Variances and variance estimators of the improved ratio estimators under adaptive cluster sampling

Abstract

Although not design-unbiased, the ratio estimator is recognized as more efficient when a certain degree of correlation exists between the variable of primary interest and the auxiliary variable. Meanwhile, the Rao–Blackwell method is another commonly used procedure to improve estimation efficiency. Various improved ratio estimators under adaptive cluster sampling (ACS) that make use of the auxiliary information together with the Rao–Blackwellized univariate estimators have been proposed in past research studies. In this article, the variances and the associated variance estimators of these improved ratio estimators are proposed for a thorough framework of statistical inference under ACS. Performance of the proposed variance estimators is evaluated in terms of the absolute relative percentage bias and the empirical mean-squared error. As expected, results show that both the absolute relative percentage bias and the empirical mean-squared error decrease as the initial sample size increases for all the variance estimators. To evaluate the confidence intervals based on these variance estimators and the finite-population Central Limit Theorem, the coverage rate and the interval width are used. These confidence intervals suffer a disadvantage similar to that of the conventional ratio estimator. Hence, alternative confidence intervals based on a certain type of adjusted variance estimators are constructed and assessed in this article.

Appendix

1.1 Derivation of the bound for $\text{ MSE }(\hat{\mu }_{r\cdot ht}) - \text{ Avar }(\hat{\mu }_{r \cdot ht})$

Let $\delta \bar{y}_k^{*} = (\hat{\mu }_{y\cdot ht} - \mu _y)/\mu _y$ and $\delta \bar{x}_k^{*} = (\hat{\mu }_{x\cdot ht} - \mu _x)/\mu _x$, the MSE of $\hat{R}$ can be expressed as

$$\begin{aligned} \text{ MSE }(\hat{R}) = E(\hat{R} - R)^2 = R^2E\left[ \{\delta \bar{y}_k^{*} - \delta \bar{x}_k^{*}\}^2\{\mu _x^2/\hat{\mu }_{x\cdot ht}^2\}\right] . \end{aligned}$$

Consider approximating $\text{ MSE }(\hat{R})$ by the first-order approximate MSE

$$\begin{aligned} \text{ Avar }(\hat{R}) = R^2E\left[ \{\delta \bar{y}_k^{*} - \delta \bar{x}_k^{*}\}^2\right] , \end{aligned}$$

thus, we obtain that

$$\begin{aligned} \text{ MSE }(\hat{R}) - \text{ Avar }(\hat{R}) = -\mu _y^2E\left[ \hat{\mu }_{x\cdot ht}^{-2}\left\{ \delta \bar{y}_k^{*} - \delta \bar{x}_k^{*}\right\} ^2\left\{ 2\delta \bar{x}_k^{*} + (\delta \bar{x}_k^{*})^2\right\} \right] . \end{aligned}$$

