  Under-coverage and nonresponse problems are jointly present in most
socio-economic surveys. The purpose of this paper is to propose a completely
design-based estimation strategy that accounts for both problems without
resorting to models but simply performing a two-step calibration. The first
calibration exploits a set of auxiliary variables only available for the units
in the sampled population to account for nonresponse. The second calibration
exploits a different set of auxiliary variables available for the whole
population, to account for under-coverage. The two calibrations are then
unified in a double-calibration estimator. Mean and variance of the estimator
are derived up to the first order of approximation. Conditions ensuring
approximate unbiasedness are derived and discussed. The strategy is empirically
checked by a simulation study performed on a set of artificial populations. A
case study is lead on Danish data coming from the European Union Statistics on
Income and Living Conditions survey. The strategy proposed is flexible and
suitable in most situations in which both under-coverage and nonresponse are
present.
