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Estimation and testing for separable variance–covariance structures

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The statistical analysis of data for a p‐variate response observed repeatedly on q occasions or of spatiotemporal data recorded at p locations by q times for n individuals may require that constraints be imposed on the modeling of the variance–covariance structure of the underlying process, not because of the repeated‐measures or spatiotemporal nature of the data but because there is not enough data otherwise to estimate the model parameters. Besides stationarity and isotropy, separability is an interesting option for that purpose because it reduces the number of variance‐covariance parameters to estimate, from pq(pq + 1)/2 to the Kronecker product of two matrices with p(p + 1)/2 and q(q + 1)/2 parameters. Originally, in the late 1980s, separability of the variance–covariance structure was assumed. Under this model, combined with the normality assumption on the underlying distribution, novel theoretical developments were thus made. The question of estimation of the parameters of a separable variance–covariance structure, more particularly by maximum likelihood, was raised from the early 1990s on, the question of testing for this structure being effectively addressed several years later. The existence and uniqueness of maximum likelihood estimators for the matrix normal distribution (i.e., the doubly multivariate normal distribution characterized by a simply separable variance–covariance structure) have been and remain questions of interest, as shown by recent results. Below, the reader is guided throughout the field of study of the separable variance–covariance structures as the author provides a fair treatment of the topic, its components, extensions (e.g., double separability), and future perspectives. This article is categorized under Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms
An example of the empirical distributions for the unmodified and biased (Λ) and modified and unbiased (Λ*) LRT statistics for simple separability of a variance–covariance structure, together with the theoretical distribution, from Manceur and Dutilleul (); the theoretical distribution fits the empirical distribution of Λ* so closely that they can barely be distinguished from each other; LRT, likelihood ratio test
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Plot of the 90th (lower curve), 95th, and 99th (higher curve) percentiles of the LRT (blue) and RST (red) statistics for simple separability of a variance–covariance structure, in relation to the natural logarithm of the sample size n; LRT, likelihood ratio test; RST, Rao's score test—The numerical values used to produce this plot come from table A1 in Filipiak et al. ()
[ Normal View | Magnified View ]

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Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms
Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data
Statistical and Graphical Methods of Data Analysis > Multivariate Analysis

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