On the Most Effective Use of Continuous Auxiliary Variables in Regression Estimation in Survey Sampling

Auxiliary variables with known population totals are extensively used in survey sampling to construct generalised regression (GR) estimators or optimal regression (OR) estimators of totals or means of study variables. This article explores the possibility of improving the efficiency of such estimators when continuous auxiliary variables are used in the regression estimation jointly with appropriate power functions of them, provided that the values of the auxiliary variables are known for all units in the population. The efficiency gain is determined analytically in the case of the OR estimator. A practical criterion for choosing the power functions that maximise the efficiency gain, involving the coefficient of determination in the regression fit of the study variable, is proposed for both the OR estimation and the more practicable, but generally less efficient, GR estimation. Furthermore, the effect of adding a power function of a continuous auxiliary variable in regression estimation is investigated when this variable is also used at the design stage. A simulation study shows that the joint use of a continuous auxiliary variable and a power function of it chosen according to the proposed criterion may improve considerably the efficiency of OR estimation, and much more the efficiency of GR estimation.