Model based inference using ranked set samples
Section 2. Sampling designs

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We consider RSS sampling designs from a super population model to draw statistical inference in a finite population setting. Let $Y$ be the characteristic of interest. The copies of $Y,$ $Y_1\text{}\mathrm{,}\dots \mathrm{,}{Y}_N\text{},$ are considered as independent identically distributed (iid) random variables from a super population. Basic assumption for this super population model can be stated as

$$\text{Model:}{Y}_1\text{}\mathrm{,}\dots \mathrm{,}{Y}_N\text{independent}\text{identically}\text{distributed}\text{with}{E}_M\left(Y_i\right)\mathrm{<mo>=</mo>}\mu \mathrm{,}{V}_M\left(Y_i\right)\mathrm{<mo>=</mo>}{\sigma }^2\text{}\mathrm{.}(2.1)$$

The subscript $M$ in model (2.1) is used to highlight that the mean and variance are computed based on a super population model, not the randomization distribution as in Ozturk (2016b). In this super population model, $\mu$ and ${\sigma }^2$ represent unknown infinite population parameters.

In super population model, a particular realization, $y_1\text{}\mathrm{,}\dots \mathrm{,}{y}_N\text{},$ of random variables $Y_1\text{}\mathrm{,}\dots \mathrm{,}{Y}_N$ from model (2.1), is considered as a finite population. Let $P^N\mathrm{<mo>=</mo>}\left\{y_1\text{}\mathrm{,}\dots \mathrm{,}{y}_N\right\}$ denotes this finite population. Ranked set sample is constructed from $P^N\text{}.$ Without loss of generality, we assume that $y_{\left(1\right)}\mathrm{<mo><</mo>}{y}_{\left(2\right)}\mathrm{<mo><</mo>}\dots \mathrm{<mo><</mo>}{y}_{\left(N\right)}$ are ordered values of $y_1\text{}\mathrm{,}\dots \mathrm{,}{y}_N$ where $y_{\left(i\right)}$ is the $i^{\text{th}}$ largest value of $Y$ in $P^N\text{}.$ Throughout the paper, $H$ and $d$ are used to denote the set and cycle sizes, respectively.

To construct a ranked set sample, one selects a set of $H$ experimental units, $y_{s_1}\text{}\mathrm{,}\dots \mathrm{,}{y}_{s_H}\text{},$ at random from $P^N$ and ranks them based on their $Y$ values in an increasing magnitude without actual measurement. Ranking process can be performed either using visual inspection or some auxiliary variables and hence subjected to ranking error. The unit that corresponds to the smallest $Y,$ $y_{\left[1\right]},$ is identified and measured where the square bracket in the subscript, [1], denotes the rank of the smallest unit (rank 1) in the set $\left\{y_{\left[1\right]}\mathrm{,}{y}_{\left[2\right]}^{*}\mathrm{,}\dots \mathrm{,}{y}_{\left[H\right]}^{*}\right\}.$ The remaining unmeasured units are denoted with $\left\{y_{\left[2\right]}^{*}\mathrm{,}\dots \mathrm{,}{y}_{\left[H\right]}^{*}\right\}.$ After $y_{\left[1\right]}$ is measured, none of the $H$ units in the set $\left\{y_{\left[1\right]}\mathrm{,}{y}_{\left[2\right]}^{*}\mathrm{,}\dots \mathrm{,}{y}_{\left[H\right]}^{*}\right\}$ are returned to the population. One then selects another set of $H$ experimental units at random from the remaining population $P^{N-H}$ and ranks them without measurement. This time, the unit that corresponds to the second smallest $Y,$ $y_{\left[2\right]}\text{},$ is identified and measured in $\left\{y_{\left[1\right]}^{*}\mathrm{,}{y}_{\left[2\right]}\mathrm{,}{y}_{\left[3\right]}^{*}\mathrm{,}\dots \mathrm{,}{y}_{\left[H\right]}^{*}\right\}.$ This process is continued until a simple random sample of size $H$ is taken from the reduced population $P^{N-H\left(H-1\right)}$ and the $H^{\text{th}}$ smallest unit is identified and measured in the set $\left\{y_{\left[1\right]}^{*}\mathrm{,}{y}_{\left[2\right]}^{*}\mathrm{,}\dots \mathrm{,}{y}_{\left[H-1\right]}^{*}\mathrm{,}{y}_{\left[H\right]}\right\}.$ This is called a cycle. A cycle selects $H$ disjoint sets, each of size $H$ and only measures $H$ units. The remaining $H\left(H-1\right)$ units are used only for ranking purposes. The cycles are repeated $d$ times to yield a ranked set sample of size $n\mathrm{<mo>=</mo>}dH$ units. A ranked set sample can then be represented as

