Optimal sampling from concomitant variables for regression problems

Accounting for an auxiliary covariate in a two-phase sampling strategy in order to reduce the experimental costs was initially proposed by Cochran (Sampling Techniques, 2nd Edition, Wiley, New York, 1963, Sampling Techniques, 3rd Edition, Wiley, New York, 1977) in the context of sample surveys. Conniffe and Moran (Biometrics 28 (1972) 1011) have extended this methodology to the estimation of linear regression functions. More recently, Conniffe (J. Econometrics 27 (1985) 179) and Causeur and Dhorne (Biometrics 54 (4) (1998) 1591) have derived two-phase sampling estimators of the linear regression function in the situation where many auxiliary covariates are available. A detailed study of the distributional aspects of these estimators is provided by Causeur (Statistics 32 (1999) 297). In the same multivariate context, this paper aims at an extension of the double-sampling strategies to monotone designs accounting for differences between the costs of subsets of covariates. In particular, the maximum-likelihood estimators are provided and asymptotic solutions for the optimal designs are derived.