Existence and uniqueness of the maximum likelihood estimator for models with a Kronecker product covariance structure

Roś, Beata; Bijma, Fetsje; Munck, Jan C. de; Gunst, Mathisca C. M. de

doi:10.1016/j.jmva.2015.05.019

Cited by 22 publications

(28 citation statements)

References 17 publications

(53 reference statements)

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“…It was claimed in Dutilleul (1999) that max[p/q, q/p] + 1 samples are needed for the existence and uniqueness of the MLE solution in the Gaussian case. However, it was later shown by a counterexample Roś et al (2016) that the uniqueness does not follow from this condition. In addition, the authors of Roś et al (2016) write that "Moreover, it is not known whether it [this condition] guaranties existence, because it is not sufficient to show that all updates of the FF algorithm have full rank as is done in Dutilleul (1999).…”

Section: Existence Uniqueness and Convergence: State Of The Artmentioning

confidence: 99%

Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness

Soloveychik

Trushin

2016

Journal of Multivariate Analysis

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We study the Gaussian and robust covariance estimation, assuming the true covariance matrix to be a Kronecker product of two lower dimensional square matrices. In both settings we define the estimators as solutions to the constrained maximum likelihood programs. In the robust case, we consider Tyler's estimator defined as the maximum likelihood estimator of a certain distribution on a sphere. We develop tight sufficient conditions for the existence and uniqueness of the estimates and show that in the Gaussian scenario with the unknown mean, p/q + q/p + 2 samples are almost surely enough to guarantee the existence and uniqueness, where p and q are the dimensions of the Kronecker product factors. In the robust case with the known mean, the corresponding sufficient number of samples is max [p/q, q/p] + 1.

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Section: Existence Uniqueness and Convergence: State Of The Artmentioning

confidence: 99%

Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness

Soloveychik

Trushin

2016

Journal of Multivariate Analysis

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“…Actually, theoretical approximations in Dutilleul () were already discussed by Srivastava et al () (see above), including that the properties of the regular exponential family of distributions cannot be applied to the matrix normal distribution because it is curved. Roś et al (, p. 348) argue that the MLE may not exist for sample sizes n between max( p / q , q / p ) + 1 and pq (without constraint; Dutilleul, ) or between max( p , q ) + 1 and pq (with constraint; Srivastava et al, ), but present (pp. 349–350) two mathematical proofs demonstrating that the MLE is not unique for n = 2 and p = q and for n = 3 and p = q = 2.…”

Section: Debate and Recent Resultsmentioning

confidence: 99%

“…No specific information was given in Mardia and Goodall (), and the question was not addressed in de Munck et al (). The stronger condition n > pq was also mentioned for existence (Roś, Bijma, de Munck, & de Gunst, ).…”

Section: Estimationmentioning

confidence: 94%

Estimation and testing for separable variance–covariance structures

Dutilleul

2018

WIREs Computational Stats

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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

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“…which is equivalent to Σ * = ∆ −1 * ⊗ Φ −1 * , where Φ * is an unknown r by r precision matrix with a,b |Φ * a,b | = r, and ∆ * is an unknown c by c precision matrix. The norm condition on Φ * is added for identifiability: see Roś et al (2016) for more on identifiability under (2). This simplification of a covariance matrix makes the conditional distributions in (1) become matrix normal (Gupta and Nagar, 2000).…”

Section: Introductionmentioning

confidence: 99%

A Penalized Likelihood Method for Classification With Matrix-Valued Predictors

Molstad

Rothman

2018

Journal of Computational and Graphical Statistics

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We propose a penalized likelihood method to fit the linear discriminant analysis model when the predictor is matrix valued. We simultaneously estimate the means and the precision matrix, which we assume has a Kronecker product decomposition. Our penalties encourage pairs of response category mean matrices to have equal entries and also encourage zeros in the precision matrix. To compute our estimators, we use a blockwise coordinate descent algorithm. To update the optimization variables corresponding to response category mean matrices, we use an alternating minimization algorithm that takes advantage of the Kronecker structure of the precision matrix. We show that our method can outperform relevant competitors in classification, even when our modeling assumptions are violated. We analyze an EEG dataset to demonstrate our method's interpretability and classification accuracy.

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Existence and uniqueness of the maximum likelihood estimator for models with a Kronecker product covariance structure

Cited by 22 publications

References 17 publications

Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness

Gaussian and robust Kronecker product covariance estimation: Existence and uniqueness

Estimation and testing for separable variance–covariance structures

A Penalized Likelihood Method for Classification With Matrix-Valued Predictors

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