Explicit estimators of parameters in the Growth Curve model with linearly structured covariance matrices

Ohlson, Martin; Rosen, Dietrich von

doi:10.1016/j.jmva.2009.12.023

Cited by 17 publications

(10 citation statements)

References 32 publications

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“…Thus, tests and estimates can also be obtained by a decomposition of the whole tensor space as done in Seid Hamid andvon Rosen (2006), von Rosen (1995) and Ohlson and von Rosen (2010).…”

Section: Modelmentioning

confidence: 99%

Test for the mean matrix in a Growth Curve model for high dimensions

Srivastava

Singull

2016

Communications in Statistics - Theory and Methods

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In this paper, we consider the problem of estimating and testing a general linear hypothesis in a general multivariate linear model, the so called Growth Curve model, when the p × N observation matrix is normally distributed with an unknown covariance matrix.The maximum likelihood estimator (MLE) for the mean is a weighted estimator with the inverse of the sample covariance matrix which is unstable for large p close to N and singular for p larger than N . We modify the MLE to an unweighted estimator and propose a new test which we compare with the previous likelihood ratio test (LRT) based on the weighted estimator, i.e., the MLE. We show that the performance of this new test based on the unweighted estimator is better than the LRT based on the MLE.For the high-dimensional case, when p is larger than N , we construct two new tests based on the trace of the variation matrices due to the hypothesis (between sum of squares) and the error (within sum of squares).To compare the performance of all four tests we compute the attained significance level and the empirical power.

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“…Thus, tests and estimates can also be obtained by a decomposition of the whole tensor space as done in Seid Hamid andvon Rosen (2006), von Rosen (1995) and Ohlson and von Rosen (2010).…”

Section: Modelmentioning

confidence: 99%

Test for the mean matrix in a Growth Curve model for high dimensions

Srivastava

Singull

2016

Communications in Statistics - Theory and Methods

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“…The paper by Ohlson and von Rosen (2010) was the first to propose a residual based procedure to obtain explicit estimators for an arbitrary linear structured covariance matrix in the classical growth curve model as an alternative to iterative methods. The idea in Ohlson and von Rosen (2010) was later on applied to the sum of two profiles model by Nzabanita et al (2012). The results in Nzabanita et al (2012) have been generalized to the extended GMANOVA model with an arbitrary number of profiles by Nzabanita et al (2015a).…”

Section: Explicit Estimators When the Covariance Matrix Is Linearly Smentioning

confidence: 99%

Bilinear and Trilinear Regression Models with Structured Covariance Matrices

Nzabanita

2015

Linköping Studies in Science and Technology. Dissertations

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Joseph Nzabanita (2015). Bilinear and Trilinear Regression Models with Structured Covariance Matrices Doctoral dissertation. This thesis focuses on the problem of estimating parameters in bilinear and trilinear regression models in which random errors are normally distributed. In these models the covariance matrix has a Kronecker product structure and some factor matrices may be linearly structured. Most of techniques in statistical modeling rely on the assumption that data were generated from the normal distribution. Whereas real data may not be exactly normal, the normal distributions serve as a useful approximation to the true distribution. The modeling of normally distributed data relies heavily on the estimation of the mean and the covariance matrix. The interest of considering various structures for the covariance matrices in different statistical models is partly driven by the idea that altering the covariance structure of a parametric model alters the variances of the model's estimated mean parameters.Firstly, we consider the extended growth curve model with a linearly structured covariance matrix. In general there is no problem to estimate the covariance matrix when it is completely unknown. However, problems arise when one has to take into account that there exists a structure generated by a few number of parameters. An estimation procedure that handles linear structured covariance matrices is proposed. The idea is first to estimate the covariance matrix when it may be used to define an inner product in a regression space and thereafter re-estimate it when it should be interpreted as a dispersion matrix. This idea is exploited by decomposing the residual space, the orthogonal complement to the design space, into orthogonal subspaces. Studying residuals obtained from projections of observations on these subspaces yields explicit consistent estimators of the covariance matrix. An explicit consistent estimator of the mean is also proposed.Secondly, we study a bilinear regression model with matrix normally distributed random errors. For those models, the dispersion matrix follows a Kronecker product structure and it can be used, for example, to model data with spatio-temporal relationships. The aim is to estimate the parameters of the model when, in addition, one covariance matrix is assumed to be linearly structured. On the basis of n independent observations from a matrix normal distribution, estimating equations, a flip-flop relation, are established.At last, the models based on normally distributed random third order tensors are studied. These models are useful in analyzing 3-dimensional data arrays. The 3-dimensional data arrays may be obtained when, for example, a single response is sampled in a 3-D space or in a 2-D space and time, multiple responses are recorded in a 2-D space or in a 1-D space and time. In some studies the analysis is done using the tensor normal model, where the focus is on the estimation of the variance-covariance matrix which has a Kronecker structure. Little attention...

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“…The paper [11] was the first to propose a residual based procedure to obtain explicit estimators for an arbitrary linear structured covariance matrix in the classical growth curve model as an alternative to iterative methods. The idea was later on applied to the sum of two profiles model in [12] and our aim here is to generalize results in [12] to the extended GMANOVA model with an arbitrary number of profiles.…”

Section: Introductionmentioning

confidence: 99%

Extended GMANOVA model with a linearly structured covariance matrix

2015

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In this paper we consider the extended generalized multivariate analysis of variance (GMANOVA) with a linearly structured covariance matrix. The main theme is to find explicit estimators for the mean and for the linearly structured covariance matrix. We show how to decompose the residual space, the orthogonal complement to the mean space, into m + 1 orthogonal subspaces and how to derive explicit estimators of the covariance matrix from the sum of squared residuals obtained by projecting observations on those subspaces. Also an explicit estimator of the mean is derived and some properties of the proposed estimators are studied.

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Explicit estimators of parameters in the Growth Curve model with linearly structured covariance matrices

Cited by 17 publications

References 32 publications

Test for the mean matrix in a Growth Curve model for high dimensions

Test for the mean matrix in a Growth Curve model for high dimensions

Bilinear and Trilinear Regression Models with Structured Covariance Matrices

Extended GMANOVA model with a linearly structured covariance matrix

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