**Sankhya: The Indian Journal of Statistics**

1995, Volume 57, Series A, Pt. 2, pp. 227--255

**RECURSIVE INFERENCE IN A GENERAL LINEAR MODEL**

By

P. BHIMASHANKARAM and DEBASIS SENGUPTA, *Indian Statistical Institute*

and

S. RAMANATHAN, *University of California*

SUMMARY. In this paper we provide exact algebraic expressions for the recalculation of the BLUE, the mean square error and several other statistical quantities of interest in a general linear model when an observation or a parameter is added or deleted. The dispersion matrix as well as the design matrix is allowed to be rank deficient by using the unified theory of least squares estimation as the point of departure. The updating formulas follow from the modifications of a generalized inverse of a symmetric matrix corresponding to the simultaneous addition or deletion of a row and a column. Statistical interpretations are given to the various special cases that arise naturally. Their relation to the changes in the rank of a key matrix is also pointed out. An extension to the mixed linear model with singular dispersion matrix is considered next. Finally we outline applications of these results in the areas of regression diagnostics and optimal design of experiments

*AMS (1991) subject classification.* Primary 62F05, secondary 62F30.

*Key words and phrases.* Recursive estimation and testing, generalized inverse of a matrix, unified theory of linear estimation, regression diagnostics.