Questionnaire Dwelling Unit-Level and Person Pair-Level Sampling ...

Appendix A: Technical Details about the Generalized
Exponential Model
A.1 Distance Function
Let Δ(w,d) denote the distance between the initial weights d { dk k∈s}
= : and the
adjusted weights w, with k being the k th unit in the sample, and s, the sample selected. The
distance function minimized under the generalized exponential model (GEM), subject to
calibration constraints, is given by
d ⎧
⎫
k ⎪
ak −l
k
uk −ak
⎪
Δ ( w,d ) = ∑ . ( − ) log + ( − ) log ,
k∈s
⎨ ak l
k
uk ak
⎬
Ak ⎪⎩
c −l
−
k k
uk
a
k ⎪⎭
(A.1.1)
ak w
k
/d
k,Ak u l k k
/ ⎣ uk ck c l k k
⎤⎦,
and l , , k
ck
and uk
are prescribed real
numbers. Let T x denote the p-vector of control totals corresponding to predictor variables (x 1 , ...,
x p ). Then, the calibration constraints for the above minimization problem are
where = = ( − ) ⎡( − )( − )
∑ .
∈ k k k
=
x
(A.1.2)
k s x da T
The solution for the above minimization problem, if it exists, is given by a GEM with
model parameters 8, i.e.,
a
k
l
λ =
( )
( u − c ) + u ( c −l
) exp{ A x′
λ}
( u − c ) + ( c −l
) exp{ A x′
λ}
k k k k k k k k
k k k k k k
.
(A.1.3)
Note that the number of parameters in GEM should be # n, where n is the size of the sample s.
This is also the dimension of vectors d and w. It follows from Equation A.1.3 that
l < a < u , k = 1, K , n.
(A.1.4)
k k k
The usual raking ratio method (see, e.g., Singh & Mohl, 1996) of weight adjustment is a
special case of GEM, such that for l = 0, u =∞ , c = 1, and k = 1, K , n, we have
and
Δ
∑
k k k
∑
( w d ) = d a log a − d ( a − 1)
, (A.1.5)
k∈s
k
a
k
k
k
k∈s
( λ ) exp ( x′
λ )
= .
The logit method of Deville and Särndal (1992) is also a special case of GEM by setting
lk
= l , uk
= u,
and c
k
= 1 for all k.
k
k
k
A-3