Design and estimation problems when estimating a regression coefficient from survey data

Summary The values of a variablex are assumed known for all elements in a finite population. Between this variable and another variableY, whose values are registered in a sample survey, there is the usual linear regression relationship. This paper considers problems of design and of estimation of the regression coefficienta and the interceptb. The followingGodambe type theorem is proved: There exists no minimum variance unbiased linear estimator ofa andb. We also derive that the usual estimators ofa andb have minimum variance if attention is restricted to the class of linear estimators unbiased in any given sample.