Pub Date : 2025-04-15DOI: 10.1016/j.jmva.2025.105445
Zbigniew Burdak , Marek Kosiek , Patryk Pagacz , Marek Słociński
A new approach to the evanescent part of a two-dimensional weak-stationary stochastic process with the past given by a half-plane is proceeded. The classical result due to Helson and Lowdenslager divides a two-parametric weak-stationary stochastic process into three parts. In this paper, we describe the most untouchable one — the evanescent part. Moreover, we point out how this part depends on the shape of the past.
{"title":"An operator theory approach to the evanescent part of a two-parametric weak-stationary stochastic process","authors":"Zbigniew Burdak , Marek Kosiek , Patryk Pagacz , Marek Słociński","doi":"10.1016/j.jmva.2025.105445","DOIUrl":"10.1016/j.jmva.2025.105445","url":null,"abstract":"<div><div>A new approach to the evanescent part of a two-dimensional weak-stationary stochastic process with the past given by a half-plane is proceeded. The classical result due to Helson and Lowdenslager divides a two-parametric weak-stationary stochastic process into three parts. In this paper, we describe the most untouchable one — the evanescent part. Moreover, we point out how this part depends on the shape of the past.</div></div>","PeriodicalId":16431,"journal":{"name":"Journal of Multivariate Analysis","volume":"209 ","pages":"Article 105445"},"PeriodicalIF":1.4,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143859192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-02DOI: 10.1016/j.jmva.2025.105444
Marouane Il Idrissi , Nicolas Bousquet , Fabrice Gamboa , Bertrand Iooss , Jean-Michel Loubes
Hoeffding’s functional decomposition is the cornerstone of many post-hoc interpretability methods. It entails decomposing arbitrary functions of mutually independent random variables as a sum of interactions. Many generalizations to dependent covariables have been proposed throughout the years, which rely on finding a set of suitable projectors. This paper characterizes such projectors under hierarchical orthogonality constraints and mild assumptions on the variable’s probabilistic structure. Our approach is deeply rooted in Hilbert space theory, giving intuitive insights on defining, identifying, and separating interactions from the effects due to the variables’ dependence structure. This new decomposition is then leveraged to define a new functional analysis of variance. Toy cases of functions of bivariate Bernoulli and Gaussian random variables are studied.
{"title":"Hoeffding decomposition of functions of random dependent variables","authors":"Marouane Il Idrissi , Nicolas Bousquet , Fabrice Gamboa , Bertrand Iooss , Jean-Michel Loubes","doi":"10.1016/j.jmva.2025.105444","DOIUrl":"10.1016/j.jmva.2025.105444","url":null,"abstract":"<div><div>Hoeffding’s functional decomposition is the cornerstone of many post-hoc interpretability methods. It entails decomposing arbitrary functions of mutually independent random variables as a sum of interactions. Many generalizations to dependent covariables have been proposed throughout the years, which rely on finding a set of suitable projectors. This paper characterizes such projectors under hierarchical orthogonality constraints and mild assumptions on the variable’s probabilistic structure. Our approach is deeply rooted in Hilbert space theory, giving intuitive insights on defining, identifying, and separating interactions from the effects due to the variables’ dependence structure. This new decomposition is then leveraged to define a new functional analysis of variance. Toy cases of functions of bivariate Bernoulli and Gaussian random variables are studied.</div></div>","PeriodicalId":16431,"journal":{"name":"Journal of Multivariate Analysis","volume":"208 ","pages":"Article 105444"},"PeriodicalIF":1.4,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-24DOI: 10.1016/j.jmva.2025.105443
Hendrik Paul Lopuhaä
We show that the limiting variance of a sequence of estimators for a structured covariance matrix has a general form, that for linear covariance structures appears as the variance of a scaled projection of a random matrix that is of radial type, and a similar result is obtained for the corresponding sequence of estimators for the vector of variance components. These results are illustrated by the limiting behavior of estimators for a differentiable covariance structure in a variety of multivariate statistical models. We also derive a characterization for the influence function of corresponding functionals. Furthermore, we derive the limiting distribution and influence function of scale invariant mappings of such estimators and their corresponding functionals. As a consequence, the asymptotic relative efficiency of different estimators for the shape component of a structured covariance matrix can be compared by means of a single scalar and the gross error sensitivity of the corresponding influence functions can be compared by means of a single index. Similar results are obtained for estimators of the normalized vector of variance components. We apply our results to investigate how the efficiency, gross error sensitivity, and breakdown point of S-estimators for the normalized variance components are affected simultaneously by varying their cutoff value.
