Concentration Inequalities for Statistical Inference

Chen, Huiming Zhang & Song Xi

doi:10.4208/cmr.2020-0041

Cited by 25 publications

(12 citation statements)

References 65 publications

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“…Then,

{false{\overset{ψ}{true} false(α_{k} false(X_{i} - μ false) false) false}}_{k = 1}^{\infty}

is uniformly integrable for the fixed

i

; then, Vitali' theorem (see theorem 4.6.3 (iii) in Durrett, 2019) gives

\lim_{n \to \infty} 𝔼 \dot{ψ} (α_{n} (X_{i} - μ)) = 1, \forall i .

Hence,

\lim_{n \to \infty} 𝔼 {normalH}_{n} : = \lim_{n \to \infty} \frac{1}{n} {falsefalse}_{\sum}^{i = 1} 𝔼 \overset{ψ}{true} false(α_{n} false(X_{i} - μ false) false) = 1

. If we can show that

{normalH}_{n} - 𝔼 {normalH}_{n} \to_{}^{ℙ} 0

, then we can get

\frac{1}{n} {falsefalse}_{\sum}^{i = 1} \overset{ψ}{true} false(α_{n} false(X_{i} - θ false) false) = {normalH}_{n} = false({normalH}_{n} - 𝔼 {normalH}_{n} false) + 𝔼 {normalH}_{n} \to_{}^{ℙ} 1

, By Hoeffding's inequality (see corollary 2.1 in Zhang & Chen, 2021),

ℙ ()(, (||, {normalH}_{n} - 𝔼 {normalH}_{n}) \geq t) = ℙ false(false| {falsefalse}_{\sum}^{i = 1}

…”

Section: Proofs Of Main Resultsmentioning

confidence: 94%

“…Lemma (Sharper maximal inequality, corollary 7.1 in Zhang & Chen, 2021). Let

{false{X_{i} false}}_{i = 1}^{n}

be identically distributed and assume

𝔼 false(false| X_{1} {false|}^{p} false) < \infty, false(p \geq 1 false)

.…”

Section: Proofs Of Main Resultsmentioning

confidence: 99%

“…Lemma 3. (Sharper maximal inequality, corollary 7.1 in Zhang & Chen, 2021). Let fX i g n i¼1 be identically distributed and assume…”

Section: Some Lemmasmentioning

confidence: 99%

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Asymptotic normality and confidence region for Catoni's Z estimator

Yao

Zhang

2022

Stat

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This paper studies the asymptotic property and confidence region of Catoni's Z estimator under the finite variance assumption. First, we investigate the CLT for Catoni's Z estimator with the asymptotically optimal variance, if the tuning parameter has order O)(n−12. Second, we propose the normal approximated confidence region of Catoni's Z estimator. The simulations show that the proposed 99% CLT‐based confidence regions are shorter than the existing non‐asymptotic confidence regions; mean squared errors of Catoni's Z estimators are typically smaller than sample means from heavy‐tailed data. Furthermore, the simulated coverage probabilities of the CLT‐based confidence region are close to the confidence levels when the sample size is moderately large. For 6‐year DOCVIS data (the number of doctor visits) in the German health care demand dataset, our CLT‐based confidence regions are tighter than non‐asymptotic confidence regions.

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“…Then,

{false{\overset{ψ}{true} false(α_{k} false(X_{i} - μ false) false) false}}_{k = 1}^{\infty}

is uniformly integrable for the fixed

i

; then, Vitali' theorem (see theorem 4.6.3 (iii) in Durrett, 2019) gives

\lim_{n \to \infty} 𝔼 \dot{ψ} (α_{n} (X_{i} - μ)) = 1, \forall i .

