Best unbiased estimates for parameters of three-level multivariate data with doubly exchangeable covariance structure

Kozioł, Arkadiusz; Roy, Anuradha; Zmyślony, Roman; Leiva, Ricardo; Fonseca, Miguel

doi:10.1016/j.laa.2017.08.005

Cited by 6 publications

(7 citation statements)

References 12 publications

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“…So, µ = X is BLUE. This coincides with the arguments of the proofs in Roy et al (2016) and Koziol et al (2017).…”

Section: C4 Derivations Of the Maximum Likelihood Estimator And Its P...supporting

confidence: 88%

“…By Result 2 from Roy et al (2016); Koziol et al (2017), (1 n ⊗ I p )Y is the complete and minimal sufficient statistic for µ, and…”

Section: C4 Derivations Of the Maximum Likelihood Estimator And Its P...mentioning

confidence: 93%

“…Second, we can prove α kk and β kk are best quadratic unbiased estimator (BQUE) for α kk and β kk for every k and k . We need to check the conditions of Theorem 2 in Zmyślony (1976), or the lines of analogous arguments in Roy et al (2016) and Koziol et al (2017), that is, given P 0 V = V P 0 and R 0 = I pn − P 0 , there exist BQUE for the parameters of quadratic covariance if and only if span (R 0 V R 0 ), i.e., the smallest linear space containing R 0 V R 0 , is a quadratic space. Since Σ(A, B, τ ) is a UB matrix, {Σ(A, B, τ )} 2 is a UB matrix, expressed by Σ 2 (A 2 , AB + BA + BP B, τ ), so span{Σ(A, B, τ )} is a quadratic subspace and the identity matrix I(τ ) belongs to it.…”

Section: C4 Derivations Of the Maximum Likelihood Estimator And Its P...mentioning

confidence: 99%

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Covariance estimation with uniform blocks

Yang¹,

Chen²,

Chen³

2023

Preprint

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Estimating a covariance matrix is central to high-dimensional data analysis. The proposed method is motivated by the dependence pattern analyses of multiple types of highdimensional biomedical data including but not limited to genomics, proteomics, microbiome, and neuroimaging data. The correlation matrices of these biomedical data all demonstrate a well-organized block pattern. In this pattern, the positive and negative pairwise correlations with large absolute values, are mainly concentrated within diagonal and off-diagonal blocks. We develop a covariance-and precision-matrix estimation framework to fully leverage the organized block pattern. We propose new best unbiased covarianceand precision-matrix estimators in closed forms, and develop theories for the asymptotic proprieties of estimators in both scenarios where the number of blocks is less or greater than the sample size. The simulation and data example analyses show that our method is robust and improves the accuracy of covariance-and precision-matrix estimation.

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“…So, µ = X is BLUE. This coincides with the arguments of the proofs in Roy et al (2016) and Koziol et al (2017).…”

Section: C4 Derivations Of the Maximum Likelihood Estimator And Its P...supporting

confidence: 88%

“…By Result 2 from Roy et al (2016); Koziol et al (2017), (1 n ⊗ I p )Y is the complete and minimal sufficient statistic for µ, and…”

Section: C4 Derivations Of the Maximum Likelihood Estimator And Its P...mentioning

confidence: 93%

Section: C4 Derivations Of the Maximum Likelihood Estimator And Its P...mentioning

confidence: 99%

See 1 more Smart Citation

Covariance estimation with uniform blocks

Yang¹,

Chen²,

Chen³

2023

Preprint

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“…Since k-SSCS matrix Γ k in ( 9) is of Jordan algebra type, following Kozioł et al [26] one can show that the above estimate Γ 3 is the best unbiased, consistent and complete estimator for Γ 3 . 2.…”

Section: The Expressions Of the ∆'Smentioning

confidence: 94%

“…Therefore, we conclude that G p 1,k is a Jordan Algebra. See Lemma 4.1 on Page 10 in Malley [24], which states that G p 1,k is a Jordan Algebra if and only if Roy et al [25] and Kozioł et al [26] for proofs that 2-SSCS and 3-SSCS covariance structures are of Jordan algebra types.…”

Section: K-sscs Covariance Structure Is Of the Jordan Algebra Typementioning

confidence: 99%

Mean Equality Tests for High-Dimensional and Higher-Order Data with k-Self Similar Compound Symmetry Covariance Structure

Leiva

Roy

2022

Symmetry

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An extension of the D2 test statistic to test the equality of mean for high-dimensional and k-th order array-variate data using k-self similar compound symmetry (k-SSCS) covariance structure is derived. The k-th order data appear in many scientific fields including agriculture, medical, environmental and engineering applications. We discuss the property of this k-SSCS covariance structure, namely, the property of Jordan algebra. We formally show that our D2 test statistic for k-th order data is an extension or the generalization of the D2 test statistic for second-order data and for third-order data, respectively. We also derive the D2 test statistic for third-order data and illustrate its application using a medical dataset from a clinical trial study of the eye disease glaucoma. The new test statistic is very efficient for high-dimensional data where the estimation of unstructured variance-covariance matrix is not feasible due to small sample size.

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