3 An explicit solution
The constrained minimisation problem stated above has an explicit solution for a distance function based on the chi-squared measure. This is discussed in subsection 1. A numerical example is examined in subsection 2.
3.1 The Chi-squared distance measure
Consider the chi-squared type of distance measure, where the aggregate distance is given by:
(6) 
The Lagrangean in (5) can be written as:
(7) 
where the 
for 
, are the Lagrange multipliers, and 
represents the 
th element of the vector of known population aggregates, 
.
Differentiation of (7) gives the set of 
first-order conditions:
(8) 
for 
, along with the 
conditions in (3). Rewriting 
as 
, where the prime indicates transposition, and multiplication of each equation in (8) by 
gives, after rearrangement:
(9) 
for .
To solve for the Lagrange multipliers, pre-multiply (9) by 
and rearrange, so that:
(10) 
Summing (10) over all 
and making use of the calibration equations, gives:
(11) 
where the term in brackets on the right hand side of (11) is a by square matrix. Hence, if this matrix can be inverted, the vector of Lagrange multipliers is given by:
(12) 
The resulting values of 
are substituted into (9) to obtain the new weights.[5]
Notes
- [5]Write (9) as and (12) as with as the symmetric matrix . Given sample observations on the variable an estimate of the population total, , can be obtained as . Substituting for gives the result in Deville and Särndal (1992, p.377) that , where . This provides the link between reweighting and the Generalised Regression (GREG) estimator. The production of asymptotic standard errors is often based on this estimator, in view of the result that other distance functions are asymptotically equivalent; see Deville and Särndal (1992, p.378). The present discussion concentrates only on reweighting.
