In sample surveys, nonresponse and the high cost of collecting information on the study variable may substantially affect the efficiency of population mean estimation. Double sampling provides a useful framework in which auxiliary information can be obtained from a relatively large first-phase sample, whereas detailed information on the study variable is collected from a smaller second-phase sample. The presence of nonresponse in the second phase, however, introduces an additional source of variation and requires an appropriate follow-up procedure. The present study proposes a joint regression estimator for estimating the finite-population mean under double sampling with nonresponse using two auxiliary variables. The proposed estimator combines the Hansen–Hurwitz nonresponse-adjusted mean of the study variable with regression corrections based on the differences between first- and second-phase auxiliary means. The two auxiliary variables are incorporated jointly through their variance-covariance structure. First-order bias and mean squared error (MSE) are derived, and the optimum regression coefficients are obtained by minimizing the first-order MSE. The resulting coefficients correspond to the joint regression coefficients based on the inverse of the auxiliary covariance matrix. A second-order bias expression is also developed symbolically. An empirical illustration based on a population of 1,023 workers, with tax as the study variable, grade level as the first auxiliary variable, and gross payment as the second auxiliary variable, is presented. The proposed estimator is evaluated for follow-up factors k=2,3,4,5. The calculated first-order MSEs of the proposed estimator are 41,787.54, 43,572.55, 41,815.78 and 48,922.39, respectively. These values are compared with source-reported MSEs of existing estimators. The results demonstrate the potential of joint regression adjustment for utilizing correlated auxiliary information under double sampling with nonresponse.

Keywords- Double sampling, Nonresponse, Joint regression estimator, Hansen–Hurwitz estimator, Population mean,