Model based inference using ranked set samples
Section 7. Concluding remarks

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We have developed a model based statistical inference for population mean and total based on RSS samples in a finite population setting where samples are constructed by using a without replacement sampling design. It is shown that the sample mean of RSS samples are model unbiased and they have smaller mean square prediction error (MSPE) than the MSPE of a simple random sample mean. We constructed unbiased estimator for the MSPE and prediction confidence interval for the population mean. A small scale simulation study showed that estimators are as good as or better than SRS estimators when the quality of ranking information in RSS sampling is low or high, respectively, and the cost ratio of obtaining a unit in RSS and a unit in SRS is not too high. The coverage probabilities of the prediction intervals are also very close to the nominal coverage probabilities. Proposed sampling designs and inferential procedures are applied to a data set containing a sheep population in an agricultural research farm.

Acknowledgements

The second author’s research is supported by The Scientific and Technological Research Council of Turkey (TUBITAK). The authors thank the Editor, Associate Editor and two referees for their constructive comments and suggestions.

Appendix

Proof of Theorem 1: We write mean square prediction error (MSPE) as

$${\text{MSPE}}_M\left({\overline{Y}}_R\right)\mathrm{<mo>=</mo>}{E}_M{\left\{{\overline{Y}}_R-\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{Y}_i\right\}}^2\mathrm{<mo>=</mo>}{E}_M{\left\{\frac{1}{dH}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^d}{Y}_{\left[h\right]i}-\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{Y}_i\right\}}^2\mathrm{.}$$

Let $Z_i,$ $i\mathrm{<mtext>\hspace{0.17em}</mtext>\; <mo>=</mo>\; <mtext>\hspace{0.17em}</mtext>\; 1,}\dots ,N-nH\text{},$ be the responses on $N-nH$ population units that are neither measured nor used in ranking in any one of the randomly selected sets of size $H$ in the construction of the RSS sample. Then the MSPE can be written

$${\text{MSEP}}_M\left({\overline{Y}}_R\right)\mathrm{<mo>=</mo>}{E}_M{\left\{\frac{1}{dH}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^d}{Y}_{\left[h\right]i}-\frac{1}{N}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^d}\left[Y_{\left[h\right]i}+{\displaystyle \sum _{h\ne h^{\prime }}^H}{Y}_{\mathrm{<mo\; stretchy="false">[</mo>}{h}^{\prime }\mathrm{<mo\; stretchy="false">]</mo>}i}^{*}\right]-\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^{N-nH}}{Z}_i\right\}}^2$$

where $Y_{\mathrm{<mo\; stretchy="false">[</mo>}{h}^{\prime }\mathrm{<mo\; stretchy="false">]</mo>}i}^{*},$ $h^{\prime }\ne \mathrm{,}h,$ are responses on unmeasured units that are used in ranking of units in a set. Hence, $Y_{\mathrm{<mo\; stretchy="false">[</mo>}{h}^{\prime }\mathrm{<mo\; stretchy="false">]</mo>}i}^{*}$ and $Y_{\left[h\right]i}$ are correlated, but they are uncorrelated with $Z_i.$ Let

$$c_{h\mathrm{,}{h}^{\prime }}\mathrm{<mo>=</mo>}\{\begin{array}{ll}\frac{N-n}{n}\hfill & h\mathrm{<mo>=</mo>}{h}^{\prime }\hfill \\ -1\hfill & h\ne h^{\prime }\text{}\mathrm{.}\hfill \end{array}$$

Using the definition of $c_{h\mathrm{,}{h}^{\prime }}\text{},$ we combine $Y_{\mathrm{<mo\; stretchy="false">[</mo>}{h}^{\prime }\mathrm{<mo\; stretchy="false">]</mo>}i}^{*}$ and $Y_{\left[h\right]i}$ under the same summation and write the MSPE as

