**Sankhya: The Indian Journal of Statistics**

1996, Volume 58, Series B, Pt. 3, pp. 338--351

**THE LINEAR ZERO FUNCTIONS APPROACH TO LINEAR MODELS**

By

POCHIRAJU BHIMASANKARAM , *Indian Statistical Institute, Calcutta*

and

DEBASIS SENGUPTA, *University of California, Santa Barbara*

SUMMARY. In this paper we develop the theory of linear models using the properties of Linear Zero Functions. It is shown that all the major results on inference can be derived using these properties and simple vector arguments. The general philosophy of the development is to enhance understanding of the problem by appealing to the intuition of the reader. A common feature of the results derived here is that they integrate the singular dispersion case into mainstream. This is achieved without use of heavy algebra. On the contrary, most of the derivations are simpler than the conventional ones. Nevertheless, the extension of some of the results to the singular dispersion case are new.

*AMS (1991) subject classification.* 62J05.

*Key words and phrases.* Linear estimation, residual sum of squares, vector space, singular model, linear restriction, nuisance parameter.