$$W_{h\mathrm{,}i\mathrm{,}H}\mathrm{<mo>=</mo>}\left\{y_{\left[1\right]i}^{*}\mathrm{,}\dots \mathrm{,}{y}_{\left[h-1\right]i}^{*}\mathrm{,}{y}_{\left[h\right]i}\mathrm{,}{y}_{\left[h+1\right]i}^{*},\dots \mathrm{,}{y}_{\left[H\right]i}^{*}\right\}\mathrm{,}h\mathrm{<mo>=</mo>1,}\dots \mathrm{,}H\mathrm{,}i\mathrm{<mo>=</mo>1,}\dots \mathrm{,}d\mathrm{,}(2.2)$$

where only $y_{\left[h\right]i},$ $h\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}H\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}d,$ are measured. The other values are used to obtain the rank of the measured values. Units in sets $W_{h\mathrm{,}i\mathrm{,}H}$ and $W_{h^{\prime }\text{}\mathrm{,}{i}^{\prime }\text{}\mathrm{,}H}$ are all independent if either $h\ne h^{\prime }$ or $i\ne i^{\prime }\text{},$ but the units in $W_{h\mathrm{,}i\mathrm{,}H}$ are all correlated since they are ranked in the same set. Under model (2.1), means, variances and covariances of judgment order statistics are given by

$$\begin{array}{ll}{E}_M\left(Y_{\left[h\right]i}\right)\hfill & \mathrm{<mo>=</mo>}{\mu }_{\left[h\right]}\mathrm{,}{\text{Var}}_M\left(Y_{\left[h\right]i}\right)\mathrm{<mo>=</mo>}{\sigma }_{\left[h\right]}^2\mathrm{,}\hfill \\ {\text{Cov}}_M\left(Y_{\left[h\right]i}\mathrm{,}{Y}_{[h^{\prime }\text{}]i}\right)\hfill & \mathrm{<mo>=</mo>}\{\begin{array}{ll}{\sigma }_{\mathrm{<mo\; stretchy="false">[</mo>}h\mathrm{,}{h}^{\prime }\mathrm{<mo\; stretchy="false">]</mo>}}\hfill & \text{if}{Y}_{\left[h\right]i}\mathrm{,}{Y}_{\mathrm{<mo\; stretchy="false">[</mo>}{h}^{\prime }\mathrm{<mo\; stretchy="false">]</mo>}i}\text{are}\text{from}\text{the}\text{same}\text{set}\hfill \\ 0\hfill & \text{otherwise}\text{.}\hfill \end{array}\hfill \end{array}$$

It should be noted that since all sets are disjoint no units can be used more than once in any one of the sets. Hence all sample units are distinct. Since the sets are independently ranked $Y_{\left[h\right]i}\text{}’\text{s}$ are mutually independent. Observations having the same rank $h,$ $Y_{\left[h\right]i},$ $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}d$ are identically distributed.

