Professor Heinz Neudecker and matrix differential calculus

Liu, Shuangzhe; Trenkler, Götz; Kollo, Tõnu; von Rosen, Dietrich; Baksalary, Oskar Maria

doi:10.1007/s00362-023-01499-w

Cited by 4 publications

(1 citation statement)

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“…Definition of the derivative with respect to matrix X of a scalar, vector and matrix function, can be also found in a recent paper by Liu et al (2023), where an insightful overview of matrix-oriented results and their applications in statistics are presented.…”

Section: Notation and Notionsmentioning

confidence: 99%

Covariance structure tests for multivariate t-distribution

Filipiak,

Kollo

2024

Stat Papers

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We derive an equation system for finding Maximum Likelihood Estimators (MLEs) for the parameters of a p-dimensional t-distribution with $$\nu $$ ν degrees of freedom, $$t_{p,\nu }$$ t p , ν , and use the MLEs for testing covariance structures for the $$t_{p,\nu }$$ t p , ν -distributed population. The likelihood ratio test (LRT), Rao score test (RST) and Wald test (WT) statistics are derived under the general null-hypothesis $$\textrm{H}_0:\varvec{\Sigma }=\varvec{\Sigma }_0$$ H 0 : Σ = Σ 0 , using a matrix derivative technique. Here the $$p\times p$$ p × p -matrix $$\varvec{\Sigma }$$ Σ is a dispersion/scale parameter. Convergence to the asymptotic chi-square distribution under the null hypothesis is examined in extensive simulation experiments. Also the convergence to the chi-square distribution is studied empirically in the situation when the MLEs of a $$t_{p,\nu }$$ t p , ν -distribution are changed to the corresponding estimators for a normal population. Type I errors and the power of the tests are also examined by simulation. In the simulation study the RST behaved more adequately than all remaining statistics in the situation when the dimensionality p was growing.

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Section: Notation and Notionsmentioning

confidence: 99%