Hence,

\lim_{n \to \infty} 𝔼 {normalH}_{n} : = \lim_{n \to \infty} \frac{1}{n} {falsefalse}_{\sum}^{i = 1} 𝔼 \overset{ψ}{true} false(α_{n} false(X_{i} - μ false) false) = 1

. If we can show that

{normalH}_{n} - 𝔼 {normalH}_{n} \to_{}^{ℙ} 0

, then we can get

\frac{1}{n} {falsefalse}_{\sum}^{i = 1} \overset{ψ}{true} false(α_{n} false(X_{i} - θ false) false) = {normalH}_{n} = false({normalH}_{n} - 𝔼 {normalH}_{n} false) + 𝔼 {normalH}_{n} \to_{}^{ℙ} 1

, By Hoeffding's inequality (see corollary 2.1 in Zhang & Chen, 2021),

ℙ ()(, (||, {normalH}_{n} - 𝔼 {normalH}_{n}) \geq t) = ℙ false(false| {falsefalse}_{\sum}^{i = 1}

…”

Section: Proofs Of Main Resultsmentioning

confidence: 94%

“…Lemma (Sharper maximal inequality, corollary 7.1 in Zhang & Chen, 2021). Let

{false{X_{i} false}}_{i = 1}^{n}

be identically distributed and assume

𝔼 false(false| X_{1} {false|}^{p} false) < \infty, false(p \geq 1 false)

.…”

Section: Proofs Of Main Resultsmentioning

confidence: 99%

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Asymptotic normality and confidence region for Catoni's Z estimator

Yao

Zhang

2022

Stat

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show abstract

“…The first part is relatively easy to analyze because the following concentration inequality gives a measure of dispersion about the weighted summation of negative binomial variables. This concentration inequality is a special case for the weighted summation of a series of random variables, which can be proved by sub-exponential concentration results in Proposition 4.2 in [31]. Lemma A2.…”

Section: Conclusion and Future Studymentioning

confidence: 86%

Heterogeneous Overdispersed Count Data Regressions via Double-Penalized Estimations

Wei

Lei

2022

Mathematics

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Recently, the high-dimensional negative binomial regression (NBR) for count data has been widely used in many scientific fields. However, most studies assumed the dispersion parameter as a constant, which may not be satisfied in practice. This paper studies the variable selection and dispersion estimation for the heterogeneous NBR models, which model the dispersion parameter as a function. Specifically, we proposed a double regression and applied a double ℓ1-penalty to both regressions. Under the restricted eigenvalue conditions, we prove the oracle inequalities for the lasso estimators of two partial regression coefficients for the first time, using concentration inequalities of empirical processes. Furthermore, derived from the oracle inequalities, the consistency and convergence rate for the estimators are the theoretical guarantees for further statistical inference. Finally, both simulations and a real data analysis demonstrate that the new methods are effective.

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“…βX 𝜃 (𝑡; βX )M 𝑡; βX (𝑡) 𝑑𝑒 𝑓 = 𝒬 𝑛 + 𝒲 4 , the random-walk Winner process 𝒲 4 imposes us to denote the following − as the25 See e.g [61]-[71]. for concentration inequalities 26.…”

mentioning

confidence: 99%

How to Game-theoretically Disentangle & Control the Efficiency of Orbital Angular Momentum

Zamanipour¹

2022

Preprint

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<p>How are we able to fix it so as to theoretically go over it, if the optimality and efficiency of the orbital angular momentum (OAM) multiplexing technique in a multi-input multi-output (MIMO) system fails in practice $-$ something that technically causes a considerable reduction in the maximum achievable value of the transceiver's degree-of-freedom? Is the aforementioned challenging issue theoretically interpretable?</p>

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Concentration Inequalities for Statistical Inference

Cited by 25 publications

References 65 publications

Asymptotic normality and confidence region for Catoni's Z estimator

Asymptotic normality and confidence region for Catoni's Z estimator

Heterogeneous Overdispersed Count Data Regressions via Double-Penalized Estimations

How to Game-theoretically Disentangle & Control the Efficiency of Orbital Angular Momentum

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