Estimator of the population mean $\mu$ based on RSS data in equation (2.2) can be defined as follows.

$${\overline{Y}}_R\mathrm{<mo>=</mo>}\frac{1}{dH}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^d}{Y}_{\left[h\right]i}\mathrm{.}(2.3)$$

It can be immediately observed that the estimator ${\overline{Y}}_R$ is model unbiased. In other words, under the model (2.1), $E_M\left({\overline{Y}}_R-{\overline{Y}}_N\right)\mathrm{<mo>=</mo>\; 0},$ where ${\overline{Y}}_N\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N}{Y}_i\mathrm{.}$

We now consider the mean square prediction error (MSPE) of the estimator ${\overline{Y}}_R$ under model (2.1)

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

Since the predictor ${\overline{Y}}_R$ is model unbiased for ${\overline{Y}}_N\text{},$ $E_M\left({\overline{Y}}_R-{\overline{Y}}_N\right)\mathrm{<mo>=</mo>\; 0},$ the mean square prediction error (MSPE) of ${\overline{Y}}_R$ is the same as ${\text{Var}}_M\left({\overline{Y}}_R-{\overline{Y}}_N\right).$

Theorem 1 Let $Y_{\left[h\right]i}\text{}\mathrm{,}h\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}H\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}d,$  be a ranked set sample from a finite population $P^N\text{}.$  Under a super population model in equation (2.1), the mean square prediction error of the estimator ${\overline{Y}}_R$  is given by

$${\sigma }_{\text{RSS}}^2\mathrm{<mo>=</mo>}{\text{MSPE}}_M\left({\overline{Y}}_R\right)\mathrm{<mo>=</mo>}\frac{N-n}{Nn}{\sigma }^2-\frac{1}{nH}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\left({\mu }_{\left[h\right]}-\mu \right)}^2\mathrm{.}(2.4)$$

We note that expression on equation (2.4) is very similar to the sample variance of an infinite population RSS sample. Only difference is due to the coefficient $\frac{N-n}{Nn}.$ In infinite population setting the fraction $\frac{N-n}{Nn}$ in equation (2.4) becomes $\frac{1}{n}.$ Hence, $\left(1-\frac{n}{N}\right)$ is the finite population correction (fpc) factor for the variance of RSS sample mean. If the sample size is not small in comparison with the population size $N,$ the fpc, $\frac{N-n}{Nn},$ makes a correction on the variance of an RSS sample mean. This correction would be substantial if $n$ is relatively large with respect to $N.$ If $n$ is small, fpc is close to 1 and the impact of finite population correction factor is minimal.

Corollary 1 Assume that $n$  and $N$  increase in such a way that the ratio $\frac{n}{N}$  approaches to a limit at $a,$   ${{\displaystyle \lim }}_{n\to \infty }\frac{n}{N}\mathrm{<mo>=</mo>}a\mathrm{.}$

i. If $a\mathrm{<mo>></mo>\; 0},$   ${\sigma }_{RSS}^2$  converges to a simple form

$$\underset{n\to \infty }{{\displaystyle \lim }}n{\sigma }_{RSS}^2\mathrm{<mo>=</mo>}\left(1-a\right){\sigma }^2-\frac{1}{H}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\left({\mu }_{\left[h\right]}-\mu \right)}^2\text{}\mathrm{,}$$

ii. if $a\mathrm{<mo>=</mo>\; 0},$   ${{\displaystyle \lim }}_{n\to \infty }n{\sigma }_{RSS}^2\mathrm{<mo>=</mo>}\frac{1}{H}{\sigma }_{\left[h\right]}^2\text{},$  which is the same as the variance of the sample mean of a balanced ranked set sample in an infinite population setting,

iii. if $a$  is strictly positive, then ${{\displaystyle \lim }}_{n\to \infty }n{\sigma }_{RSS}^2\mathrm{<mo><</mo>}\frac{1}{H}{\sigma }_{\left[h\right]}^2.$

The corollary indicates that when sample and population sizes grow at a certain rate, variance of sample mean of an RSS $\left({\sigma }_{\text{RSS}}^2\right)$ sample in a finite population setting reduces to simple form. If $a$ is strictly positive, variance of an RSS sample mean is smaller than the variance of an RSS sample mean in an infinite population setting.

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