{"title":"Asymptotics of estimators for structured covariance matrices","authors":"Hendrik Paul Lopuhaä","doi":"10.1016/j.jmva.2025.105443","DOIUrl":"10.1016/j.jmva.2025.105443","url":null,"abstract":"<div><div>We show that the limiting variance of a sequence of estimators for a structured covariance matrix has a general form, that for linear covariance structures appears as the variance of a scaled projection of a random matrix that is of radial type, and a similar result is obtained for the corresponding sequence of estimators for the vector of variance components. These results are illustrated by the limiting behavior of estimators for a differentiable covariance structure in a variety of multivariate statistical models. We also derive a characterization for the influence function of corresponding functionals. Furthermore, we derive the limiting distribution and influence function of scale invariant mappings of such estimators and their corresponding functionals. As a consequence, the asymptotic relative efficiency of different estimators for the shape component of a structured covariance matrix can be compared by means of a single scalar and the gross error sensitivity of the corresponding influence functions can be compared by means of a single index. Similar results are obtained for estimators of the normalized vector of variance components. We apply our results to investigate how the efficiency, gross error sensitivity, and breakdown point of S-estimators for the normalized variance components are affected simultaneously by varying their cutoff value.</div></div>","PeriodicalId":16431,"journal":{"name":"Journal of Multivariate Analysis","volume":"208 ","pages":"Article 105443"},"PeriodicalIF":1.4,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, Kundu (2017) proposed a multivariate skewed distribution, termed the Geometric-Normal (GN) distribution, by compounding the multivariate normal distribution with the geometric distribution. This distribution is a viable alternative to Azzalini’s multivariate skew-normal distribution and possesses several desirable properties. This paper introduces a novel class of asymmetric distributions by compounding the geometric distribution with scale mixtures of normal distributions. This class constitutes a special case of the continuous mixtures of multivariate normal distributions introduced by Arellano-Valle and Azzalini (2021). The proposed multivariate distributions exhibit high flexibility, featuring heavy tails, multi-modality, and the ability to model skewness. We have also derived several properties of this class and discussed specific examples to illustrate its applications. The expectation–maximization algorithm was employed to calculate the maximum likelihood estimates of the unknown parameters. Simulation experiments have been performed to show the effectiveness of the proposed algorithm. For illustrative purposes, we have provided one multivariate data set where it has been observed that there exist members in the proposed class of models that can provide better fit compared to skew-normal, skew-t, and generalized hyperbolic distribution. In another example, it was demonstrated that when data generated from a heavy-tailed skew-t distribution is contaminated with noise, the proposed distributions offer a better fit compared to the skew-t distribution.
{"title":"Geometric scale mixtures of normal distributions","authors":"Deepak Prajapati , Sobhan Shafiei , Debasis Kundu , Ahad Jamalizadeh","doi":"10.1016/j.jmva.2025.105430","DOIUrl":"10.1016/j.jmva.2025.105430","url":null,"abstract":"<div><div>Recently, Kundu (2017) proposed a multivariate skewed distribution, termed the Geometric-Normal (GN) distribution, by compounding the multivariate normal distribution with the geometric distribution. This distribution is a viable alternative to Azzalini’s multivariate skew-normal distribution and possesses several desirable properties. This paper introduces a novel class of asymmetric distributions by compounding the geometric distribution with scale mixtures of normal distributions. This class constitutes a special case of the continuous mixtures of multivariate normal distributions introduced by Arellano-Valle and Azzalini (2021). The proposed multivariate distributions exhibit high flexibility, featuring heavy tails, multi-modality, and the ability to model skewness. We have also derived several properties of this class and discussed specific examples to illustrate its applications. The expectation–maximization algorithm was employed to calculate the maximum likelihood estimates of the unknown parameters. Simulation experiments have been performed to show the effectiveness of the proposed algorithm. For illustrative purposes, we have provided one multivariate data set where it has been observed that there exist members in the proposed class of models that can provide better fit compared to skew-normal, skew-t, and generalized hyperbolic distribution. In another example, it was demonstrated that when data generated from a heavy-tailed skew-t distribution is contaminated with noise, the proposed distributions offer a better fit compared to the skew-t distribution.</div></div>","PeriodicalId":16431,"journal":{"name":"Journal of Multivariate Analysis","volume":"208 ","pages":"Article 105430"},"PeriodicalIF":1.4,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143682852","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-12DOI: 10.1016/j.jmva.2025.105424
Arash A. Foroushani, Sévérien Nkurunziza
