4. The effect of using estimated cost parameters

David G. Steel and Robert Graham Clark

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In practice, $c_i$ are not known precisely. Suppose that estimates ${\widehat{c}}_i=b_i{c}_i$ are used instead. Using the auxiliary variable and the estimated costs in the optimal probabilities implies ${\pi }_i\propto z_i^{1/2}{\widehat{c}}_i^{-1/2}.$ To make comparisons for the same expected costs,

$${\pi }_i=C_f\frac{z_i^{1/2}{\widehat{c}}_i^{-1/2}}{{\displaystyle {\sum }_{j\in U}{z}_j^{1/2}{\widehat{c}}_j^{-1/2}{c}_j}}\mathrm{.}$$

The resulting anticipated variance is

$$A{V}_{ests}={\sigma }^2{C}_f^{-1}\left({\displaystyle \sum _{i\in U}}{\widehat{c}}_i^{1/2}{z}_i^{1/2}\right)\left({\displaystyle \sum _{j\in U}}{z}_j^{1/2}{\widehat{c}}_j^{-1/2}{c}_j\right)-{\sigma }^2{\displaystyle \sum _{i\in U}}{z}_i\mathrm{.}(4.1)$$

If we assume that the values of $b_i$ are unrelated to the values of $c_i$ and $z_i,$ then

$$A{V}_{ests}={\sigma }^2{C}_f^{-1}{\left({\displaystyle \sum _{i\in U}}{c}_i^{1/2}{z}_i^{1/2}\right)}^2{N}^{-2}\left({\displaystyle \sum _{i\in U}}{b}_i^{-1/2}\right)\left({\displaystyle \sum _{i\in U}}{b}_i^{1/2}\right)-{\sigma }^2{\displaystyle \sum _{i\in U}}{z}_i,(4.2)$$

see Appendix for details. If the coefficient of variation of $b_i$ is small, then a Taylor Series approximation gives $N^{-2}{\displaystyle \sum }{b}_i^{-1/2}{\displaystyle \sum }{b}_i^{1/2}\approx 1+\left(1/4\right)C_b^2.$ Applying this, and the same approximations as in Subsection 3.1, (4.2) becomes

$$A{V}_{ests}=\frac{{\sigma }^2{C}_f^{-1}{N}^2\overline{c}\overline{z}\left(1+\frac{1}{4}{C}_b^2\right)}{\left(1+\frac{1}{4}{C}_c^2\right)\left(1+\frac{1}{4}{C}_z^2\right)}.(4.3)$$

See Appendix for details.

Comparing (4.3) and (3.7), the effect of using estimated cost parameters rather than no costs at all is to multiply the anticipated variance by $\left[1+\left(1/4\right)C_b^2\right]/\left[1+\left(1/4\right)C_c^2\right].$ Therefore cost information is worth using provided $C_b<C_c.$ The coefficient of variation of the error factors has to be less than that of the true unit costs over the population.

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