4.3  Some distance functions

One reason why the chi-squared distance function produces a solution is that no constraints are placed on the size of the adjustment to each of the survey weights. It is therefore also possible for the calibrated weights to become negative. However, Deville and Särndal (1992) suggested the following simple modification to the chi-squared function, although the explicit solution for the chi-squared case is no longer available and the iterative method must be used.

Suppose it is required to constrain the proportionate changes to certain limits, different for increases compared with decreases in the weights. Define and such that The objective is to ensure that, for increases, the proportionate change, is less than or that . For decreases, the aim is to ensure that (or the negative of the proportional change) is less than so that .

For the chi-squared distance function, it has been seen that where and solves for . Hence if is outside the specified range, it is necessary to set it to the relevant limit, either or rather than allow it to take the value generated. Since it is clear that the limits are exceeded if and if . In each case where the value of has to be set to the relevant limit, the corresponding value of is zero. This approach ensures that weights are kept within the range, . Hence, negative values of are avoided simply by setting to be positive.[9]

It has been seen above that the solution procedure requires only an explicit form for the inverse function from which its derivative can be obtained. It is not necessary to start from a specification of . Deville and Särndal (1992) suggest the simple form:

(23)    

The gradient function, is given by solving (23) for so that:

(24)    

and the form of the distance function can be obtained by integrating (24).[10] This is referred to as Case A, and its properties are given in the first row of Table 3. The second row of the table provides details of Case B, where , and the final row gives the corresponding properties of the basic chi-squared function.[11] A feature of these functions is that they do not require any parameters to be set.

Table 3 – Alternative distance functions

Case
A
B
Chi-squared

Deville and Särndal (1992) also suggest the use of an inverse function of the form:[12]

(25)    

where and are as defined above and:

(26)    

Thus and , so that the limits of are and This function therefore has the property that adjustments to the weights are kept within the range, , although, unlike the chi-squared modification, no checks have to be made during computation.

The derivative required in the computation of the Hessian is therefore:

(27)    

Since solves for (25) can be rearranged, by collecting terms in , to give:

(28)    

so that the gradient of the distance function is:

(29)    

The special nature of this gradient function is illustrated by the line D-S in Figure 1, which shows the profile of (29) for the wide range where and The first characteristic of the S-D function that is evident is the restriction of to the range specified. Figure 1 also shows the function for the other cases discussed above. In all cases, the slope is zero (corresponding to a turning point of the distance function) when . Given the quadratic U-shaped nature of the chi-squared distance function, the gradient increases at a constant rate, being negative in the range . Cases A and B also imply U-shaped distance functions, but with the gradient increasing more sharply for and more slowly than the chi-square function in the range .

Figure 1 – Alternative gradient functions

The distance function is given by integrating (29) with respect to . It is most convenient to apply the variate transformation , so that , and it is required to obtain:

(30)    

Using the result that:

(31)    

and:

(32)    

substitution and rearrangement gives multiplied by:

(33)    

plus a term , which, since it is a constant, may be dropped without loss.[13] Examples of this distance function are shown in Figure 2.[14]

Figure 2 – Deville-Särndal distance functions