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:

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_iy_i

w_i = d_i(1 + q_i λ^T x_i)

Note from here that Σ_N x_i = Σ_S w_ix_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_iq_i] subject to Σ_N x_i = Σ_S w_ix_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_ix_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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