In this paper, we introduce a class of improved estimators for the mean parameter matrix of a multivariate normal distribution with an unknown variance–covariance matrix. In particular, the main results of Chételat and Wells (2012) are established in their full generalities and we provide the corrected version of their Theorem 2. Specifically, we generalize the existing results in three ways. First, we consider a parametric estimation problem which encloses as a special case the one about the vector parameter. Second, we propose a class of James–Stein matrix estimators and, we establish a necessary and a sufficient condition for any member of the proposed class to have a finite risk function. Third, we present the conditions for the proposed class of estimators to dominate the maximum likelihood estimator. On the top of these interesting contributions, the additional novelty consists in the fact that, we extend the methods suitable for the vector parameter case and the derived results hold in the classical case as well as in the context of high and ultra-high dimensional data.
{"title":"Improved Gaussian mean matrix estimators in high-dimensional data","authors":"Arash A. Foroushani, Sévérien Nkurunziza","doi":"10.1016/j.jmva.2025.105424","DOIUrl":"10.1016/j.jmva.2025.105424","url":null,"abstract":"<div><div>In this paper, we introduce a class of improved estimators for the mean parameter matrix of a multivariate normal distribution with an unknown variance–covariance matrix. In particular, the main results of Chételat and Wells (2012) are established in their full generalities and we provide the corrected version of their Theorem 2. Specifically, we generalize the existing results in three ways. First, we consider a parametric estimation problem which encloses as a special case the one about the vector parameter. Second, we propose a class of James–Stein matrix estimators and, we establish a necessary and a sufficient condition for any member of the proposed class to have a finite risk function. Third, we present the conditions for the proposed class of estimators to dominate the maximum likelihood estimator. On the top of these interesting contributions, the additional novelty consists in the fact that, we extend the methods suitable for the vector parameter case and the derived results hold in the classical case as well as in the context of high and ultra-high dimensional data.</div></div>","PeriodicalId":16431,"journal":{"name":"Journal of Multivariate Analysis","volume":"208 ","pages":"Article 105424"},"PeriodicalIF":1.4,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143828601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-06DOI: 10.1016/j.jmva.2025.105433
Kristi Kuljus , Han Bao , Bo Ranneby
This article considers the maximum spacing (MSP) method for multivariate observations, nearest neighbour balls are used as a multidimensional analogue to univariate spacings. Compared to the previous studies, a broader class of MSP estimators corresponding to different information-type measures is studied. The studied class of estimators includes also the estimator corresponding to the Kullback–Leibler information measure obtained with the logarithmic function. Consistency of the MSP estimators is proved when the assigned model class is correct, that is the true density belongs to the assigned class. The behaviour of the MSP estimator under different divergence measures is studied and the advantage of using MSP estimators corresponding to different information measures in the context of model validation is illustrated in simulation examples.
{"title":"Maximum spacing estimation for multivariate observations under a general class of information-type measures","authors":"Kristi Kuljus , Han Bao , Bo Ranneby","doi":"10.1016/j.jmva.2025.105433","DOIUrl":"10.1016/j.jmva.2025.105433","url":null,"abstract":"<div><div>This article considers the maximum spacing (MSP) method for multivariate observations, nearest neighbour balls are used as a multidimensional analogue to univariate spacings. Compared to the previous studies, a broader class of MSP estimators corresponding to different information-type measures is studied. The studied class of estimators includes also the estimator corresponding to the Kullback–Leibler information measure obtained with the logarithmic function. Consistency of the MSP estimators is proved when the assigned model class is correct, that is the true density belongs to the assigned class. The behaviour of the MSP estimator under different divergence measures is studied and the advantage of using MSP estimators corresponding to different information measures in the context of model validation is illustrated in simulation examples.</div></div>","PeriodicalId":16431,"journal":{"name":"Journal of Multivariate Analysis","volume":"208 ","pages":"Article 105433"},"PeriodicalIF":1.4,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143631729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-05DOI: 10.1016/j.jmva.2025.105431
Yuzo Maruyama , Takeru Matsuda
This is a follow-up paper of Polson and Scott (2012, Bayesian Analysis), which claimed that the half-Cauchy prior is a sensible default prior for a scale parameter in hierarchical models. For estimation of a -variate normal mean under the quadratic loss, they demonstrated that the Bayes estimator with respect to the half-Cauchy prior seems to be minimax through numerical experiments. In this paper, we theoretically establish the minimaxity of the corresponding Bayes estimator using the interval arithmetic.
