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# Multivariate Models

• George A. F. Seber
Part of the Springer Series in Statistics book series (SSS)

## Abstract

Up till now we have been considering various univariate linear models of the form $$y_{i} =\theta _{i} +\varepsilon _{i}$$ (i = 1, 2, , n), where $$E[\varepsilon _{i}] = 0$$ and the $$\varepsilon _{i}$$ are independently and identically distributed. We assumed G that $$\boldsymbol{\theta }\in \varOmega$$, where Ω is a p-dimensional vector space in $$\mathbb{R}^{n}$$. A natural extension to this is to replace the response variable y i by a 1 × d row vector of response variables y i ′, and replace the vector y = (y i ) by the data matrix
$$\displaystyle{\mathbf{Y} = \left (\begin{array}{c} \mathbf{y}_{1}' \\ \mathbf{y}_{2}'\\ \vdots \\ \mathbf{y}_{n}' \end{array} \right ) = (\mathbf{y}^{(1)},\mathbf{y}^{(2)},\ldots,\mathbf{y}^{(d)}),}$$
say. Here y(j) (j = 1, 2, , d) represents n independent observations on the jth variable of y. Writing $$\mathbf{y}^{(j)} =\boldsymbol{\theta } ^{(j)} + \mathbf{u}^{(j)}$$ with E[u(j)] = 0, we now have d univariate models, which will generally not be independent, and we can combine them into one equation giving us
$$\displaystyle{\mathbf{Y} =\boldsymbol{\varTheta } +\mathbf{U},}$$
where $$\boldsymbol{\varTheta }= (\boldsymbol{\theta }^{(1)},\boldsymbol{\theta }^{(2)},\ldots,\boldsymbol{\theta }^{(d)})$$, $$\mathbf{U} = (\mathbf{u}^{(1)},\mathbf{u}^{(2)},\ldots,\mathbf{u}^{(d)})$$, and E[U] = 0. Of particular interest are vector extensions of experimental designs where each observation is replaced by a vector observation. For example, we can extend the randomized block design
$$\displaystyle{\theta _{ij} =\mu +\alpha _{i} +\tau _{j}\quad (i = 1,2,\ldots,I;j = 1,2,\ldots,J),}$$

## References

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## Copyright information

© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• George A. F. Seber
• 1
1. 1.Department of StatisticsThe University of AucklandAucklandNew Zealand