Sample Covariance Matrices of Heavy-Tailed Distributions

Tikhomirov, Konstantin

doi:10.1093/imrn/rnx067

Cited by 32 publications

(30 citation statements)

References 35 publications

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“…and linear functionals exhibit an L p -L 2 norm equivalence for some p > 4 [17]. This extra independence-the fact that there are more independent random vectors (the m rows of the random matrix) than the dimension n of the underlying space, is crucial when trying to control the way the matrix acts of S n−1 (see, for example, the proofs in [1,12,17]).…”

Section: Here and In What Follows Denotementioning

confidence: 99%

Random embeddings with an almost Gaussian distortion

Bartl¹,

Mendelson²

2021

Preprint

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Let X be a symmetric, isotropic random vector in R m and let X1..., Xn be independent copies of X. We show that under mild assumptions on X 2 (a suitable thin-shell bound) and on the tail-decay of the marginals X, u , the random matrix A, whose columns are Xi/ √ m exhibits a Gaussian-like behaviour in the following sense: for an arbitrary subset of T ⊂ R n , the distortion sup t∈T | At 2 2 − t 2 2 | is almost the same as if A were a Gaussian matrix.A simple outcome of our result is that if X is a symmetric, isotropic, log-concave random vector and n ≤ m ≤ c1(α)n α for some α > 1, then with high probability, the extremal singular values of A satisfy the optimal estimate: 1 − c2(α) n/m ≤ λmin ≤ λmax ≤ 1 + c2(α) n/m.

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Section: Here and In What Follows Denotementioning

confidence: 99%

Random embeddings with an almost Gaussian distortion

Bartl¹,

Mendelson²

2021

Preprint

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“…Results of this type have been studied extensively [1,14,33,36,39,45], however (to the best of our knowledge) the focus was on random matrix ensembles that had some additional special structure, like iid rows/columns. Unfortunately such special structure need not exist in our setting.…”

Section: On the Smallest Singular Value Of General Random Matrix Ense...mentioning

confidence: 99%

On Monte-Carlo methods in convex stochastic optimization

Bartl¹,

Mendelson²

2021

Preprint

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We develop a novel procedure for estimating the optimizer of general convex stochastic optimization problems of the form min x∈X E[F (x, ξ)], when the given data is a finite independent sample selected according to ξ. The procedure is based on a median-of-means tournament, and is the first procedure that exhibits the optimal statistical performance in heavy tailed situations: we recover the asymptotic rates dictated by the central limit theorem in a non-asymptotic manner once the sample size exceeds some explicitly computable threshold. Additionally, our results apply in the high-dimensional setup, as the threshold sample size exhibits the optimal dependence on the dimension (up to a logarithmic factor). The general setting allows us to recover recent results on multivariate mean estimation and linear regression in heavy-tailed situations and to prove the first sharp, non-asymptotic results for the portfolio optimization problem.

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“…Recently, different copulas and new approaches of introducing HT distribution have been studied due to the importance of HT distributions in the financial sector (for example, Bladt et al [14]; Tikhomirov [15]; Lugosi and Mendelson [16]; and Yousri et al [17]). For more information about the usefulness of statistical distributions, one can refer to studies by Ramos et al [18,19]; He et al [20]; Alfaer et al [21]; Afify et al [22]; and Al Mutairi et al [23].…”

Section: Introductionmentioning

confidence: 99%

A New Exponential-X Family: Modeling Extreme Value Data in the Finance Sector

Ahmad

Mahmoudi

Roozegar

et al. 2021

Mathematical Problems in Engineering

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In this paper, a family of statistical models, namely, a new exponential-X family is proposed. A subcase of the introduced family, called the new exponential-Weibull (NE-Weibull) model, is studied. The NE-Weibull model is very competent and possesses heavy-tailed properties. The maximum likelihood estimators of its parameters are derived. The consistency and efficiency of these estimators are assessed in a brief simulation study. Finally, the effectiveness of the NE-Weibull distribution is illustrated by modeling real insurance claims data. The practical analysis shows that the NE-Weibull distribution outclassed other distributions and it can be a better choice for modeling data in the finance sector.

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Sample Covariance Matrices of Heavy-Tailed Distributions

Abstract: Let p > 2, B ≥ 1, N ≥ n and let X be a centered n-dimensional random vector with the identity covariance matrix such that sup a∈S n−1 E| X, a | p ≤ B. Further, let X 1 , X 2 , . . . , X N be independent copies of X, and Σ N :T be the sample covariance matrix. We prove that

Cited by 32 publications

References 35 publications

Random embeddings with an almost Gaussian distortion

Random embeddings with an almost Gaussian distortion

On Monte-Carlo methods in convex stochastic optimization

A New Exponential-X Family: Modeling Extreme Value Data in the Finance Sector

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