本文是 Polson 和 Scott（2012，《贝叶斯分析》）的后续论文，他们声称半考奇先验是层次模型中规模参数的合理默认先验。对于二次损失下的 p 变量正态均值估计，他们通过数值实验证明了关于半考奇先验的贝叶斯估计器似乎是最小的。本文从理论上利用区间算术建立了相应贝叶斯估计器的最小性。
{"title":"Minimaxity under the half-Cauchy prior","authors":"Yuzo Maruyama , Takeru Matsuda","doi":"10.1016/j.jmva.2025.105431","DOIUrl":"10.1016/j.jmva.2025.105431","url":null,"abstract":"<div><div>This is a follow-up paper of Polson and Scott (2012, Bayesian Analysis), which claimed that the half-Cauchy prior is a sensible default prior for a scale parameter in hierarchical models. For estimation of a <span><math><mi>p</mi></math></span>-variate normal mean under the quadratic loss, they demonstrated that the Bayes estimator with respect to the half-Cauchy prior seems to be minimax through numerical experiments. In this paper, we theoretically establish the minimaxity of the corresponding Bayes estimator using the interval arithmetic.</div></div>","PeriodicalId":16431,"journal":{"name":"Journal of Multivariate Analysis","volume":"208 ","pages":"Article 105431"},"PeriodicalIF":1.4,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-28DOI: 10.1016/j.jmva.2025.105426
Shiyuan Deng , He Tang , Shuyang Bai
We investigate the estimation of multivariate extreme models with a discrete spectral measure using spherical clustering techniques. The primary contribution involves devising a method for selecting the order, that is, the number of clusters. The method consistently identifies the true order, i.e., the number of spectral atoms, and enjoys intuitive implementation in practice. Specifically, we introduce an extra penalty term to the well-known simplified average silhouette width, which penalizes small cluster sizes and small dissimilarities between cluster centers. Consequently, we provide a consistent method for determining the order of a max-linear factor model, where a typical information-based approach is not viable. Our second contribution is a large-deviation-type analysis for estimating the discrete spectral measure through clustering methods, which serves as an assessment of the convergence quality of clustering-based estimation for multivariate extremes. Additionally, as a third contribution, we discuss how estimating the discrete measure can lead to parameter estimations of heavy-tailed factor models. We also present simulations and real-data studies that demonstrate order selection and factor model estimation.
