Estimating linear statistical relationships.

*(English)*Zbl 0542.62039This 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.

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 |