The generalized exponential model for sampling weight calibration for extreme values, nonresponse, and poststratification

The generalized exponential model for sampling weight calibration for extreme values, nonresponse, and poststratification was written by R.E. Folsom and A.C. Singh in 2000. It was part of the proceedings of the American Statistical Association Section on Survey Research Methods.

The authors build on Deville and Särndal (1992) (notated DS in this article). Establishing some notation:

• individuals are indexed by k

• the sample of individuals is s

• y_k is the study variable

• x_k is a vector of p predictor variables

• T_x is a vector of p (i.e., same dimension as x_k) post-stratification controls

• d_k is a design weight

• w_k is an adjusted weight, i.e. d_ka_k

• λ be a vector of p (i.e., same dimension as x_k) parameters

The logit-type SD model for the aforementioned weight adjustments is:

where l and u are set as lower/upper bounds, for which the only requirement is that l < 1 < u since 1 is implicitly the 'center'; and where A = (u - l)/(u - 1)(1 - l). The λ parameters are then estimated iteratively through satisfying the equation Σ_s x_kd_ka_k(λ) - T_x = 0 and minimizing Δ(w_k,d_k):

The authors extend this model for centers other than 1, variable bounds, and post-stratification controls designed to address problems other than sampling error. They call this the generalized exponential model (GEM).

• bounds and center are set such that l_k < c_k < u_k

• it follows that A_k = (u_k - l_k)/(u_k - c_k)(c_k - l_k)

• 

• Σ_s x_kd_ka_k(λ) - T̃_k = 0 (note the tilde over T)

• 

The λ parameters can be estimated through Newton-Raphson iterative steps.

• Let v denote the iteration.

• Set initial values as φ_k⁽⁰⁾ = 1 and λ⁽⁰⁾ = 0.

• Within each iteration, let Γ_φ,v = diag(d_kφ_k^(v)) for convenience.

• Update the parameters as λ^(v) = λ^(v-1) + (X'Γ_φ,v-1X)^-1 (T_x - T̂_x^(v-1)).

• For subsequent iterations, φ_k^(v) = (u_k - a_k^(v))(a_k^(v) - l_k)/(u_k - c_k)(c_k - l_k).

• Iterate until ||T_x - T̂_x^(v)|| increases, instead of decreasing.

Reading Notes

Section 4 gets into asymptotic properties, will need to come back to this too.

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