On ordinary least-squares methods for sample surveys

Abstract

The performance of the ordinary least-squares (LS) method for two-stage sampling in regression analysis is studied. It is shown that the best linear unbiased estimator (BLUE) can be approximated by a polynomial in intracluster correlation. In particular, the least-squares estimator (LSE) is a zero-order approximation to the BLUE. To provide some insights into the approximation of the BLUE by the LSE, an upper bound for the difference between LSE and the first-order approximation to the BLUE is derived. Furthermore, bounds for the difference between the covariance matrices of the LSE and the BLUE are derived. Similar idea is applied to compare the LS and the BLU predictors of population total under a superpopulation model. For example, when the design matrix is ill-conditioned and/or the condition number of the covariance matrix of the error vector is large, the LSE has a poor performance compared to the BLUE.