Model based inference using ranked set samples
Section 3. Unbiased estimators

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In this section, we construct an unbiased estimator for ${\sigma }_{\text{RSS}}^2\text{}.$ By rewriting the estimator for ${\sigma }_{\text{RSS}}^2$ in a slightly different form, we obtain

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

Let

$$\begin{array}{ll}{T}_1^{*}\hfill & \mathrm{<mo>=</mo>}\frac{1}{2d^2{H}^2}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\displaystyle \sum _{h\ne h^{\prime }}^H}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^d}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^d}{\left(Y_{\left[h\right]i}-Y_{\mathrm{<mo\; stretchy="false">[</mo>}{h}^{\prime }\mathrm{<mo\; stretchy="false">]</mo>}j}\right)}^2\hfill \\ T_2^{*}\hfill & \mathrm{<mo>=</mo>}\frac{1}{2d\left(d-1\right)H^2}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^d}{\displaystyle \sum _{j\ne i}^d}{\left(Y_{\left[h\right]i}-Y_{\left[h\right]j}\right)}^2\text{}\mathrm{.}\hfill \end{array}$$

Using these definitions, one can easily establish the following result.

Theorem 2 Let $Y_{\left[h\right]i}\text{},$   $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n\mathrm{,}h\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}H$  be an RSS sample of set size $H$  from a finite population. An unbiased estimator of ${\sigma }_{RSS}^2$  is given by

$${\widehat{\sigma }}_{RSS}^2\mathrm{<mo>=</mo>}{T}_2^{*}\left(\frac{H}{n}\right)-\left(T_1^{*}+T_2^{*}\right)\frac{1}{N}\mathrm{.}(3.1)$$

Theorem 2 indicates that the variance estimator is unbiased for any sample and set sizes regardless of the quality of ranking information. Unbiased estimator of the variance of ${\overline{Y}}_R$ allows us to construct confidence interval for population mean and total. Using normal approximation, $\left(1-\alpha \right)\mathrm{100\; <mi>\%</mi>}$ confidence interval for the population mean is given by

$${\overline{Y}}_R\pm t_{n-H\text{}\mathrm{,}\alpha /2}{\widehat{\sigma }}_{\text{RSS}}^2\text{}\mathrm{,}$$

where $t_{df\text{}\mathrm{,}a}$ is the $a^{\text{th}}$ upper quantile of $t-$ distribution with degrees of freedom $df\text{}.$ The degrees of freedom $n-H$ is suggested to account the heterogeneity among $H$ judgment classes. The choice of $df\mathrm{<mo>=</mo>}n-H$ is also suggested in Ahn, Lim and Wang (2014) in infinite population setting.

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