ABOUT THE CONSTRUCTION OF ORTHOGONAL PROJECTORS

Antonio Francisco IEMMA[1]

Rudy PALM2

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)GX', where X+, X 1and (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.

 

 



[1] Departamento de Matemática - Escola Superior de Agricultura " Luiz Vaz de Queiróz" - ESALQ - USP - 13418-260 - Piracicaba - SP - Brasil.

[2] Bureau de Statistique et D’Informatique – Faculté des Sciences Agronomiques – 5030 – Gembloux – Bélgique.