ABOUT THE CONSTRUCTION OF ORTHOGONAL
PROJECTORS

Antonio Francisco IEMMA[1]

Rudy PALM^{2}

Jean-Jacques CLAUSTRIAUX[2]

§ ABSTRACT: If in the analysis of the
data one uses the linear model of the type y=Xq + e , e ~ N(f , Is ^{2}) ( Gauss-Markov), then the orthogonal projector P of the y
vector (vector of observations) on the space spanned by the columns of X is
obtained just doing P = XX^{+ }= XX ^{1} = X(X'X)^{G}X',
where X^{+}, X ^{1}and (X'X)^{G} are the Moore-Penrose
generalized inverse of X , any Least Square Generalized Inverse and any
Generalized Inverse of X'X. Of course P is invariant and defined as above
becomes the matrice of quadratic form y' Py associate to the sum of square of
all parameters of the model. On the other hand , to obtain specific orthogonal
projection for the quadratic form associate to a single parameter or to particular
hypotheses on one or more parameters, it is not obvious and in the case of
unbalanced sample it is too difficult. This study shows a criteria to obtain
orthogonal projections for any linear model with Gauss-Markov structure, any
interest hypothesis and any level of unbalancedness of the sample.

§ KEYWORDS: Orthogonal projectors;
hypothesis tests; unbalanced data.