Improved Horvitz-Thompson estimator in survey sampling

Section 4. Extension to the ratio estimator

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When an auxiliary variable is available, the ratio estimator is usually used to estimate the population total. In this section, we extend the IHT estimator to the case of ratio estimation.

4.1  Improved ratio estimator

Denote the ratio between the population totals of $Y$ and $Z$ as

$$R\mathrm{<mo>=</mo>}{t}_y/t_z\mathrm{,}$$

where $t_y$ and $t_z$ are the totals of the finite populations $Y$ and $Z,$ respectively. Let ${\widehat{t}}_{y\pi }={\displaystyle {\sum }_s}\frac{y_k}{{\pi }_k},$ ${\widehat{t}}_{z\pi }\mathrm{<mo>=</mo>}{\displaystyle {\sum }_s}\frac{z_k}{{\pi }_k},$ ${\widehat{t}}_{y\pi }^{*}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_s}\frac{y_k}{{\pi }_k^{*}},$ and ${\widehat{t}}_{z\pi }^{*}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_s}\frac{z_k}{{\pi }_k^{*}}.$ The classical estimator and our modified estimator of $R$ are given by

$$\widehat{R}\mathrm{<mo>=</mo>}{\widehat{t}}_{y\pi }/{\widehat{t}}_{z\pi }\mathrm{,}\text{and}{\widehat{R}}^{*}\mathrm{<mo>=</mo>}{\widehat{t}}_{y\pi }^{*}/{\widehat{t}}_{z\pi }^{*}.$$

We assume that the population total $t_z$ is known. To estimate the population total $t_y$ of $Y,$ the classical ratio estimator is given by

$${\widehat{Y}}_R\mathrm{<mo>=</mo>}{t}_z\cdot {\widehat{t}}_{y\pi }/{\widehat{t}}_{z\pi }\mathrm{.}$$

Alternatively, our improved ratio estimator of $t_y$ based on the modified inclusion probabilities is expressed as

$${\widehat{Y}}_R^{*}\mathrm{<mo>=</mo>}{t}_z\cdot {\widehat{t}}_{y\pi }^{*}/{\widehat{t}}_{z\pi }^{*}\mathrm{.}$$

4.2  Properties of the improved ratio estimator

To show theoretically that the improved ratio estimator ${\widehat{Y}}_R^{*}$ is more efficient than the classical ratio estimator ${\widehat{Y}}_R,$ we need the following regularity conditions:

Condition C.3. ${{\displaystyle \lim }}_{N\to \infty }\frac{n}{N}\mathrm{<mo>=</mo>}c\mathrm{,}$  where $c\in \left(\mathrm{0,}1\right)$  is a constant.

Condition C.4. ${{\displaystyle \max }}_{i\ne j\ne k\in U}\left({\pi }_{ijk}-{\pi }_{ij}{\pi }_k\right)\mathrm{<mo>=</mo>}O\left(n^{-1}\right)\mathrm{,}$  and

$$\underset{i\ne j\ne k\ne l\in U}{{\displaystyle \max }}\left({\pi }_{ijkl}-4{\pi }_{ijk}{\pi }_l+6{\pi }_{ij}{\pi }_k{\pi }_l-3{\pi }_i{\pi }_j{\pi }_k{\pi }_l\right)\mathrm{<mo>=</mo>}O\left(n^{-2}\right)\mathrm{.}$$

Condition C.3 is a common condition. The same condition is used in Breidt and Opsomer (2000). Condition C.4 is a mild assumption on the third-order and fourth-order inclusion probabilities. In Appendix A.5, we present some frequent examples which satisfy Condition C.4.

Comparing our improved estimators with the classical estimators, we have the following result.

Theorem 4. If Conditions C.1-C.4 are satisfied, and $c_1\le z_k\le c_2$  for all $k\in U$  with $c_1$  and $c_2$  some positive constants, then

$$MSE\left({\widehat{R}}^{*}\right)\le MSE\left(\widehat{R}\right)+o\left(n^{-1}\right)\mathrm{.}$$

Furthermore,

$$MSE\left(N^{-1}{\widehat{Y}}_R^{*}\right)\le MSE\left(N^{-1}{\widehat{Y}}_R\right)+o\left(n^{-1}\right)\mathrm{.}$$

Proof. See Appendix A.4.

Like Theorem 3, Theorem 4 shows that the proposed method improves the classical ratio estimators with a tolerance of order $o\left(n^{-1}\right).$

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