Generalized ratio-type and exponential ratio-type estimators for mean in presence of non-response using follow-up procedure based on three-phase and four-phase sampling

Hansen and Hurwitz (1946) advised a follow-up procedure based on two-phase sampling which is reasonable when respondents and non-respondents are not similar. Such a follow-up procedure eliminates the non-response bias if all the units selected in the 2nd-phase sample from the non-respondent's group will respond. However, since in most of the real-life situations all units don't respond, this hints Hansen and Hurwitz (1946) estimator will be no more unbiased. This study presents a follow-up procedure based on three-phase, and four-phase sampling separately, and it shows that sample mean estimator based on the proposed procedures is now unbiased which is further discussed with the weighting procedure and the variance expressions separately. Furthermore, the auxiliary variable is introduced to estimate population mean more efficiently. Empirical and simulation results show that the proposed estimators are more efficient. A real-life application of mixed-mode survey is provided to show applicability of the proposed sampling procedures.