Multi-step Sequential and Accelerated Sequential Methodologies for a Replicable Linear Model

Abstract

We consider a sequence of observable p-dimensional independent and identically distributed N p (A θ, Σ) observations X l, X 2, … where A p × q is a known matrix with rank q(1<q<p), θ q × 1 consists of unknown regression parameters, and Σ is an unknown positive definite matrix. Chatterjee (Chatterjee, S.K. Multi-step sequential procedures for a replicable linear model with correlated variables. In Probability Statistics and Design of Experiments (Proceedings of the R. C. Bose Symposium 1988); Bahadur R.R., Ed.; Wiley Eastern: New Delhi, 1990; 217–226) gave a multi-step sequential methodology to construct a fixed-size confidence region for θ. This methodology used the largest characteristic root of an appropriate conditional dispersion matrix. We propose new multi-step sequential and accelerated methodologies, which are more efficient in lowering the oversampling rate substantially. Accelerated versions of these estimation techniques further reduce sampling operations and these preserve crucial properties associated with the original multi-step methodology of Chatterjee (1988). A substantial part of this investigation includes thorough comparisons of all available methodologies for moderate sample sizes. Some limited robustness issues are addressed. An illustration with realistic data on scores of college students on the college level examination program (CLEP) is also included.