Hence, if $x_0$ is a positive lower bound of $\hat{\mu }_{x \cdot ht}$, then

$$\begin{aligned}&\left| \text{ MSE }(\hat{\mu }_{r\cdot ht}) - \text{ Avar }(\hat{\mu }_{r \cdot ht})\right| \\&\quad \le \left( \frac{\mu _x\mu _y}{x_0}\right) ^2 \left| E\left[ \{\delta \bar{y}_k^{*} - \delta \bar{x}_k^{*}\}^2\{2\delta \bar{x}_k^{*} + \left( \delta \bar{x}_k^{*}\right) ^2\}\right] \right| \\&\quad =\left( \frac{\mu _x\mu _y}{x_0}\right) ^2\Biggl |2 \left[ E(\delta \bar{x}_k^{*})^3 - 2E(\delta \bar{x}_k^{*})^2(\delta \bar{y}_k^{*}) + E(\delta \bar{x}_k^{*})(\delta \bar{y}_k^{*})^2\right] \\&\qquad + E(\delta \bar{x}_k^{*})^4 - 2E(\delta \bar{x}_k^{*})^3(\delta \bar{y}_k^{*}) + E(\delta \bar{x}_k^{*})^2(\delta \bar{y}_k^{*})^2\Biggr |\\&\quad = \left( \frac{\mu _x\mu _y}{x_0N}\right) ^2\Biggl |\sum \limits _{i=1}^K \sum \limits _{j=1}^K\sum \limits _{k=1}^K\left( \frac{\alpha _{ijk} - \alpha _i\alpha _j\alpha _k}{\alpha _i\alpha _j\alpha _k}\right) \Bigl (2\frac{x_i^{*}}{\mu _x}\frac{x_j^{*}}{\mu _x}\frac{x_k^{*}}{\mu _x} - 4\frac{x_i^{*}}{\mu _x}\frac{x_j^{*}}{\mu _x}\frac{y_k^{*}}{\mu _y} + 2\frac{x_i^{*}}{\mu _x}\frac{y_j^{*}}{\mu _y}\frac{y_k^{*}}{\mu _y}\Bigr ) \\&\qquad +\sum \limits _{i=1}^K\sum \limits _{j=1}^K\sum \limits _{k=1}^K\sum \limits _{l=1}^K \left( \frac{\alpha _{ijkl}- \alpha _i\alpha _j\alpha _k\alpha _l}{\alpha _i\alpha _j\alpha _k\alpha _l} \right) \Bigl (\frac{x_i^{*}}{\mu _x}\frac{x_j^{*}}{\mu _x}\frac{x_k^{*}}{\mu _x}\frac{x_l^{*}}{\mu _x} -2\frac{x_i^{*}}{\mu _x}\frac{x_j^{*}}{\mu _x} \frac{x_k^{*}}{\mu _x}\frac{y_l^{*}}{\mu _y} +\frac{x_i^{*}}{\mu _x}\frac{x_j^{*}}{\mu _x} \frac{y_k^{*}}{\mu _y}\frac{y_l^{*}}{\mu _y}\Bigr )\Biggr |, \end{aligned}$$

where $\alpha _{ijk}$ and $\alpha _{ijkl}$ are the initial intersection probabilities of networks $i,j,k$ and $i,j,k,l$, respectively.

1.2 Derivation of the variance and variance estimator of $\hat{\mu }_{r\cdot ht(s)}$

The variance of $\hat{\mu }_{r\cdot ht(s)}$ is

$$\begin{aligned} \text{ Avar }\left( \hat{\mu }_{r \cdot ht(s)}\right)&= \text{ var }\left( \hat{\mu }_{u \cdot ht}\right) - \text{ E }\left( \text{ var }(\hat{\mu }_{u \cdot ht}|d^+)\right) \nonumber \\&= \text{ var }\left( \hat{\mu }_{u \cdot ht}\right) - \text{ E }\left( \hat{\mu }_{u \cdot ht}-\hat{\mu }_{u \cdot ht(s)}\right) ^2. \end{aligned}$$

(19)

On substitution the analytical forms of $\hat{\mu }_{u \cdot ht}$ (Thompson 1990) and $\hat{\mu }_{u \cdot ht(s)}$ (Dryver and Thompson 2005) into the second term of (19), we then have

$$\begin{aligned} \text{ E }\left( \hat{\mu }_{u \cdot ht}-\hat{\mu }_{u \cdot ht(s)}\right) ^{\!\!2}&= \text{ E }\left[ \frac{1}{n_0^2}\left( \sum \limits _{k\in s_0\cap F_2}u^*_k-\frac{\phi _{s_0}}{\phi _s}\sum \limits _{k\in F_2}u^*_k\right) ^2\right] \\&= \sum \limits _{d^+}\frac{P_{d^+}}{L_{d^+}}\Biggl [\sum \limits _{s_0\in \mathcal{S }}\frac{I_{s_0}}{n_0^2} \biggl (\sum \limits _{k\in s_0\cap F_2}u^{*2}_k+ 2\sum \limits _{k,l \in F_2}\sum \limits _{l < k}u^*_ku^*_l\biggr )\\&+ \sum \limits _{s_0\in \mathcal{S }}\frac{I_{s_0}}{n_0^2} \biggl (\!-2 \Bigl (\frac{\phi _{s_0}}{\phi _s} \sum \limits _{k\in s_0\cap F_2}u^*_k\sum \limits _{k\in F_2}u^*_k\Bigr )+\Bigl (\frac{\phi _{s_0}}{\phi _s}\sum \limits _{k\in F_2}u^*_k\Bigr )^2\biggr )\!\Biggr ]. \end{aligned}$$

