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Estimating linear statistical relationships. (English) Zbl 0542.62039
This survey paper describes the statistical analysis of a collection of related models, defined as follows. For $$i=1,...,n$$ the observable p by vector $$X_ i$$ is decomposed as $$Z_ i+U_ i$$, where the nonobservable $$Z_ i$$ is the ”systematic” part and the nonobservable $$U_ i$$ is the ”random error”. The systematic part varies in a linear space of dimension less than p. Each component of $$U_ i$$ has zero mean, and the covariance matrix of the components of $$U_ i$$ is denoted by C. $$U_ 1,...,U_ n$$ are mutually independent and are independent of $$(Z_ 1,...,Z_ n).$$
The cases discussed are given by the Cartesian product of the two sets of conditions (1,2) and (a,b,c):
(1) $$Z_ 1,...,Z_ n$$ are nonrandom parameters. (2) $$Z_ 1,...,Z_ n$$ are random.
(a) $$C=\sigma^ 2I$$, where I is the p by p identity matrix and $$\sigma^ 2$$ is unknown. (b) C is diagonal but not necessarily equal to $$\sigma^ 2I$$. (c) C is unrestricted, so that replicated observations are needed to estimate it.
Through most of the paper, it is assumed that the $$U_ i$$ are normally distributed, and if the $$Z_ i$$ are random they are normally distributed. Maximum likelihood estimators of the coefficients of the equations determining the linear space of $$Z_ i$$ and of the components of C are derived and analyzed. Such estimators do not exist in some cases.
Reviewer: L.Weiss

##### MSC:
 62H12 Estimation in multivariate analysis 62H25 Factor analysis and principal components; correspondence analysis 62-02 Research exposition (monographs, survey articles) pertaining to statistics
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