Sampling

Liu

Yang

2012

Adv. Appl. Probab.

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Importance sampling is a widely used variance reduction technique to compute sample quantiles such as value at risk. The variance of the weighted sample quantile estimator is usually a difficult quantity to compute. In this paper we present the exact convergence rate and asymptotic distributions of the bootstrap variance estimators for quantiles of weighted empirical distributions. Under regularity conditions, we show that the bootstrap variance estimator is asymptotically normal and has relative standard deviation of order O(n −1/4 ).

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“…For independent and identically distributed random variables Z 1 , … , Z n , de la Peña et al (2009) gave an extensive account of the asymptotic properties of the self-normalized statistic

\sum_{i = 1}^{n} Z_{i} / \sqrt false(\sum_{i = 1}^{n} Z_{i}^{2} false)

. In this section, we present a unified self-normalized central limit theorem for μ̂ n ( x ).…”

Section: Unified Approaches For Sparse and Dense Datamentioning

confidence: 99%

Unified inference for sparse and dense longitudinal models

Kim

Zhao

2012

Biometrika

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Summary In longitudinal data analysis, statistical inference for sparse data and dense data could be substantially different. For kernel smoothing estimate of the mean function, the convergence rates and limiting variance functions are different under the two scenarios. The latter phenomenon poses challenges for statistical inference as a subjective choice between the sparse and dense cases may lead to wrong conclusions. We develop self-normalization based methods that can adapt to the sparse and dense cases in a unified framework. Simulations show that the proposed methods outperform some existing methods.

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Self-Normalized Processes

Cited by 99 publications

References 83 publications

High-dimensional limit theorems for random vectors in ℓpn-balls

High-dimensional limit theorems for random vectors in ℓpn-balls

The Convergence Rate and Asymptotic Distribution of the Bootstrap Quantile Variance Estimator for Importance Sampling

Unified inference for sparse and dense longitudinal models

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