= 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 [[CalibrationEstimatorsInSurveySampling|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,,'') [[Statistics/PostStratification|post-stratification]] controls
 * ''d,,k,,'' is a [[Statistics/DesignWeights|design weight]]
 * ''w,,k,,'' is an adjusted weight, i.e. ''d,,k,,a,,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:

{{attachment:sd.svg}}

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,,k,,d,,k,,a,,k,,(λ) - T,,x,, = 0'' and minimizing ''Δ(w,,k,,,d,,k,,)'':

{{attachment:dist.svg}}

The authors extend this model for centers other than 1, variable bounds, and post-stratification controls designed to address problems other than [[Statistics/SurveyInference#Sampling_Error|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,,)''
 * {{attachment:sd2.svg}}
 * ''Σ,,s,, x,,k,,d,,k,,a,,k,,(λ) - T̃,,k,, = 0'' (note the tilde over ''T'')
 * {{attachment:dist2.svg}}

The ''λ'' parameters can be estimated through [[Analysis/NewtonRaphsonMethod|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,,k,,φ,,k,,^(v)^)'' for convenience.
 * Update the parameters as ''λ^(v)^ = λ^(v-1)^ + (X'Γ,,φ,v-1,,X)^-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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