= The calibration approach in survey theory and practice =

'''The calibration approach in survey theory and practice''' was written by Carl-Erik Särndal in 2007. It was published in ''Survey Methodology'' (vol. 33, no. 2).

The author introduces the GREG estimator as:

{{attachment:greg.svg}}

where each individual ''i'' in set ''S'' (complete survey responses) has a measured outcome ''y,,i,,'', but '''all''' cases (up to ''N'') have a predicted outcome ''ŷ,,i,,''. The predictions do not need to be true; the model is "assisting only".

That assisting model is based on auxilliary data '''''x''',,i,,'' known for all ''i''. It is also based on scaling factors ''q,,i,,'' are set for all ''i''. A fair starting point is 1. Only "outrageous choices" lead to a biased estimator. This model should include fixed effects for the strata. These are used to calculate:

'''''B''',,q,, = (Σ,,N,, q,,i,, '''x''',,i,, '''x''',,i,,^T^)^-1^ Σ,,N,, q,,i,, '''x''',,i,, '''y''',,i,,)''

This model is then used for the subset of cases ''S'', and with the base weights ''d,,i,,'', such that it will be notated as '''''B''',,dq,,'':

'''''B''',,dq,, = (Σ,,S,, d,,i,, q,,i,, '''x''',,i,, '''x''',,i,,^T^)^-1^ Σ,,S,, d,,i,, q,,i,, '''x''',,i,, '''y''',,i,,)''

''ŷ,,i,, = '''x''',,i,,^T^ '''B''',,dq,,''

Given an accurate prediction, the residuals (''y,,i,, - ŷ,,i,,'') are very small. Base weights then have a minimized impact on the estimator, making it very consistent.

The estimator can be re-written as a weighted sum:

''Ŷ,,GREG,, = w,,i,,y,,i,,''

''w,,i,, = d,,i,,(1 + q,,i,, '''λ'''^T^ '''x''',,i,,)''

{{attachment:lambda.svg}}

Note from here that ''Σ,,N,, '''x''',,i,, = Σ,,S,, w,,i,,'''x''',,i,,''; this is the core of calibration.

The author then introduces the calibrated estimator as being about a minimization problem of ''Σ,,S,, [(w,,i,,-d,,i,,)^2^ / 2d,,i,,q,,i,,]'' subject to ''Σ,,N,, '''x''',,i,, = Σ,,S,, w,,i,,'''x''',,i,,''. This comes from conceptualizing a distance between design weights and calibrated weights, ''G(d,,i,,,w,,i,,)'', and specifically minimizing chi-square distance.

In this specific formulation, GREG and calibration are equivalent. In any formulations, as long as the GREG estimator satisfies ''Σ,,N,, '''x''',,i,, = Σ,,S,, w,,i,,'''x''',,i,,'', the GREG estimator is a valid calibrated estimator that is specifically optimized for the outcome ''y''. They otherwise represent very different approaches to the problem.

The author continues on with many further remarks. I struggle to wrap my head around the math.



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