$$\begin{array}{ll}{\text{MSPE}}_M\left({\overline{Y}}_R\right)\hfill & \mathrm{<mo>=</mo>}\frac{1}{N^2}\text{var}\left\{{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^d}{\displaystyle \sum _{h^{\prime }\mathrm{<mo>=</mo>\; 1}}^H}{c}_{h\mathrm{,}{h}^{\prime }}{Y}_{\left[h\right]i}\right\}+\text{var}\left\{\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^{N-nH}}{Z}_i\right\}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{1}{N^2}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\text{var}\left[{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^d}{\displaystyle \sum _{h^{\prime }\mathrm{<mo>=</mo>\; 1}}^H}{c}_{h\mathrm{,}{h}^{\prime }}{Y}_{\left[h\right]i}\right]+\text{var}\left\{\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^{N-nH}}{Z}_i\right\}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{d}{N^2}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\displaystyle \sum _{h^{\prime }\mathrm{<mo>=</mo>\; 1}}^H}{\left(c_{h\mathrm{,}{h}^{\prime }}\right)}^2{\sigma }_{\mathrm{<mo\; stretchy="false">[</mo>}{h}^{\prime }\mathrm{<mo\; stretchy="false">]</mo>}}^2+\frac{d}{N^2}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\left({\displaystyle \sum _{h^{\prime }\mathrm{<mo>=</mo>\; 1}}^H}{\displaystyle \sum _{t\ne h^{\prime }}^H}{c}_{h\mathrm{,}{h}^{\prime }}{c}_{h\mathrm{,}t}{\sigma }_{\mathrm{<mo\; stretchy="false">[</mo>}{h}^{\prime }\text{}\mathrm{,}t\mathrm{<mo\; stretchy="false">]</mo>}}\right)\hfill \\ \hfill & +\text{var}\left\{\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^{N-nH}}{Z}_i\right\}\hfill \\ \hfill & \mathrm{<mo>=</mo>}A+B+\frac{\left(N-nH\right){\sigma }^2}{N^2}\mathrm{.}(\text{A}.1)\hfill \end{array}$$

The expression $A$ reduces to

$$\begin{array}{ll}A\hfill & \mathrm{<mo>=</mo>}\frac{d}{N^2}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\left(c_{h\mathrm{,}h}\right)}^2{\sigma }_{\left[h\right]}^2+\frac{d}{N^2}{\displaystyle \sum _h}{\displaystyle \sum _{h^{\prime }\ne h}^H}{\left(c_{h\mathrm{,}{h}^{\prime }}\right)}^2{\sigma }_{\mathrm{<mo\; stretchy="false">[</mo>}h\mathrm{,}{h}^{\prime }\mathrm{<mo\; stretchy="false">]</mo>}}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{d}{N^2}\left[{\left(\frac{N-n}{n}\right)}^2+\left(H-1\right)\right]{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\sigma }_{\left[h\right]}^2\text{}\mathrm{.}\hfill \end{array}$$

In a similar fashion, the expression $B$ reduces to

$$\begin{array}{ll}B\hfill & \mathrm{<mo>=</mo>}\frac{d}{N^2}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\left[{\displaystyle \sum _{t\ne h^{\prime }\text{}\mathrm{,}h}^H}\left({\displaystyle \sum _{h^{\prime }\mathrm{<mo>=</mo>\; 1}}^H}{c}_{h\mathrm{,}{h}^{\prime }}{c}_{h\mathrm{,}t}{\sigma }_{\mathrm{<mo\; stretchy="false">[</mo>}{h}^{\prime }\text{}\mathrm{,}t\mathrm{<mo\; stretchy="false">]</mo>}}+c_{h\mathrm{,}h}{c}_{h\mathrm{,}t}{\sigma }_{\left[h\mathrm{,}t\right]}\right)+{\displaystyle \sum _{h^{\prime }\ne h}^H}{c}_{h\mathrm{,}{h}^{\prime }}{c}_{h\mathrm{,}h}{\sigma }_{\mathrm{<mo\; stretchy="false">[</mo>}{h}^{\prime }\text{}\mathrm{,}h\mathrm{<mo\; stretchy="false">]</mo>}}\right]\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{d}{N^2}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\left[{\displaystyle \sum _{h^{\prime }\ne h}^H}{\displaystyle \sum _{t\ne h^{\prime }\text{}\mathrm{,}h}^H}{\sigma }_{\mathrm{<mo\; stretchy="false">[</mo>}{h}^{\prime }\text{}\mathrm{,}t\mathrm{<mo\; stretchy="false">]</mo>}}-2\left(\frac{N-n}{n}\right)\left({\displaystyle \sum _{t\mathrm{<mo>=</mo>\; 1}}^H}{\sigma }_{\left[h\mathrm{,}t\right]}-{\sigma }_{\left[h\mathrm{,}h\right]}\right)\right]\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{d}{N^2}\left[\left(H^2-2H\right){\sigma }^2-\left(H-2\right){\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\sigma }_{\left[h\right]}^2-2\left(\frac{N-n}{n}\right)\left(H{\sigma }^2-{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\sigma }_{\left[h\right]}^2\right)\right]\mathrm{.}\hfill \end{array}$$