{"title":"On estimation and order selection for multivariate extremes via clustering","authors":"Shiyuan Deng , He Tang , Shuyang Bai","doi":"10.1016/j.jmva.2025.105426","DOIUrl":"10.1016/j.jmva.2025.105426","url":null,"abstract":"<div><div>We investigate the estimation of multivariate extreme models with a discrete spectral measure using spherical clustering techniques. The primary contribution involves devising a method for selecting the order, that is, the number of clusters. The method consistently identifies the true order, i.e., the number of spectral atoms, and enjoys intuitive implementation in practice. Specifically, we introduce an extra penalty term to the well-known simplified average silhouette width, which penalizes small cluster sizes and small dissimilarities between cluster centers. Consequently, we provide a consistent method for determining the order of a max-linear factor model, where a typical information-based approach is not viable. Our second contribution is a large-deviation-type analysis for estimating the discrete spectral measure through clustering methods, which serves as an assessment of the convergence quality of clustering-based estimation for multivariate extremes. Additionally, as a third contribution, we discuss how estimating the discrete measure can lead to parameter estimations of heavy-tailed factor models. We also present simulations and real-data studies that demonstrate order selection and factor model estimation.</div></div>","PeriodicalId":16431,"journal":{"name":"Journal of Multivariate Analysis","volume":"208 ","pages":"Article 105426"},"PeriodicalIF":1.4,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143579875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-27DOI: 10.1016/j.jmva.2025.105425
Andreas H. Hamel, Thi Khanh Linh Ha
Expectile regions–like depth regions in general–capture the idea of centrality of multivariate distributions. If an order relation is present for the values of random vectors and a decision maker is interested in dominant/best points with respect to this order, centrality is not a useful concept. Therefore, cone expectile sets are introduced which depend on a vector preorder generated by a convex cone. This provides a way of describing and clustering a multivariate distribution/data cloud with respect to an order relation. Fundamental properties of cone expectiles are established including dual representations of both expectile regions and cone expectile sets. It is shown that set-valued sublinear risk measures can be constructed from cone expectile sets in the same way as in the univariate case. Inverse functions of cone expectiles are defined which should be considered as ranking functions related to the initial order relation rather than as depth functions. Finally, expectile orders for random vectors are introduced and characterized via expectile ranking functions.
{"title":"Set-valued expectiles for ordered data analysis","authors":"Andreas H. Hamel, Thi Khanh Linh Ha","doi":"10.1016/j.jmva.2025.105425","DOIUrl":"10.1016/j.jmva.2025.105425","url":null,"abstract":"<div><div>Expectile regions–like depth regions in general–capture the idea of centrality of multivariate distributions. If an order relation is present for the values of random vectors and a decision maker is interested in dominant/best points with respect to this order, centrality is not a useful concept. Therefore, cone expectile sets are introduced which depend on a vector preorder generated by a convex cone. This provides a way of describing and clustering a multivariate distribution/data cloud with respect to an order relation. Fundamental properties of cone expectiles are established including dual representations of both expectile regions and cone expectile sets. It is shown that set-valued sublinear risk measures can be constructed from cone expectile sets in the same way as in the univariate case. Inverse functions of cone expectiles are defined which should be considered as ranking functions related to the initial order relation rather than as depth functions. Finally, expectile orders for random vectors are introduced and characterized via expectile ranking functions.</div></div>","PeriodicalId":16431,"journal":{"name":"Journal of Multivariate Analysis","volume":"208 ","pages":"Article 105425"},"PeriodicalIF":1.4,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143549608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-25DOI: 10.1016/j.jmva.2025.105429
Benoît Oriol , Alexandre Miot
This work addresses large dimensional covariance matrix estimation with unknown mean. The empirical covariance estimator fails when dimension and number of samples are proportional and tend to infinity, settings known as Kolmogorov asymptotics. When the mean is known, Ledoit and Wolf (2004) proposed a linear shrinkage estimator and proved its convergence under those asymptotics. To the best of our knowledge, no formal proof has been proposed when the mean is unknown. To address this issue, we propose to extend the linear shrinkage and its convergence properties to translation-invariant estimators. We expose four estimators respecting those conditions, proving their properties. Finally, we show empirically that a new estimator we propose outperforms other standard estimators.
{"title":"Ledoit-Wolf linear shrinkage with unknown mean","authors":"Benoît Oriol , Alexandre Miot","doi":"10.1016/j.jmva.2025.105429","DOIUrl":"10.1016/j.jmva.2025.105429","url":null,"abstract":"<div><div>This work addresses large dimensional covariance matrix estimation with unknown mean. The empirical covariance estimator fails when dimension and number of samples are proportional and tend to infinity, settings known as Kolmogorov asymptotics. When the mean is known, Ledoit and Wolf (2004) proposed a linear shrinkage estimator and proved its convergence under those asymptotics. To the best of our knowledge, no formal proof has been proposed when the mean is unknown. To address this issue, we propose to extend the linear shrinkage and its convergence properties to translation-invariant estimators. We expose four estimators respecting those conditions, proving their properties. Finally, we show empirically that a new estimator we propose outperforms other standard estimators.</div></div>","PeriodicalId":16431,"journal":{"name":"Journal of Multivariate Analysis","volume":"208 ","pages":"Article 105429"},"PeriodicalIF":1.4,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143508247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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