Since each set of the sampled edge units given $d^+$ has an equal initial selection probability, and out of the $\left( {\begin{array}{c}\phi _s\\ \phi _{s_0}\end{array}}\right)$ possible combinations for choosing the sampled edge units, all of the $i$ edge units appear $\left( {\begin{array}{c}\phi _s-i\\ \phi _{s_0}-i\end{array}}\right)$ times. Hence, out of the $L_{d^+}$ combinations, any of the $i$ edge units appears $\frac{\left( {\begin{array}{c}\phi _s-i\\ \phi _{s_0}-i\end{array}}\right) }{\left( {\begin{array}{c}\phi _s\\ \phi _{s_0}\end{array}}\right) }\times L_{d^+}$ times. Thus we have

$$\begin{aligned} \sum \limits _{s_0\in \mathcal{S }}I_{s_0}\sum \limits _{k\in s_0\cap F_2} u^{*2}_k&= L_{d^+}\frac{\left( {\begin{array}{c}\phi _s-1\\ \phi _{s_0}-1\end{array}}\right) }{\left( {\begin{array}{c}\phi _s\\ \phi _{s_0}\end{array}}\right) }\sum \limits _{k\in F_2}u^{*2}_k,\\ \sum \limits _{s_0\in \mathcal{S }}I_{s_0}\sum \limits _{k,l \in F_2} \sum \limits _{l < k}u^*_ku^*_l&= L_{d^+}\frac{\left( {\begin{array}{c}\phi _s-2\\ \phi _{s_0}-2\end{array}}\right) }{\left( {\begin{array}{c}\phi _s\\ \phi _{s_0}\end{array}}\right) }\sum \limits _{k,l \in F_2} \sum \limits _{l < k}u^*_ku^*_l, \end{aligned}$$

and

$$\begin{aligned} \sum \limits _{s_0\in \mathcal{S }}I_{s_0}\sum \limits _{k\in s_0\cap F_2}u^{*}_k = L_{d^+}\frac{\left( {\begin{array}{c}\phi _s-1\\ \phi _{s_0}-1\end{array}}\right) }{\left( {\begin{array}{c}\phi _s\\ \phi _{s_0}\end{array}}\right) }\sum \limits _{k\in F_2}u^{*}_k. \end{aligned}$$

Consequently,

$$\begin{aligned} \text{ E }\left( \text{ var }(\hat{\mu }_{u \cdot ht}|d^+)\right)&= \text{ E }\left( \hat{\mu }_{u \cdot ht}-\hat{\mu }_{u \cdot ht(s)}\right) ^2\\&= \sum \limits _{d^+}\frac{P_{d^+}}{L_{d^+}}\frac{1}{\left( n_0\phi _s\right) ^2}\left( \phi _{s_0}\left( \phi _s-\phi _{s_0} \right) \sum \limits _{k\in F_2}u^{*2}_k+ 2\frac{\phi _{s_0}\left( \phi _{s_0}-\phi _s\right) }{\phi _s-1}\sum \limits _{k,l \in F_2}\sum \limits _{l < k}u^*_ku^*_l\right) \end{aligned}$$

and hence

$$\begin{aligned} \widehat{\text{ var }}\left( \hat{\mu }_{u\cdot ht}|d^+\right) = \frac{1}{\left( n_0\phi _s\right) ^2}\left( \phi _{s_0}\left( \phi _s-\phi _{s_0}\right) \sum \limits _{k \in F_2} \hat{u}^{*2}_k +2\frac{\phi _{s_0}\left( \phi _{s_0}-\phi _s\right) }{\phi _s-1}\sum \limits _{k,l \in F_2}\sum \limits _{l < k}\hat{u}^*_k\hat{u}^*_l\right) . \end{aligned}$$