By inserting expressions $A$ and $B$ in equation (A.1), we conclude that

$$\begin{array}{ll}{\text{MSPE}}_M\left({\overline{Y}}_R\right)\hfill & \mathrm{<mo>=</mo>}\frac{d}{N^2}\left[{\left(\frac{N-n}{n}\right)}^2+\left(H-1\right)\right]{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\sigma }_{\left[h\right]}^2\hfill \\ \hfill & +\frac{d}{N^2}\left[\left(H^2-2H\right){\sigma }^2-\left(H-2\right){\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\sigma }_{\left[h\right]}^2-2\left(\frac{N-n}{n}\right)\left(H{\sigma }^2-{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\sigma }_{\left[h\right]}^2\right)\right]\hfill \\ \hfill & +\left(\frac{N-nH}{N^2}\right){\sigma }^2\hfill \\ \hfill & \mathrm{<mo>=</mo>}\left(\frac{N-n}{Nn}\right){\sigma }^2-\frac{1}{nH}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\left({\mu }_{\left[h\right]}-\mu \right)}^2\hfill \end{array}$$

which completes the proof. Note that to establish the last equality we used the fact that ${\sigma }^2\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{h\mathrm{<mo>=</mo>\; 1}}^H}{\sigma }_{\left[h\right]}^2/H+{\displaystyle {\sum }_{h\mathrm{<mo>=</mo>\; 1}}^H}{\left({\mu }_{\left[h\right]}-\mu \right)}^2/H\text{}.$

Proof of Theorem 2: We first look at the expected values of $T_1^{*}$ and $T_2^{*}$ under the super population model in equation (2.1)

$$\begin{array}{ll}E\left(T_1^{*}\right)\hfill & \mathrm{<mo>=</mo>}\frac{1}{H}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\left({\mu }_{\left[h\right]}-\mu \right)}^2+\frac{H-1}{H^2}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\sigma }_{\left[h\right]}^2\mathrm{<mo>=</mo>}{\sigma }^2-\frac{1}{H^2}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\sigma }_{\left[h\right]}^2\hfill \\ E\left(T_2^{*}\right)\hfill & \mathrm{<mo>=</mo>}\frac{1}{H^2}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\sigma }_{\left[h\right]}^2\mathrm{.}\hfill \end{array}$$

It is now easy to establish that $E\left(T_1^{*}+T_2^{*}\right)\mathrm{<mo>=</mo>}{\sigma }^2\text{}.$ The proof is then completed by inserting these expressions in equation (3.1).

Proof of Theorem 3: We sketch the proof for $\text{RE}\left({\overline{Y}}_{\text{RC}}\mathrm{,}{\overline{Y}}_{\text{SRS}}\right).$ From the total cost function, we write

$$n_S\mathrm{<mo>=</mo>}\frac{C}{c_i+c_{qy}}\text{and}{n}_R\mathrm{<mo>=</mo>}\frac{C}{H{c}_i+H{c}_{qx}+c_{qy}},$$

where $C$ is the fixed total cost. Using these expressions, we have

$$\begin{array}{ll}\text{RE}\left({\overline{Y}}_{\text{RC}}\text{}\mathrm{,}\text{SRS}\right)\hfill & \mathrm{<mo>=</mo>}\frac{n_R\left(N-n_S\right)}{n_S\left(N-n_r-ND\right)}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{N\left(c_i+c_{qy}\right)-C}{N\left(H{c}_i+H{c}_{qx}+c_{qy}\right)-C-ND\left(H{c}_i+H{c}_{qx}+c_{qy}\right)}\mathrm{.}\hfill \end{array}$$

We now establish that $\text{RE}\left({\overline{Y}}_{RC}\text{}\mathrm{,}\text{SRS}\right)\ge 1$ if and only if

$$\begin{array}{ll}{c}_i+c_{qy}\hfill & \ge \left(H{c}_i+H{c}_{qx}+c_{qy}\right)\left(1-D\right)\mathrm{<mo>=</mo>}\frac{H{c}_i+H{c}_{qx}+c_{qy}}{\text{RP}}\hfill \\ \text{RP}\hfill & \ge \frac{H{c}_i+H{c}_{qx}+c_{qy}}{c_i+c_{qy}}\hfill \end{array}$$

which completes the proof.

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