4  Alternative Distance Functions

The chi-squared distance function is convenient because it enables an explicit solution for the calibrated weights to be obtained, requiring only matrix inversion. However, a modified form of the same approach can be applied to a range of alternative distance functions, as shown in this section. These functions belong to a class of functions having two features: the first derivative with respect to can be expressed as a function of and its inverse can be obtained explicitly. An interactive solution procedure is required for the calculation of the Lagrange multipliers. The general case of this class is presented in subsection 1. An iterative approach based on Newton’s method is described in subsection 2. Several weighting functions are described in subsection 3 and illustrated in subsection 4.

4.1  The general case

The Lagrangean for the general case, stated in section 2, was written as:

(13)    

Suppose that has the property, shared with the chi-square distance function, that the differential with respect to can be expressed as a function of the ratio , so that:

(14)    

The first-order conditions for minimisation can therefore be written as:

(15)    

Write the inverse function of as so that if say, then In the case of the chi-square distance function used above, , and the inverse takes a simple linear form. In general, from (15) the values of are expressed as:

(16)    

If the inverse function, can be obtained explicitly, equation (16) can be used to compute the calibrated weights, given a solution for the vector, .

As before, the Lagrange multipliers can be obtained by post-multiplying (16) by the vector , summing over all and using the calibration equations, so that:

(17)    

Finally, subtracting from both sides of (17) gives:

(18)    

The term is of course a scalar, and the left hand side is a known vector. In general, (18) is nonlinear in the vector and so must be solved using an iterative procedure, as described in the following subsection.

4.2  An iterative procedure

Writing the equations in (18) can be written as:

(19)    

for . The roots can be obtained using Newton’s method, described in the Appendix. This involves the following iterative sequence, where denotes the value of in the th iteration:[7]

(20)    

The Hessian matrix and the vector on the right hand side of (20) are evaluated using .

The elements are given by:

(21)    

which can be written as:

(22)     

Starting from arbitrary initial values, the matrix equation in (20) is used repeatedly to adjust the values until convergence is reached, where possible.

As mentioned earlier, the application of the approach requires that it is limited to distance functions for which the form of the inverse function, can be obtained explicitly, given the specification for . Hence, the Hessian can easily be evaluated at each step using an explicit expression for . As these expressions avoid the need for the numerical evaluation of and for each individual at each step, the calculation of the new weights can be expected to be relatively quick, even for large samples.[8] However, it must be borne in mind that a solution does not necessarily exist, depending on the distance function used and the adjustment required to the vector .