3 Auxiliary information: Ratio estimator

Kelly Cristina M. Gonçalves, Fernando A. S. Moura et Helio S. Migon

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In many practical situations, it is possible to have information about an auxiliary variate $x_i$ (correlated with $y_i$ ) for all the population units, or at least for each unit in the sample, plus the population mean, $\overline{X}.$ In practice, $x_i$ is often the value of $y_i$ at some previous time when a complete census was taken. This approach is used in situations where the expected value and the variance of $y_i$ is proportional to $x_i,$ so in the BLE setup, we replace some hypotheses about the $y’\text{s}$ with ones about the first two moments of the rate $y_i/x_i.$To the best of our knowledge, the new ratio estimator proposed below is a novel contribution in sampling survey theory.

The new ratio estimator is obtained as a particular case of model (2.4) and with the hypothesis of exchangeability, used in Bayes linear approach, applied to the rate $y_i/x_i$ for all $i=\mathrm{1,}\dots \mathrm{,}N,$as described below:

$$E\left(\frac{y_i}{x_i}\right)=m\mathrm{,}V\left(\frac{y_i}{x_i}\right)=v\text{and}\text{Cov}\left(\frac{y_i}{x_i}\mathrm{,}\frac{y_j}{x_j}\right)=c\mathrm{,}i\mathrm{,}j=\mathrm{1,}\dots \mathrm{,}N\mathrm{,}\forall i\ne j\mathrm{.}\text{ (3}\text{.1)}$$

Applying the general result established in (2.10) to (3.1) with $X=\left(x_1\mathrm{,}\dots \mathrm{,}{x}_N\right)\prime ,$ the vector $N\times 1$ of auxiliary variables, $a=m,$ $R=c$ and $V={\sigma }^2\text{diag}\left(x_1\mathrm{,}\dots \mathrm{,}{x}_N\right),$ where ${\sigma }^2=v-c,$we obtain the BLE of T and its associated variance as follows:

$$\begin{array}{c}{\widehat{T}}_{ra}=n{\overline{y}}_s+\left(N-n\right)\widehat{\mu }{\overline{x}}_{\overline{s}}\text{and}\\ \widehat{V}\left({\widehat{T}}_{ra}\right)=\left(N-n\right){\overline{x}}_{\overline{s}}{\sigma }^2+{\left(N-n\right)}^2{\overline{x}}_{\overline{s}}{}^2{\left(c^{-1}+{\sigma }^{-2}n{\overline{x}}_s\right)}^{-1}\mathrm{,}\text{where}\\ \widehat{\mu }=\omega \frac{{\overline{y}}_s}{{\overline{x}}_s}+\left(1-\omega \right)m\text{and}\omega =\frac{{\sigma }^{-2}n{\overline{x}}_s}{\left(c^{-1}+{\sigma }^{-2}n{\overline{x}}_s\right)}\mathrm{,}\end{array}$$

where ${\overline{x}}_{\overline{s}}=\left(N\overline{X}-n{\overline{x}}_s\right)/\left(N-n\right)$ is mean of $x’\text{s}$ for the non-sample units. Letting $v\to \infty$ and $n\to \infty ,$ but keeping ${\sigma }^2$ fixed, we recover the ratio type estimator, found in the design-based approach: ${\widehat{T}}_{ra}=N\overline{X}\left({\overline{y}}_s/{\overline{x}}_s\right)\mathrm{.}$

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